From The Classical Beta Distribution To Generalized Beta ...
271
Supervisor: Prof. J.A.M Ottieno FROM THE CLASSICAL BETA DISTRIBUTION TO GENERALIZED BETA DISTRIBUTIONS Title A project submitted to the School of Mathematics, University of Nairobi in partial fulfillment of the requirements for the degree of Master of Science in Statistics. By Otieno Jacob I56/72137/2008 School of Mathematics University of Nairobi July, 2013
Transcript of From The Classical Beta Distribution To Generalized Beta ...
Supervisor: Prof. J.A.M Ottieno
FROM THE CLASSICAL BETA DISTRIBUTION
TO GENERALIZED BETA DISTRIBUTIONS
Title
A project submitted to the School of Mathematics, University of Nairobi in partial fulfillment of
the requirements for the degree of Master of Science in Statistics. By
Otieno Jacob
I56/72137/2008
School of Mathematics
University of Nairobi
July, 2013
ii
FROM THE CLASSICAL BETA DISTRIBUTION
TO GENERALIZED BETA DISTRIBUTIONS
Half Title
iii
Declaration
This project is my original work and has not been presented for a degree in any other University
Signature ________________________
Otieno Jacob
This project has been submitted for examination with my approval as the University Supervisor
Signature ________________________
Prof. J.A.M Ottieno
iv
Dedication
I dedicate this project to my wife Lilian; sons Benjamin, Vincent, John; daughter Joy and friends.
A special feeling of gratitude to my loving Parents, Justus and Margaret, my aunt Wilkista, my
Grandmother Nereah, and my late Uncle Edwin for their support and encouragement.
v
Acknowledgement
I would like to thank my family members, colleagues, friends and all those who helped and
supported me in writing this project. My sincere appreciation goes to my supervisor, Prof. J.A.M
Ottieno. I would not have made it this far without his tremendous guidance and support. I would
like to thank him for his valuable advice concerning the presentation of this work before the
Master of Science Project Committee of the School of Mathematics at the University of Nairobi
and for his encouragement and guidance throughout. I would also like to thank Dr. Kipchirchir
for his questions and comments which were beneficial for finalizing the project.
I would like to recognize the financial support from the Higher Education Loans Board (HELB)
without which this study might not have been accomplished. Most of all, I would like to thank
the Almighty God for always being there for me.
Table of Contents
Title.......................................................................................................................................................... i
Half Title ................................................................................................................................................. ii
Declaration ........................................................................................................................................... iii
Dedication ............................................................................................................................................. iv
Acknowledgement ................................................................................................................................. v
Table of Contents .................................................................................................................................. 1
Executive Summary............................................................................................................................. 16
1 Chapter I: General Introduction ................................................................................................. 18
1.1 Introduction ........................................................................................................................... 18
1.2 Problem Statement ................................................................................................................ 19
1.3 Objective................................................................................................................................ 19
1.4 Literature Review ................................................................................................................... 20
2 Chapter II: The Classical Beta Distribution and its Special Cases ........................................... 27
2.1 Beta and Gamma Functions.................................................................................................... 27
2.2 Construction of the Classical Beta Distribution ....................................................................... 31
2.2.1 From the beta function ................................................................................................... 31
2.2.2 From stochastic processes .............................................................................................. 32
2.2.3 From ratio transformation of two independent gamma variables ................................... 32
2.2.4 From the rth order statistic of the Uniform distribution ................................................... 33
2.3 Properties of the Classical Beta Distribution ........................................................................... 34
2.3.1 rth-order moments .......................................................................................................... 34
2.3.2 The first four moments ................................................................................................... 34
2.3.3 Mode ............................................................................................................................. 36
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2.3.4 Variance ......................................................................................................................... 37
2.3.5 Skewness ........................................................................................................................ 37
2.3.6 Kurtosis .......................................................................................................................... 39
2.4 Shapes of Classical Beta Distribution ...................................................................................... 41
2.5 Applications of the Classical Beta Distribution ........................................................................ 45
2.6 Special Cases of the Classical Beta Distribution ....................................................................... 45
2.6.1 Power Distribution Function ........................................................................................... 45
2.6.2 Properties of power distribution ..................................................................................... 46
2.6.3 Shape of Power distribution ........................................................................................... 47
2.6.4 Uniform Distribution Function ........................................................................................ 48
2.6.5 Properties of the Uniform distribution ............................................................................ 48
2.6.6 Shape of Uniform distribution ........................................................................................ 50
2.6.7 Arcsine Distribution Function.......................................................................................... 51
2.6.8 Properties of the arcsine distribution .............................................................................. 51
2.6.9 Shape of Arcsine distribution .......................................................................................... 54
2.6.10 Triangular shaped distributions (a=1, b=2) ...................................................................... 54
2.6.11 Properties of the Triangular shaped distribution (a=1, b=2) ............................................ 54
2.6.12 Shape of triangular distribution (a=1, b=2) ...................................................................... 57
2.6.13 Triangular shaped distributions (a=2, b=1) ...................................................................... 57
2.6.14 Properties of the Triangular shaped distribution (a=2, b=1) ............................................ 57
2.6.15 Shape of triangular distribution (a=2, b=1) ...................................................................... 59
2.6.16 Parabolic shaped distribution ......................................................................................... 60
2.6.17 Properties of parabolic shaped distribution .................................................................... 60
2.6.18 Shape of Parabolic distribution ....................................................................................... 62
2.7 Transformation of Special Case of the Classical Beta Distribution ........................................... 62
2.7.1 Wigner Semicircle Distribution Function ......................................................................... 62
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2.7.2 Properties of Wigner Semicircle ...................................................................................... 63
2.7.3 Shape of Wigner semicircle distribution .......................................................................... 65
3 Chapter III: Type II Two Parameter Inverted Beta Distribution .............................................. 66
3.1 Construction of Beta Distribution of the Second Kind ............................................................. 66
3.2 Properties of Beta Distribution of the Second Kind ................................................................. 67
3.2.1 rth-order moments .......................................................................................................... 67
3.2.2 Mean .............................................................................................................................. 69
3.2.3 Mode ............................................................................................................................. 70
3.2.4 Variance ......................................................................................................................... 71
3.2.5 Skewness ........................................................................................................................ 72
3.2.6 Kurtosis .......................................................................................................................... 73
3.2.7 Shapes of Beta distribution of the second kind ............................................................... 75
3.3 Application of Beta Distribution of the Second Kind ............................................................... 75
3.4 Special Cases of Beta Distribution of the Second Kind............................................................. 75
3.4.1 Lomax (Pareto II) distribution function ........................................................................... 75
3.4.2 Properties of Lomax distribution..................................................................................... 76
3.4.3 Shape of Lomax distribution ........................................................................................... 77
3.4.4 Log-Logistic (Fisk) distribution function ........................................................................... 77
3.4.5 Properties of Log-logistic distribution ............................................................................. 78
3.4.6 Shape of Log-logistic distribution .................................................................................... 79
3.5 Limiting Distribution ............................................................................................................... 79
4 Chapter IV: The Three Parameter Beta Distributions ............................................................... 80
4.1 Introduction ........................................................................................................................... 80
4.2 Case I: Libby and Novick Three Parameter Beta Distribution ................................................... 80
4.3 Case II: McDonaldโs Three Parameter Beta I Distribution ........................................................ 83
4.4 Gamma Distribution as a Limiting Distribution ....................................................................... 84
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4.5 Case III ................................................................................................................................... 85
4.5.1 3 Parameter Beta Distribution Derived from the Beta Distribution of the Second Kind .... 85
4.5.2 Case IV ........................................................................................................................... 86
4.5.3 Case V ............................................................................................................................ 86
4.5.4 Case VI ........................................................................................................................... 87
5 Chapter V: The Four Parameter Beta Distributions .................................................................. 88
5.1 Introduction ........................................................................................................................... 88
5.2 Generalized Four Parameter Beta Distribution of the First Kind .............................................. 88
5.2.1 Properties of generalized beta distribution of the first kind ............................................ 90
5.3 Applications of Generalized Beta Distribution of the First Kind ............................................... 93
5.4 Special Cases of Generalized Beta Distribution of the First Kind .............................................. 93
5.4.1 Pareto type 1 distribution function ................................................................................. 94
5.4.2 Properties of Pareto type 1 distribution .......................................................................... 94
5.4.3 Stoppa (generalized Pareto type 1) distribution function ................................................ 97
5.4.4 Properties of Stoppa distribution .................................................................................... 98
5.4.5 Kumaraswamy distributions ......................................................................................... 100
5.4.6 Properties of Kumaraswamy distribution ...................................................................... 101
5.4.7 Generalized Power distribution Function ...................................................................... 103
5.4.8 Properties of power distribution ................................................................................... 103
5.4.9 Generalized Gamma Distribution Function ................................................................... 105
5.4.10 Unit Gamma Distribution Function ............................................................................... 106
5.5 Generalized Four Parameter Beta Distribution of the Second Kind ....................................... 107
5.5.1 Properties of generalized beta distribution of the second kind ..................................... 108
5.6 Special Case of the Generalized Beta Distribution of the Second Kind................................... 113
5.6.1 Singh-Maddala (Burr 12) Distribution ........................................................................... 113
5.6.2 Properties of Singh-Maddala distribution ..................................................................... 114
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5.6.3 Shapes of Singh-Maddala distribution........................................................................... 117
5.6.4 Generalized Lomax (Pareto type II) Distribution ............................................................ 118
5.6.5 Properties of Generalized Lomax distribution ............................................................... 118
5.6.6 Inverse Lomax Distribution ........................................................................................... 121
5.6.7 Properties of Inverse Lomax distribution ...................................................................... 122
5.6.8 Dagum (Burr type 3) Distribution .................................................................................. 123
5.6.9 Properties of Dagum distribution .................................................................................. 123
5.6.10 Para-logistic distribution ............................................................................................... 125
5.6.11 General Fisk (Log-Logistic) Distribution Function ........................................................... 125
5.6.12 Properties of Fisk (log-logistic) distribution ................................................................... 126
5.6.13 Fisher (F) Distribution Function ..................................................................................... 127
5.6.14 Properties of F distribution ........................................................................................... 127
5.6.15 Half Studentโs t distribution .......................................................................................... 129
5.6.16 Logistic Distribution Function ....................................................................................... 130
5.6.17 Generalized Gamma Distribution Function ................................................................... 130
5.6.18 Properties of Generalized Gamma distribution ............................................................. 131
5.6.19 Gamma Distribution Function ....................................................................................... 134
5.6.20 Properties of Gamma distribution................................................................................. 134
5.6.21 Inverse Gamma distribution ......................................................................................... 136
5.6.22 Properties of Inverse Gamma distribution .................................................................... 137
5.6.23 Weibull Distribution Function ....................................................................................... 140
5.6.24 Properties of Weibull distribution ................................................................................. 140
5.6.25 Exponential distribution Function ................................................................................. 141
5.6.26 Properties of exponential distribution .......................................................................... 142
5.6.27 Chi-Squared distribution function ................................................................................. 143
5.6.28 Properties of Chi-squared distribution .......................................................................... 143
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5.6.29 Rayleigh distribution function ....................................................................................... 145
5.6.30 Properties of Rayleigh distribution ................................................................................ 145
5.6.31 Maxwell distribution function ....................................................................................... 146
5.6.32 Properties of Maxwell distribution................................................................................ 147
5.6.33 Half Standard normal function ...................................................................................... 149
5.6.34 Properties of Half Standard normal distribution............................................................ 149
5.6.35 Half-normal distribution function ................................................................................. 150
5.6.36 Properties of Half-normal distribution .......................................................................... 151
5.6.37 Half Log-normal distribution function ........................................................................... 152
5.6.38 Properties of Half Log-normal distribution .................................................................... 152
5.7 Four Parameter Generalized Beta Distribution Function ....................................................... 153
5.7.1 Properties of the four parameter generalized beta distribution .................................... 154
6 Chapter VI: The Generalized Beta Distribution Based on Transformation Technique .......... 155
6.1 Introduction ......................................................................................................................... 155
6.2 Five Parameter Generalized Beta Distribution Function Due to McDonald ............................ 155
6.3 Properties of the Five Parameter Generalized Beta Distribution ........................................... 157
6.3.1 rthโorder moments ....................................................................................................... 157
6.3.2 Mean ............................................................................................................................ 158
6.3.3 Variance ....................................................................................................................... 158
6.4 Special Cases of the Five Parameter Generalized Beta Distribution Function ........................ 158
6.4.1 The four parameter generalized beta distribution of the first kind ................................ 158
6.4.2 The four parameter generalized beta distribution of the second kind ........................... 159
6.4.3 The four parameter generalized beta distribution......................................................... 159
6.4.4 The four parameter generalized Logistic distribution function ...................................... 159
6.4.5 The three parameter Inverse Beta distribution of the first kind ..................................... 160
6.4.6 The three parameter Stoppa (generalized pareto type I) distribution function .............. 160
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6.4.7 The three parameter Singh-Maddala (Burr12) distribution function .............................. 160
6.4.8 The three parameter Dagum (Burr3) distribution function ............................................ 161
6.4.9 The two parameter Classical Beta distribution .............................................................. 161
6.4.10 The two parameter Pareto type I distribution function ................................................. 161
6.4.11 The two parameter Kumaraswamy distribution function .............................................. 162
6.4.12 The two parameter generalized Power distribution function ........................................ 162
6.4.13 The two parameter Beta distribution of the second kind .............................................. 162
6.4.14 The two parameter Inverse Lomax distribution function ............................................... 163
6.4.15 The two parameter General Fisk (Log-Logistic) distribution function ............................. 163
6.4.16 The two parameter Fisher (F) distribution function ....................................................... 163
6.4.17 The one parameter Power distribution function ........................................................... 164
6.4.18 The one parameter Wigner semicircle distribution function ......................................... 164
6.4.19 The one parameter Lomax (pareto type II) distribution function ................................... 165
6.4.20 The one parameter half studentโs (t) distribution function............................................ 165
6.4.21 Uniform distribution function ....................................................................................... 165
6.4.22 Arcsine distribution function ........................................................................................ 166
6.4.23 Triangular shaped distribution functions....................................................................... 166
6.4.24 Parabolic shaped distribution functions ........................................................................ 166
6.4.25 Log-Logistic (Fisk) distribution function ......................................................................... 167
6.4.26 Logistic distribution function ........................................................................................ 167
6.5 Construction of the Five Parameter Weighted Generalized Beta Distribution Function.......... 169
6.6 Properties of Five Parameter Weighted Generalized Beta Distribution ................................. 170
6.6.1 rth-order moments ........................................................................................................ 170
6.6.2 Mean ............................................................................................................................ 170
6.6.3 Variance ....................................................................................................................... 170
6.6.4 Skewness ...................................................................................................................... 171
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6.6.5 Kurtosis ........................................................................................................................ 171
6.7 Special Cases of the Five Parameter Weighted Generalized Beta Distribution Function ........ 172
6.7.1 Weighted beta of the second distribution function ....................................................... 172
6.7.2 Weighted Singh-Maddala (Burr12) distribution function ............................................... 172
6.7.3 Weighted Dagum (Burr3) distribution function ............................................................. 172
6.7.4 Weighted Generalized Fisk (Log-Logistic) distribution function ..................................... 173
6.8 Construction of the Five Parameter Exponential Generalized Beta Distribution Function ...... 175
6.9 Properties of Five Parameter Exponential Generalized Beta Distribution .............................. 176
6.9.1 Moment-generating function ....................................................................................... 176
6.9.2 Mean of the exponential generalized beta distribution ................................................. 177
6.10 Special Cases of Exponential Generalized Beta Distribution .................................................. 177
6.10.1 Exponential generalized beta of the first kind ............................................................... 178
6.10.2 Exponential generalized beta of the second kind .......................................................... 178
6.10.3 Exponential generalized gamma ................................................................................... 179
6.10.4 Generalized Gumbell distribution ................................................................................. 179
6.10.5 Exponential Burr 12 ...................................................................................................... 179
6.10.6 Exponential power distribution function ....................................................................... 180
6.10.7 Two parameter exponential distribution function ......................................................... 180
6.10.8 One parameter exponential distribution function ......................................................... 180
6.10.9 Exponential Fisk (logistic) distribution ........................................................................... 181
6.10.10 Standard logistic distribution .................................................................................... 181
6.10.11 Exponential weibull distribution ............................................................................... 181
7 Chapter VII: Generalized Beta Distributions Based on Generated Distributions ................... 184
7.1 Introduction ......................................................................................................................... 184
7.2 Beta Generator Approach .................................................................................................... 184
7.3 Various Beta Generated Distributions ................................................................................... 185
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7.3.1 Beta-Normal Distribution - (Eugene et. al (2002)) ......................................................... 185
7.3.2 Beta-Exponential distribution โ (Nadarajah and Kotz (2006)) ........................................ 186
7.3.3 Beta-Weibull Distribution โ (Famoye et al. (2005)) ....................................................... 188
7.3.4 Beta-Hyperbolic Secant Distribution โ (Fischer and Vaughan. (2007)) ........................... 189
7.3.5 Beta-Gamma โ (Kong et al. (2007)) ............................................................................... 189
7.3.6 Beta-Gumbel Distribution โ (Nadarajah and Kotz (2004)) .............................................. 190
7.3.7 Beta-Frรจchet Distribution (Nadarajah and Gupta (2004)) .............................................. 191
7.3.8 Beta-Maxwell Distribution (Amusan (2010)) ................................................................. 192
7.3.9 Beta-Pareto Distribution (Akinsete et.al (2008))............................................................ 193
7.3.10 Beta-Rayleigh Distribution (Akinsete and Lowe (2009)) ................................................. 194
7.3.11 Beta-generalized-logistic of type IV distribution (Morais (2009)) ................................... 195
7.3.12 Beta-generalized-logistic of type I distribution (Morais (2009)) ..................................... 196
7.3.13 Beta-generalized-logistic of type II distribution (Morais (2009)) .................................... 196
7.3.14 Beta-generalized-logistic of type III distribution (Morais (2009)) ................................... 197
7.3.15 Beta-beta prime distribution (Morais (2009))................................................................ 197
7.3.16 Beta-F distribution (Morais (2009)) ............................................................................... 198
7.3.17 Beta-Burr XII distribution (Paranaรญba et.al (2010)) ......................................................... 198
7.3.18 Beta-Dagum distribution (Condino and Domma (2010)) ............................................... 200
7.3.19 Beta-(fisk) log logistic distribution ................................................................................. 201
7.3.20 Beta-generalized half normal Distribution (Pescim, R.R, et.al (2009)) ............................ 201
7.4 Beta Exponentiated Generated Distributions........................................................................ 203
7.4.1 Exponentiated generated distribution .......................................................................... 203
7.4.2 Exponentiated exponential distribution (Gupta and Kundu (1999)) ............................... 204
7.4.3 Exponentiated Weibull distribution (Mudholkar et.al. (1995)) ...................................... 204
7.4.4 Exponentiated Pareto distribution (Gupta et.al (1998))................................................. 205
7.4.5 Beta exponentiated generated distribution .................................................................. 205
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7.4.6 Generalized beta generated distributions ..................................................................... 205
7.4.7 Generalized beta exponential distribution (Barreto-Souza et al. (2010)) ....................... 206
7.4.8 Generalized beta-Weibull distribution .......................................................................... 207
7.4.9 Generalized beta-hyperbolic secant distribution ........................................................... 208
7.4.10 Generalized beta-normal distribution ........................................................................... 208
7.4.11 Generalized beta-log normal distribution ..................................................................... 209
7.4.12 Generalized beta-Gamma distribution .......................................................................... 210
7.4.13 Generalized beta-Frรจchet distribution .......................................................................... 210
8 Chapter VIII: Generalized Beta Distributions Based on Special Functions ............................ 212
8.1 Introduction ......................................................................................................................... 212
8.2 Confluent Hypergeometric Function ..................................................................................... 212
8.3 The Differential Equations ................................................................................................... 213
8.4 Gauss Hypergeometric Function. .......................................................................................... 217
8.5 The Differential Equations ................................................................................................... 217
8.6 Appell Function .................................................................................................................... 220
8.7 Nadarajah and Kotzโs Power Beta Distribution ..................................................................... 221
8.8 Compound Confluent Hypergeometric Function ................................................................... 221
8.9 Beta-Bessel Distribution (A.K Gupta and S.Nadarajah (2006)) ............................................... 222
9 Chapter IX: Summary and Conclusion .................................................................................... 224
9.1 Summary.............................................................................................................................. 224
9.2 Conclusion ........................................................................................................................... 225
10 Appendix ................................................................................................................................ 226
A.1 Classical Beta Distribution .................................................................................................... 226
A.2 Power Distribution (b=1 in table A.1) .................................................................................... 226
A.3 Uniform Distribution (a=b=1 in table A.1) ............................................................................. 227
A.4 Arcsine Distribution .............................................................................................................. 227
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A.5 Triangular Shaped (a=1, b=2 in table A.1) ............................................................................. 228
A.6 Triangular Shaped (a=2, b=1 in table A.1) ............................................................................. 228
A.7 Parabolic Shaped.................................................................................................................. 228
A.8 Wigner Semicircle ................................................................................................................ 229
A.9 Beta Distribution of the Second Kind .................................................................................... 229
A.10 Lomax (Pareto II) (a=1 in table A.9)....................................................................................... 230
A.11 Log Logistic (Fisk) (a=b=1 in table A.9) .................................................................................. 230
A.12 Libby and Novick (1982) Beta Distribution ............................................................................ 231
A.13 McDonaldโs (1995) beta distribution of the first kind............................................................ 231
A.14 Three Parameter Beta Distribution ....................................................................................... 231
A.15 Chotikapanich et.al (2010) Beta of the Second Kind Distribution .......................................... 231
A.16 General Three Parameter Beta Distribution .......................................................................... 231
A.17 Three Parameter Beta Distribution ....................................................................................... 232
A.18 Generalized Four Parameter Beta Distribution of the First Kind ............................................ 232
A.19 Pareto Type I (b=1, p=-1 in table A.1) ................................................................................... 233
A.20 Stoppa (Generalized Pareto Type I) (a=1, p=-p in table A.1) .................................................. 234
A.21 Kumaraswamy (a=1, q=1 in table A.1) .................................................................................. 235
A.22 Generalized Power (b=1, p=1 in table A.1) ............................................................................ 236
A.23 Generalized Four Parameter Beta Distribution of the Second Kind ....................................... 237
A.24 Singh Maddala (Burr 12) ....................................................................................................... 238
A.25 Generalized Lomax (Pareto II) .............................................................................................. 239
A.26 Inverse Lomax (b=1, p=1 in table A.23) ................................................................................. 240
A.27 Dagum (Burr 3) (b=1 in table A.23) ....................................................................................... 240
A.28 Para-Logistic (a=1, b = p in table A.23) .................................................................................. 241
A.29 General Fisk (Log โLogistic) .................................................................................................. 241
A.30 Fisher (F) .............................................................................................................................. 241
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A.31 Half Studentโs t .................................................................................................................... 242
A.32 Logistic ................................................................................................................................. 242
A.33 Generalized Gamma ............................................................................................................. 242
A.34 Gamma ................................................................................................................................ 243
A.35 Inverse Gamma .................................................................................................................... 243
A.36 Weibull ................................................................................................................................ 244
A.37 Exponential .......................................................................................................................... 244
A.38 Chi-Squared ......................................................................................................................... 244
A.39 Rayleigh ............................................................................................................................... 245
A.40 Maxwell ............................................................................................................................... 245
A.41 Half Standard Normal ........................................................................................................... 245
A.42 Half Normal .......................................................................................................................... 246
A.43 Half Log Normal ................................................................................................................... 246
A.44 Four Parameter Generalized Beta Distribution ..................................................................... 246
A.45 Five Parameter Generalized Beta Distribution ...................................................................... 247
A.46 Five Parameter Weighted Generalized Beta Distribution ...................................................... 248
A.47 Weighted Beta of the Second Kind ....................................................................................... 248
A.48 Weighted Singh-Maddala (Burr12) ....................................................................................... 248
A.49 Weighted Dagum (Burr3) ..................................................................................................... 248
A.50 Weighted Generalized Fisk (Log-Logistic) .............................................................................. 249
A.51 Five Parameter Exponential Generalized Beta Distribution ................................................... 249
A.52 Exponential Generalized Beta Distribution of the First Kind .................................................. 249
A.53 Exponential Generalized Beta Distribution of the Second Kind ............................................. 249
A.54 Exponential Generalized Gamma (Generalized Gompertz) ................................................... 250
A.55 Generalized Gumbell ............................................................................................................ 250
A.56 Exponential Power ............................................................................................................... 250
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A.57 Exponential (Two Parameter) ............................................................................................... 250
A.58 Exponential (One Parameter) ............................................................................................... 251
A.59 Exponential Fisk (Logistic)..................................................................................................... 251
A.60 Standard Logistic .................................................................................................................. 251
A.61 Exponential Weibull ............................................................................................................. 251
A.62 Beta Generator .................................................................................................................... 251
A.63 ith Order Statistics (a=i, b=n-i+1 in table A.62) ....................................................................... 252
A.64 Beta-Normal (Substituting G(X) and g(X) in table A.62) ......................................................... 252
A.65 Beta-Exponential (Substituting G(X) and g(X) in table A.62) .................................................. 252
A.66 Beta-Weibull (Substituting G(X) and g(X) in table A.62) ........................................................ 252
A.67 Beta-Hyperbolic Secant (Substituting G(X) and g(X) in table A.62) ........................................ 253
A.68 Beta-Gamma (Substituting G(X) and g(X) in table A.62) ........................................................ 253
A.69 Beta-Gumbel (Substituting G(X) and g(X) in table A.62) ........................................................ 253
A.70 Beta-Frรจchet (Substituting G(X) and g(X) in table A.62) ......................................................... 254
A.71 Beta-Maxwell (Substituting G(X) and g(X) in table A.62) ....................................................... 254
A.72 Beta-Pareto (Substituting G(X) and g(X) in table A.62) .......................................................... 254
A.73 Beta-Rayleigh (Substituting G(X) and g(X) in table A.62) ....................................................... 255
A.74 Beta-Generalized-Logistic of Type IV (Substituting G(X) and g(X) in table A.62) ..................... 255
A.75 Beta-Generalized-Logistic of Type I (Substituting q=1 in table A.74) ...................................... 255
A.76 Beta-Generalized-Logistic of Type II (Substituting q=1 in table A.74) ..................................... 255
A.77 Beta-Generalized-Logistic of Type III (Substituting q=1 in table A.74) .................................... 255
A.78 Beta-Beta Prime ................................................................................................................... 256
A.79 Beta-F (Substituting G(X) and g(X) in table A.62) ................................................................... 256
A.80 Beta-Burr XII (Substituting G(X) and g(X) in table A.62) ......................................................... 256
A.81 Beta-Dagum (Substituting G(X) and g(X) in table A.62) ......................................................... 256
A.82 Beta-Fisk (Log-Logistic) (Substituting G(X) and g(X) in table A.62) ......................................... 257
14
A.83 Beta-Generalized Half Normal (Substituting G(X) and g(X) in table A.62) .............................. 257
A.84 Generalized Beta Generator ................................................................................................. 257
A.85 Exponentiated Exponential (Substituting G(X) and g(X) in table A.84) ................................... 257
A.86 Exponentiated Weibull (Substituting G(X) and g(X) in table A.84) ......................................... 258
A.87 Exponentiated Pareto (Substituting G(X) and g(X) in table A.84) ........................................... 258
A.88 Generalized Beta Exponential (Substituting G(X) and g(X) in table A.84) ............................... 258
A.89 Generalized Beta Weibull (Substituting G(X) and g(X) in table A.84) ..................................... 258
A.90 Generalized Beta Hyperbolic Secant (Substituting G(X) and g(X) in table A.84) ..................... 259
A.91 Generalized Beta Normal (Substituting G(X) and g(X) in table A.84) ...................................... 259
A.92 Generalized Beta Log Normal (Substituting G(X) and g(X) in table A.84) ............................... 259
A.93 Generalized Beta Gamma (Substituting G(X) and g(X) in table A.84) ..................................... 259
A.94 Generalized Beta Frรจchet (Substituting G(X) and g(X) in table A.84) ..................................... 260
A.95 Confluent Hypergeometric ................................................................................................... 260
A.96 Gauss Hypergeometric ......................................................................................................... 260
A.97 Appell .................................................................................................................................. 260
11 References .............................................................................................................................. 262
List of Figures
Fig.1. 1: General Framework based on transformation ........................................................................... 21
Fig.1. 2: Beta Generated Distributions ................................................................................................... 24
Fig.1. 3: Generalized Beta Distributions Based on Special Functions ..................................................... 26
Fig.2. 1 : Left Skewedness ..................................................................................................................... 42
Fig.2. 2: Case of beta unimodal distribution with ๐ > 0 ......................................................................... 43
Fig.2. 3: Case of beta unimodal distribution with ๐ > 0 ......................................................................... 43
Fig.2. 4: j-Shaped beta distribution ....................................................................................................... 44
Fig.2. 5: inverted j-Shaped beta distribution ........................................................................................... 44
15
Fig.2. 6: Power distribution ................................................................................................................... 48
Fig.2. 7: Uniform distribution on the interval (0,1) ................................................................................. 50
Fig.2. 8: Arcsine distribution ................................................................................................................. 54
Fig.2. 9: Triangular shaped distribution.................................................................................................. 57
Fig.2. 10: Trangular shaped distribution ................................................................................................. 59
Fig.2. 11: Parabolic distribution ............................................................................................................. 62
Fig.2. 12: Wigner semicircle distribution ............................................................................................... 65
Fig.3. 1: Beta distribution of the second kind ......................................................................................... 75
Fig.3. 2: Lomax distribution .................................................................................................................. 77
Fig.3. 3: Log-logistic distribution........................................................................................................... 79
16
Executive Summary
This Masterโs project considers on one hand, the construction of the two parameter classical beta
distribution on the domain (0,1) and its generalization and on the other hand, the construction of
the extended two parameter beta distribution on the domain (0,โ) and its generalization. It
provides a comprehensive mathematical treatment of these two parameter beta distributions, their
constructions, properties, shapes and special cases.
There are various ways of constructing distributions in general. Four ways of constructing the
classical beta distribution considered in this work are constructions from,
i. The definition of the beta function;
ii. A Poisson process;
iii. The transformation of a ratio of two independent gamma variables; and
iv. Order statistic.
Properties derived include the expressions of the rth
moments, first four moments, mode,
coefficient of skewness and coefficient of kurtosis. Special cases such as Power, Uniform,
Arcsine, Triangular shaped, Parabolic shaped and Wigner Semicircle distributions are given. By
the transformation technique a two parameter inverted beta distribution was obtained and its
special cases such as Lomax (Pareto II) and Log-logistic (Fisk) distributions were constructed.
The work also looked at three methods of generalizing the classical beta distribution i.e. through:
1. Transformation technique
2. Generator Approach
3. Use of Special functions
The transformation technique resulted into the following:
i. Three parameter beta distributions of both first and second kinds. In particular, the Libby-
Novick and McDonaldโs distributions.
ii. Generalized four parameter beta distributions of the first and second kinds. Particular
cases are the McDonald four parameter beta distribution of the first kind with its special
17
cases, the McDonald four parameter beta distribution of the second kind with its special
cases and the four parameter generalized beta distribution.
iii. The five parameter generalized beta distribution due to McDonald and Xu (1995).
Generalized distributions based on generator approach due to the work of Eugene et al. (2002)
are classified into:
Beta generated distributions
Exponentiated generated distributions
Generalized beta generated distributions
The generalized beta distributions based on the use of special functions considered includes the
Confluent hypergeometric distribution and the Gauss Hypergeometric distribution from the
Confluent hypergeometric function and the Gauss Hypergeometric function respectively. Other
distributions are based on the Appell function and the Bessel function.
Tables showing special cases of beta distributions and their generalizations are given in the
appendix.
18
1 Chapter I: General Introduction
1.1 Introduction
In statistical distributions, a vast activity has been observed in generalizing classical
distributions by adding more parameters to make them more flexible in analyzing
empirical data. Various methods have been developed to achieve this purpose. Among
such methods are the transformation technique, generator approach, mixture
(compounding) and use of special functions. In this work we examine the case of
classical beta distribution. The term generalized used in this work has the notion of
adding more parameters to the existing distribution in order to give a more universal
distribution as opposed to the well known existing generalized distributions.
Historically, the work on beta distributions started as early as 1676 when Sir Isaac
Newton wrote a letter to Henry Oldenberg as given by Dutka, J., (1981). In his letter
dated 24th October, 1676, Newton evaluated ๐ฆ๐๐ฅ
๐ฅ
0 as a series. The particular case of the
result was applicable to the evaluation of the classical beta distribution. The two
parameter classical beta distributions are among the celebrated Pearson system of
statistical distributions derived by Karl Pearson in the 1890s in connection with his work
on evolution.
A handbook of beta distribution and its applications was published in 2004 edited by
Gupta and Nadarajah. The book enumerates the properties of beta distributions and
related mathematical notions. It demonstrates the applications in the fields of economics,
quality control, soil science and biomedicine. It further discusses the centrality of beta
distributions in Bayesian inference, the beta-binomial model and applications of the beta-
binomial distribution. However, immediately after that publication, further work has been
done in that area. There is therefore a need to re-examine the works on beta distribution.
19
1.2 Problem Statement
Recent developments focus on new techniques for building meaningful distributions, a
remarkable development has been observed in generalizing the classical beta distribution.
There has been a growing interest in the applications of these distributions in the analysis
of empirical data and for many statistical procedures. However, their applications are
limited in important ways as they could not fit data very well. The main trend is therefore
to add more parameters to make them more flexible in order to fit the existing empirical
data.
1.3 Objective
The objective of this project is to review various methods that have been used to
construct the beta distribution from the classical to generalized beta distributions. The
specific objectives are:
to study the classical two parameter beta distribution within (0,1) domain and
(0,โ) domain
to generalize the two parameter beta distributions using
i. transformation technique
ii. generator approach
iii. some special functions
20
1.4 Literature Review
This section reviews various methods that have been used in constructing beta
distributions. There are three classes of generalized beta distributions:
i. Those based on transformations
The generalized beta distributions based on transformation has largely been contributed
by McDonald (1984). McDonald and his associates contributed in the development of the
generalized beta distributions as an income distribution and in unifying the various
research activities in closely related fields.
McDonald and Xu (1995) introduced a five parameter beta distribution which nest the
generalized beta and gamma distributions and include more than thirty distributions as
limiting or special cases.
McDonald and Richards (1987a, b) presented a four parameter generalized beta
distribution which includes as special cases three and two parameter beta, generalized
gamma, Weibull, power function, Pareto, lognormal, half-normal, uniform and others.
Libby and Novick (1982) proposed a three parameter generalized beta distribution for
utility fitting.
The figure 1.1 below illustrates the transformation from the two parameter classical beta
distribution to the five parameters generalized beta distribution where the transformations
are given as follows:
T1: is the transformation changing a beta type I (classical) distribution to a two
parameter beta type II (Inverted Beta) distribution
T2: transforms a 2 parameter to 3 parameter beta distributions
T3: transforms a 3 parameter to a 4 parameter beta distributions
T4: transforms a 4 parameter to 5 parameter beta distributions
Tiโs are defined in the coming chapters
21
Fig.1. 1: General Framework based on transformation
Beta Type I
2 Parameters
Beta Type II
2 Parameters
Generalized beta II
3 Parameters
Generalized beta I
3 Parameters
Generalized beta I
4 Parameters
Generalized beta II
4 Parameters
Generalized beta
5 Parameters
T1
T2 T2
T3 T3
T4 T4
22
ii. Generator approach based on CDF
The latest work involving the class of generalized beta distributions was introduced by
Eugene et.al (2002) that defined the beta-normal distribution and studied some of its
properties. Since then, several authors have been studying particular cases of this class of
distributions. An advantage of the beta-normal distribution over the normal distribution is
that the beta normal can be unimodal and bimodal.
The bimodal region for the beta-normal distribution was studied by Famoye et. al (2003).
Gupta and Nadarajah (2004) later derived a more general expression for the nth
moment
of the beta-normal distribution.
Nadarajah and Kotz (2004) defined the beta-Gumbel distribution, which has greater tail
flexibility than the Gumbel distribution.
Nadarajah and Gupta (2004) defined the beta-Frรจchet distribution and Barreto-Souza
et.al. (2008) presented some additional mathematical properties.
Nadarajah and Kotz (2006) defined the beta-exponential distribution and gave a detailed
presentation.
Famoye et.al (2005) defined the beta-Weibull and Lee et.al (2007) made some
applications to censored data.
Kong et.al (2007) proposed the beta-gamma distribution.
The four parameter beta-pareto distribution was defined and studied by Akinsete et al
(2008).
Akinsete and Lowe (2009) proposed the beta-Rayleigh distribution.
Amusan (2010) defined the beta-Maxwell distribution and studied some of its properties.
The figure 1.2 below illustrates how the generator approach has been applied in the two
parameters classical beta distribution to derive other generalized beta distributions.
iii. Generalized beta generated
Barreto-Souza et al. (2010) proposed the generalized beta exponential distribution and
discussed maximum likelihood estimation of its parameters.
23
iv. Beta-exponentiated distributions
Since 1995, the exponentiated distributions have been widely studied in statistics with the
development of various classes of these distributions. Mudholkar et.al. (1995) proposed
the exponentiated Weibull distribution.
Gupta et al. (1998) introduced the exponentiated Pareto distribution.
Gupta and Kundu (1999) introduced the exponentiated exponential distribution as a
generalization of the standard exponential distribution.
24
Fig.1. 2: Beta Generated Distributions
Classical beta distribution
Beta Normal
Eugene et al. (2002)
Beta Rayleigh
Akinsete and Lowe (2009)
Beta Maxwell
Amusan (2010)
Beta Pareto
Akinsete et.al (2008)
Beta Gamma
Kong et.al (2007)
Properties of Beta Frรจchet
Barreto-Souza et.al (2008)
Beta Frรจchet
Nadarajah and Gupta (2004)
Application on Beta Weibull
Lee et al. (2007
Beta Weibull
Famoye & Olumolade (2005)
Beta Gumbel
Kotz and Nadarajah (2004)
nth
moment of beta normal
Gupta and Nadarajah (2004)
Bimodal beta normal
Famoye et al (2003)
Beta exponential
Kotz and Nadarajah (2006)
Generalized Order Statistics
Jones (2004)
Original Original
Beta Hyperbolic Secant
Fischer and Vaughan (2007)
Beta Generalized Half Normal
Pescim R.R et.al (2009)
Beta Dagum
Condino and Domma (2010)
Beta Burr XII
Paranaiba et.al (2010)
Beta (Fisk) Log Logistic
Paranaiba et.al (2010)
Beta Generalized Logistic of Type IV
Morais (2009)
Beta Generalized Logistic of Type I
Morais (2009)
Beta Beta Prime
Morais (2009)
Beta F
Morais (2009)
25
v. Based on special functions
Beta function is a special function. Other special functions have been used to generalize
the beta distribution.
Armero and Bayarri (1994) suggested Gauss hypergeometric function distribution in
connection with marginal prior/posterior distribution. They obtained the generalized beta
distribution by dividing the classical beta distribution by certain algebraic function.
Nadarajah and Kotz (2004) also studied the beta distribution based on the Gauss
hypergeometric function.
Gordy (1988a) introduced confluent hypergeometric distribution and applied it to auction
theory. He (Gordy (1988b)) also introduced the compound confluent hypergeometric
distribution which contains McDonald and Xuโs generalized beta, Gauss hypergeometric
and confluent hypergeometric as special cases. Gordy generalized the classical beta
distribution using the confluent hypergeometric function in the way Armero and Bayarri
(1994) used Gauss hypergeometric function to generalize the beta to the Gauss
hypergeometric distribution.
Nadarajah and Kotz (2006) introduced ๐น1 beta distribution based on Appell function of
the first kind. The distribution is very flexible and contains several of the known
generalization of the classical beta distribution as particular cases.
The figure 1.3 below illustrates the relationship between the two parameters classical beta
distribution and some of the special functions.
26
Fig.1. 3: Generalized Beta Distributions Based on Special Functions
Classical beta
distribution
Gauss hypergeometric
Armero/Bayarri (1994)
Confluent
hypergeometric function
Gordy (1988a)
Compound confluent
hypergeometric function
Gordy (1988b)
Appell Function, ๐น1
Nadarajah/Kotz (2006)
Generalized beta
McDonald/Xu (1995)
Gauss
hypergeometric
Confluent
hypergeometric
Beta Bessel
Gupta and Nadarajah (2006)
27
2 Chapter II: The Classical Beta Distribution
and its Special Cases
2.1 Beta and Gamma Functions
Beta distributions are based on gamma functions. A brief description of these two
functions is therefore in order.
i. Beta Function
The beta function is a special function denoted by ๐ต ๐, ๐ and defined as
๐ต ๐, ๐ = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐๐ฅ
๐
๐
, ๐ > 0, ๐ > 0
ii. Gamma Function
The gamma function is a special function denoted by ฮ and defined as
๐ค ๐ = ๐ก๐โ1๐โ๐ก๐๐กโ
0
iii. Relationship between the Beta function and the Gamma function
There are two different methods for deriving the relationship between the beta function
and the gamma function:
i. The formula of the beta function can be derived as a definite integral whose
integrand depends on two variables ๐ and ๐ by reversing the order of integration of a
double integral as follows:
ฮ(๐) ฮ(๐) = ๐ก๐โ1๐โ๐ก๐๐ก
โ
๐
๐ข๐โ1๐โ๐ข๐๐ข
โ
๐
28
= ๐ก๐โ1๐ข๐โ1๐โ(๐ก+๐ข)๐๐ก๐๐ข
โ
๐
โ
๐
Using the transformation ๐ข = ๐ก๐ฃ we obtain
ฮ ๐ ฮ ๐ = ๐ก๐โ1(๐ก๐ฃ)๐โ1๐โ(๐ก+๐ก๐ฃ)๐ก๐๐ก๐๐ฃ
โ
๐
โ
๐
= ๐ก๐โ1๐ก๐๐ฃ๐โ1๐โ(๐ก+๐ก๐ฃ)๐๐ก๐๐ฃ
โ
๐
โ
๐
= ๐ก๐+๐โ1๐ฃ๐โ1๐โ๐ก(1+๐ฃ)๐๐ก๐๐ฃ
โ
๐
โ
๐
Using the transformation ๐ค = ๐ก ๐ฃ + 1 again gives
= ๐ค
๐ฃ + 1
๐+๐โ1
๐ฃ๐โ1๐โ๐ค (๐ฃ + 1)โ1๐๐ค๐๐ฃ
โ
๐
โ
๐
= ๐ฃ๐โ1๐๐ฃ ๐ค๐+๐โ1๐โ๐ค
(๐ฃ + 1)๐+๐๐๐ค
โ
๐
โ
๐
= ๐ฃ๐โ1
(๐ฃ + 1)๐+๐๐๐ฃ
โ
๐
๐ค๐+๐โ1๐โ๐ค๐๐ค
โ
๐
= ๐ฃ๐โ1
๐ฃ + 1 ๐+๐๐๐ฃ
โ
๐
ฮ(a + b)
From which it follows that
๐ต ๐, ๐ = ฮ(๐) ฮ(๐)
ฮ(๐ + ๐)=
๐ฃ๐โ1
๐ฃ + 1 ๐+๐๐๐ฃ
โ
๐
Finally, using the transformation ๐ฃ =๐ฅ
1 โ ๐ฅ we get the alternative formulation
29
๐ต ๐, ๐ =
๐ฅ1 โ ๐ฅ
๐โ1
๐ฅ
1 โ ๐ฅ + 1 ๐+๐
1
(1 โ ๐ฅ)2๐๐ฅ
โ
๐
= (
๐ฅ1 โ ๐ฅ)๐โ1
1
1 โ ๐ฅ ๐+๐
1
(1 โ ๐ฅ)2๐๐ฅ
โ
๐
= ๐ฅ๐โ1(1 โ ๐ฅ)โ๐+1+๐+๐โ2 ๐๐ฅ
โ
๐
= ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1 ๐๐ฅ
1
๐
which is the beta function.
ii. Another method for deriving the relationship between beta function and gamma
function is by using the polar co-ordinates as shown below.
ฮ(๐) ฮ(๐) = ๐ฅ๐โ1๐โ๐ฅ๐๐ฅ
โ
๐
๐ฆ๐โ1๐โ๐ฆ๐๐ฆ
โ
๐
= ๐โ(๐ฅ+๐ฆ)๐ฅ๐โ1
โ
๐
๐ฆ๐โ1๐๐ฅ๐๐ฆ
โ
๐
let ๐ฅ = s2 and ๐ฆ = t2
therefore d๐ฅ = 2๐ ๐๐ and ๐๐ฆ = 2๐ก๐๐ก
now
ฮ(a)ฮ(๐) = ๐โ(๐ 2+t2)s2(aโ1)
โ
๐
๐ก2(๐โ1). 2๐ ๐๐ 2๐ก๐๐ก
โ
๐
= 4 ๐โ(๐ 2+t2)s2aโ1
โ
๐
๐ก2๐โ1๐๐ ๐๐ก
โ
๐
Next, let ๐ = ๐๐ ๐๐๐ ๐๐๐ ๐ก = ๐๐๐๐ ๐
Therefore ๐๐ = ๐๐๐๐ ๐ ๐๐ ๐๐๐ ๐๐ก = โ๐๐ ๐๐๐๐๐
30
๐ฝ =
๐๐
๐๐
๐๐ก
๐๐๐๐
๐๐
๐๐ก
๐๐
= ๐๐ถ๐๐ ๐ โ๐๐๐๐๐๐๐๐๐ ๐ถ๐๐ ๐
= ๐๐ถ๐๐ 2๐ + ๐๐๐๐2๐ = ๐
โด ๐ค(๐)๐ค(๐) = 4 ๐โ๐2
๐/2
๐
(๐๐๐๐๐)2๐โ1(๐๐ถ๐๐ ๐)2๐โ1
โ
๐
๐ฝ ๐๐๐๐
= 4 ๐โ๐2
๐/2
๐
๐2๐โ1+2๐โ1(๐๐๐๐)2๐โ1(๐ถ๐๐ ๐)2๐โ1
โ
๐
๐๐๐๐๐
= 2 2 (๐๐๐๐)2๐โ1
๐/2
๐
๐ถ๐๐ ๐ 2๐โ1๐๐ ๐โ๐2๐2(๐+๐โ1)
โ
๐
๐๐๐
Recall that ๐ต(๐, ๐) = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ11
0๐๐ฅ
Let ๐ฅ = ๐๐๐2๐ โน ๐๐ฅ = 2๐๐๐๐๐ถ๐๐ ๐๐๐
โด ๐ต ๐, ๐ = ๐๐๐๐ 2๐โ2
๐2
๐
๐ถ๐๐ ๐ 2๐โ2 . 2๐๐๐๐๐ถ๐๐ ๐๐๐
= 2 ๐๐๐๐ 2๐โ1
๐2
๐
๐ถ๐๐ ๐ 2๐โ1๐๐
โด ฮ a ฮ(b) = 2 ๐ต ๐, ๐ ๐โ๐2๐2(๐+๐โ1)
โ
๐
๐๐๐
= ๐ต ๐, ๐ ๐โ๐2๐2(๐+๐โ1)2๐
โ
๐
๐๐
Let ๐ข = ๐2 โน ๐๐ข = 2๐๐๐
โด ฮ a ฮ(b) = ๐ต ๐, ๐ ๐โ๐ข๐ข๐+๐โ1
โ
๐
๐๐ข
= ๐ต ๐, ๐ ฮ(๐ + ๐)
โด ๐ต ๐, ๐ =ฮ a ฮ b
ฮ(๐ + ๐)
31
2.2 Construction of the Classical Beta Distribution
2.2.1 From the beta function
By definition
๐ต ๐, ๐ = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐๐ฅ
๐
๐
, ๐ > 0, ๐ > 0
Normalizing the beta function gives
1 = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐๐ฅ
๐ต ๐, ๐
1
0
Thus the probability density function of the beta distribution with shape parameters a
and b is expressed as
๐ ๐ฅ; ๐, ๐ =๐ฅ๐โ1(1 โ ๐ฅ)๐โ1
๐ต(๐, ๐) , 0 < ๐ฅ < 1, ๐ > 0, ๐ > 0 _______(2.1)
The cumulative distribution function of equation 2.1 is given by
๐น ๐ฅ =1
๐ต(๐, ๐) ๐ก๐โ1(1 โ ๐ก)๐โ1
๐ฅ
0
๐๐ก, 0 < ๐ก < 1, ๐ > 0, ๐ > 0_______(2.2)
Where the parameters ๐ and ๐ are positive real quantities and the variable x satisfies
0 โค ๐ฅ โค 1.The quantity ๐ต(๐, ๐) is the beta function. Equation (2.1) is also known as the
standard beta or classical beta distribution.
32
2.2.2 From stochastic processes
Consider a Poisson process with arrival rate of ๐ events per unit time. Let ๐ค๐ denote the
waiting time until the kth
arrival of an event and ๐ค๐ denote the waiting time until the sth
arrival, ๐ > ๐. then, ๐ค๐ and ๐ค๐ โ ๐ค๐ are independent gamma random variables with
๐ค๐~gamma (๐,1
๐) and ๐ค๐ โ ๐ค๐~ gamma (s โ k, 1/ฮป).
The proportion of the time taken by the first k arrivals in the time needed for the first
arrival is
๐ค๐
๐ค๐ =
๐ค๐
๐ค๐+(๐ค๐ โ๐ค๐)~ ๐๐๐ก๐ ๐, ๐ โ ๐
In simple terms, a continuous beta distribution ๐ต(๐, ๐) can be derived from a Poisson
process such that if ๐ + ๐ events occur in a time interval, then the fraction of that
interval until the ๐๐ก๐ event occurs has a ๐ต(๐, ๐) distribution.
2.2.3 From ratio transformation of two independent gamma variables
Let ๐1 and ๐2 be independent gamma variables with joint pdf
๐ ๐ฅ1, ๐ฅ2 =1
ฮ ฮฑ ฮ(ฮฒ)๐ฅ1
๐ผโ1๐ฅ2๐ฝโ1๐โ๐ฅ1โ๐ฅ2 , 0 < ๐ฅ1 < โ, 0 < ๐ฅ2 < โ
where ๐ผ > 0, ๐ฝ > 0
Let ๐1 = ๐1 + ๐2 and ๐2= ๐1
๐1+๐2
๐ฆ1 = ๐1 (๐ฅ1, ๐ฅ2) = ๐ฅ1 + ๐ฅ2
๐ฆ2 = ๐2 (๐ฅ1, ๐ฅ2) =๐ฅ1/(๐ฅ1 + ๐ฅ2)
๐ฅ1 = ๐1(๐ฆ1 , ๐ฆ2)= ๐ฆ1๐ฆ2
๐ฅ2 = ๐2(๐ฆ1, ๐ฆ2)= ๐ฆ1(1 โ ๐ฆ2)
๐ฝ = ๐ฆ1 ๐ฆ2
1 โ ๐ฆ2 โ๐ฆ1 = โ๐ฆ1
The transformation is one-to-one and maps ๐, the first quadrant of the ๐ฅ1๐ฅ2 plane onto
โฌ = {(๐ฆ1 , ๐ฆ2): 0 < ๐ฆ1 < 1, 0 < ๐ฆ2 < 1}.
33
The joint pdf of ๐1, ๐2 is
๐(๐ฆ1 , ๐ฆ2) = ๐ฆ1/ฮ(ฮฑ)ฮ(ฮฒ) ๐ฆ1๐ฆ2 ๐ผโ1[๐ฆ1 1 โ ๐ฆ2 ]๐ฝโ1๐โ๐ฆ1
=๐ฆ2
๐ผโ1 1 โ ๐ฆ2 ๐ฝโ1
ฮ(ฮฑ)ฮ(ฮฒ)๐ฆ1
๐ผ+๐ฝโ1๐โ๐ฆ1 , (๐ฆ1,๐ฆ2) โ โฌ.
Because โฌ is a rectangular region and because g (๐ฆ1,๐ฆ2) can be factored into a function
of ๐ฆ1 and a function of ๐ฆ2, it follows that ๐1 and ๐2 are statistically independent.
The marginal pdf of ๐2 is
๐๐2(๐ฆ2) =
๐ฆ2๐ผโ1 1 โ ๐ฆ2
๐ฝโ1
ฮ(ฮฑ)ฮ(ฮฒ) ๐ฆ1
๐ผ+๐ฝโ1โ
0
๐โ๐ฆ1๐๐ฆ1
=ฮ ฮฑ + ฮฒ
ฮ(ฮฑ)ฮ(ฮฒ)๐ฆ2
๐ผโ1 1 โ ๐ฆ2 ๐ฝโ1_______(2.3)
for 0 < ๐ฆ2 < 1.
This is the pdf of a beta distribution with parameters ฮฑ and ฮฒ.
Also, since ๐(๐ฆ1 , ๐ฆ2) = ๐๐1(๐ฆ1) ๐๐2
(๐ฆ2) it implies that
๐๐1(๐ฆ1) =
1
ฮ ฮฑ + ฮฒ ๐ฆ1
๐ผ+๐ฝโ1๐โ๐ฆ1
for 0 < ๐ฆ1 < โ.
Thus, ๐1 has a gamma distribution with parameter values ฮฑ + ฮฒ and 1.
2.2.4 From the rth order statistic of the Uniform distribution
Let ๐1, โฆ , ๐๐ be independent and identically distributed random sample from the
standard Uniform distribution ๐ข(0,1). Let ๐๐ be the rth
order statistics from this sample.
Then the probability distribution of ๐๐ is a beta distribution with parameters r and n-r+1.
34
2.3 Properties of the Classical Beta Distribution
2.3.1 rth-order moments
To obtain the rth
-order moments of the classical beta distribution, we evaluate
E Xr = ๐ฅ๐๐ ๐ฅ; ๐, ๐
1
0
๐๐ฅ
= ๐ฅ๐ ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1
๐ต(๐, ๐)
1
0
๐๐ฅ
=1
B(๐, ๐) ๐ฅ๐+๐โ1(1 โ ๐ฅ)๐โ1
1
0
๐๐ฅ
= 1
๐ต(๐, ๐)
๐ฅ(๐+๐)โ1(1 โ ๐ฅ)๐โ1
๐ต(๐ + ๐, ๐)
1
0
. ๐ต(๐ + ๐, ๐)๐๐ฅ
=๐ต(๐ + ๐, ๐)
๐ต(๐, ๐)
=ฮ(a + r) ฮ(b)
ฮ(a + b + r) .
ฮ(a + b)
ฮ(a)ฮ(b)
=ฮ(a + r) ฮ(a + b)
ฮ(a + b + r) ฮ(a) ________________(2.4)
2.3.2 The first four moments
The mean is found by substituting r = 1 in equation (2.4)
i. e., ๐1 โก ฮผ = E ๐ =ฮ(a + 1) ฮ(a + b)
ฮ(a + b + 1)ฮ(a)
=aฮ(a) ฮ(a + b)
(a + b)ฮ(a + b) ฮ(a)
=a
a + b (๐ถ๐๐๐๐ ๐ก๐๐๐ ๐๐๐๐๐, 1996)_________________________(2.5)
35
๐2 โก ๐ธ ๐2 =ฮ(a + b) ฮ(a + 2)
ฮ a + b + 2 ฮ(a)
= ฮ(a + b) (a + 1)ฮ(a + 1)
(a + b + 1)ฮ(a + b + 1) ฮ(a)
=ฮ a + b (a + 1) aฮ(a
a + b + 1 (a + b)ฮ(a + b) ฮ(a)
=(a + 1) a
a + b + 1 (a + b)
=a(a + 1)
(a + b) a + b + 1
= ๐1
(a + 1)
a + b + 1 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก. ๐๐. , 2010)____________(2.6)
๐3 โก ๐ธ ๐3 =ฮ(a + b) ฮ(a + 3)
ฮ(a + b + 3) ฮ(a)
=ฮ(a + b) a + 2 a + 1 aฮ(a)
a + b a + b + 2 a + b + 1 ฮ(a + b) ฮ(a)
=(a + 2)(a + 1)a
a + b + 2 a + b + 1 (a + b)
= ๐2
a + 2
a + b + 2 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)_______________(2.7)
๐4 โก ๐ธ ๐4 =ฮ(a + b) ฮ(a + 4)
ฮ(a + b + 4) ฮ(a)
=ฮ a + b a + 3 (a + 2) a + 1 aฮ(a)
a + b (a + b + 1) a + b + 2 a + b + 3 ฮ(a + b) ฮ(a)
=(a + 3)(a + 2)(a + 1)a
(a + b + 3) a + b + 2 a + b + 1 (a + b)
36
= a(a + 1) (a + 2)(a + 3)
(a + b) a + b + 1 a + b + 2 a + b + 3
= ๐3
(a + 3)
a + b + 3
2.3.3 Mode
Mode is obtained by solving
d
dx๐(๐ฅ; ๐, ๐) = 0
d
dx ๐ฆ๐โ1 1 โ ๐ฅ ๐โ1
๐ต ๐, ๐ = 0
(๐ โ 1)๐ฅ๐โ2 1 โ ๐ฅ ๐โ1 + ๐ โ 1 (โ1) 1 โ ๐ฅ ๐โ2๐ฅ๐โ1
๐ต ๐, ๐ = 0
(๐ โ 1)๐ฅ๐โ2 1 โ ๐ฅ ๐โ1 = ๐ โ 1 1 โ ๐ฅ ๐โ2๐ฅ๐โ1
(๐ โ 1)๐ฅ๐โ1๐ฅโ1 1 โ ๐ฅ ๐โ1 = ๐ โ 1 1 โ ๐ฅ ๐โ1(1 โ ๐ฅ)โ1๐ฅ๐โ1
(๐ โ 1)๐ฅโ1 = ๐ โ 1 (1 โ ๐ฅ)โ1
๐ โ 1
๐ โ 1=
๐ฅ
1 โ ๐ฅ, ๐ > 1, ๐ > 1
1 โ ๐ฅ ๐ โ 1 = ๐ฅ(๐ โ 1)
๐ โ 1 โ ๐ฅ๐ + ๐ฅ = ๐ฅ๐ โ ๐ฅ
๐ฅ โ๐ + 1 โ ๐ + 1 = 1 โ ๐
๐ฅ =(๐ โ 1)
(๐ + ๐ โ 2), ๐ฅ = 0 ๐๐ ๐ = 1 ๐๐๐ ๐ฅ = 1 ๐๐ ๐ = 1________________(2.8)
37
2.3.4 Variance
Since variance can be expressed as
Var X = ๐ธ ๐2 โ ๐ธ ๐ 2
= m2 โ ๐12
We can now get the variance as
Var X = ๐2 =a(a + 1)
(a + b)(a + b + 1)โ
a
a + b
2
=a a + 1 (a + b)2 โ a2 a + b (a + b + 1)
(a + b)(a + b + 1)(a + b)2
=a a + 1 (a + b) โ a2 a + b a + b + 1
a + b + 1 a + b 2
=a3 + a2b + a2 + ab โ a3 โ a2b + a2
a + b + 1 a + b 2
=ab
a + b + 1 a + b 2(๐ถ๐๐๐๐ ๐ก๐๐๐ ๐๐๐๐๐, 1996)____________(2.9)
2.3.5 Skewness
Skewness (the third central moment) can be expressed as
Skewness = ๐ข3 = ๐ธ(๐ โ ๐ธ(๐))3
= ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
= m3 โ 3๐2๐1+2๐13,
This can be manipulated algebraically as follows:
๐ข3 =a a + 1 (a + 2)
a + b a + b + 1 (a + b + 2)โ 3
a2(a + 1)
(a + b)2(a + b + 1)+ 2
a
a + b
3
38
=a a + 1 (a + 2) โ a2 a + 1 (a + b + 2)
(a + b)2(a + b + 1)(a + b + 2)+
2a3
(a + b)3
= a3 + 3a2 + 2a a + b โ 3a4 โ 3a3b โ 9a3 โ 3a2b โ 6a2
a + b 2 a + b + 1 a + b + 2 +
2a3
(a + b)3
=a4 + 3a3 + 2a2 + a3b + 3a2b + 2ab โ 3a4 โ 3a3b โ 9a3 โ 3a2b โ 6a2
a + b 2 a + b + 1 a + b + 2
+2a3
(a + b)3
= โ2a4 โ 6a3 โ 4a2 โ 2a3b + 2ab
a + b 2 a + b + 1 a + b + 2 +
2a3
(a + b)3
= (a + b)(โ2a4 โ 6a3 โ 4a2 โ 2a3b + 2ab) + (2a4 + 2a3b + 2a3)(a + b + 2)
a + b 3 a + b + 1 a + b + 2
= โ2a5 โ 6a4 โ 2a4b โ 4a3 + 2a2bโ2a4b โ 6a3b โ 2a3b2 โ 4a2b + 2ab2+2a5+2a4b
a + b 3 a + b + 1 a + b + 2
+4a4 + 2a4b + 2a3b2 + 4a3b + 2a4 + 2a3b + 4a3
a + b 3 a + b + 1 a + b + 2
=2ab2 โ 2a2b
a + b 3 a + b + 1 a + b + 2
=2ab(b โ a)
a + b 3 a + b + 1 a + b + 2 (๐ถ๐๐๐๐ ๐ก๐๐๐ ๐๐๐๐๐, 1996)_____________________(2.10)
The standardized skewness can be expressed as
๐ธ ๐ โ ๐ธ ๐
๐
3
39
=1
๐3
2ab b โ a
a + b 3 a + b + 1 a + b + 2
=1
ab
a + b + 1 a + b 2
3 2ab b โ a
a + b 3 a + b + 1 a + b + 2
= a + b + 1 a + b 2
๐๐. a + b + 1 a + b
ab
2ab b โ a
a + b 3 a + b + 1 a + b + 2
= a + b + 1
ab
2 b โ a
a + b + 2
=2 b โ a
a + b + 2
a + b + 1
ab______________(2.11)
2.3.6 Kurtosis
Kurtosis can be expressed as
Kurtosis = ๐ข4 = ๐ธ ๐4 โ 4๐ธ ๐ ๐ธ ๐3 + 6{๐ธ ๐ }2๐ธ ๐2 โ 3 ๐ธ ๐ 4
= m4 โ 4๐1๐3+6๐12๐2 โ 3๐1
4,
=a a + 1 a + 2 a + 3
a + b a + b + 1 a + b + 2 a + b + 3
โ4a a a + 1 a + 2
a + b a + b a + b + 1 a + b + 2
+6 a
a + b
2 a(a + 1)
a + b (a + b + 1)โ 3
a
a + b
4
= a2 + a a2 + 5a + 6
a + b a + b + 1 a + b + 2 a + b + 3
โ4a2 a2 + 3a + 2
(a + b)2 a + b + 1 a + b + 2
40
+6a4 + 6a3
(a + b)3(a + b + 1)โ 3
a
a + b
4
= a + b a4 + 5a3 + 6a2 + a3 + 5a2 + 6a โ 4a4 + 12a3 + 8a2 (a + b + 3)
(a + b)2 a + b + 1 a + b + 2 a + b + 3
+6 a + b (a4 + a3) โ 3a4(a + b + 1)
(a + b)4(a + b + 1)
=
a5 + 5a4 + 6a3 + a4 + 5a3 + 6a2 + a4b + 5a3b + 6a2b + a3b + 5a2b + 6ab โ
(4a5 + 4a4b + 12a4 + 12a4 + 12a3b + 36a3 + 8a3 + 8a2b + 24a2)
(a + b)2 a + b + 1 a + b + 2 a + b + 3
+(6a5 + 6a4 + 6a4 + 6a3b) โ 3a5 โ 3a4b โ 3a4)
(a + b)4(a + b + 1)
=โ3a5 โ 18a4 โ 33a3 โ 18a2 โ 3a4b โ 6a3b โ 6a3b + 3a2b + 6ab
(a + b)2 a + b + 1 a + b + 2 a + b + 3
+3a5 + 3a4 + 3a4q + 6a3b
(a + b)4(a + b + 1)
= a2 + 2ab + b2 โ3a5 โ 18a4 โ 33a3 โ 18a2 โ 3a4b โ 6a3b โ 6a3b + 3a2b + 6ab
a + b 4 a + b + 1 a + b + 2 a + b + 3
+ a2 + b2 + 2ab + 5a + 5b + 6 3a5 + 3a4 + 3a4b + 6a3b
a + b 4 a + b + 1 a + b + 2 a + b + 3
=
โ3a7 โ 18a6 โ 33a5 โ 18a4 โ 3a6b โ 6a5b + 3a4b โ 6a6b + 36a5b โ 66a4b
โ36a3b โ 6a5b2
a + b 4 a + b + 1 a + b + 2 a + b + 3
+ โ12a4b2 + 6a3b2 + 12a2b2 โ 3a5b2 โ 18a4b2 โ 33a3a2 โ 18a2b2 โ 3a4b3
โ6a3b3 + 3a2b3
a + b 4 a + b + 1 a + b + 2 a + b + 3
+ 6ab3 + 3a7 + 3a6 + 3a6b + 6a5b + 3a5b2 + 3a4b2 + 3a4b3 + 6a3b3 + 6a6b
+6a5b + 6a5b2
a + b 4 a + b + 1 a + b + 2 a + b + 3
41
+ 12a4b2 + 15a6 + 15a5 + 15a5b + 30a4b + 15a5b + 15a4b2 + 30a3b2 + 18a5
+18a4 + 18a4b + 36a3b
a + b 4 a + b + 1 a + b + 2 a + b + 3
=6a3b + 3a3b2 โ 6a2b2 + 3a2b3 + 6ab3
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3๐๐[2a2 + a2b โ 2ab + ab2 + 2b2]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)_____________(2.12)
The standardized kurtosis can be expressed as
๐ธ ๐ โ ๐ธ ๐
๐
4
=1
๐4
3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=1
ab
a + b + 1 a + b 2 2
3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
= a + b + 1 2 a + b 4
(๐๐)2
3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
= a + b + 1
๐๐
3[a2(b + 2) โ 2ab + b2(a + 2)]
a + b + 2 a + b + 3 ____________________________(2.13)
The standardized skewness and kurtosis are often denoted by ๐ฝ1 , ๐ฝ2 in literature.
2.4 Shapes of Classical Beta Distribution
The shape of the distribution is governed by the two parameters a and b, and can be
summarized as follows;
i. When a=b, the distribution is symmetrical about ยฝ, the case of arcsine distribution,
uniform distribution and parabolic distribution, and the skewness is zero, as a and b
become larger, the distribution becomes more peaked and the variance decreases
42
ii. When ๐ โ ๐, the distribution is skewed with the degree of skewness increasing as ๐
and ๐ become more unequal. In particular, if ๐ > ๐, then the skewness, ๐ข3 > 0 and
the distribution is skewed to the right as in figure 2.2 and if ๐ > ๐, then the
distribution is skewed to the left as in figure 2.1.
iii. When a=b=1, the distribution becomes uniform distribution and the density function
has a constant value 1 over the interval (0,1) for the random variable X.
iv. Figures 2.2 and 2.3 below shows two cases of the beta unimodal distributions with
๐ > 0 and ๐ > 0. If a and/or b is less than one i.e ๐ โ 1 ๐ โ 1 < 1
๐ 0 โ โ and /or f(1) โ โ and the distribution is J- shaped. Figure 2.4 and 2.5
below show the beta distribution of two cases a=10, b=1 and a=1, b=10 respectively.
v. The Classical beta distribution is U shaped if a < 1, and b < 1 as shown in figure 2.1
below.
Fig.2. 1 : Left Skewedness
0
1
2
3
4
5
6
0
0.0
4
0.0
8
0.1
2
0.1
6
0.2
0.2
4
0.2
8
0.3
2
0.3
6
0.4
0.4
4
0.4
8
0.5
2
0.5
6
0.6
0.6
4
0.6
8
0.7
2
0.76 0.
8
0.8
4
0.8
8
0.9
2
0.9
6 1
Beta(0.66,0.5)
43
Fig.2. 2: Case of beta unimodal distribution with ๐ > 0
Fig.2. 3: Case of beta unimodal distribution with ๐ > 0
0
2
4
6
8
10
12
14
00.
035
0.07
0.10
50.
140.
175
0.21
0.24
50.
280.
315
0.35
0.38
50.
420.
455
0.49
0.52
50.
560.
595
0.63
0.66
50.
70.
735
0.77
0.80
50.
840.
875
0.91
0.94
50.
98
Beta(2,30)
0
2
4
6
8
10
12
14
00.
035
0.0
70.
105
0.1
40.
175
0.2
10.
245
0.2
80.
315
0.3
50.
385
0.4
20.
455
0.4
90.
525
0.5
60.
595
0.6
30.
665
0.7
0.73
50
.77
0.80
50
.84
0.87
50
.91
0.94
50
.98
Beta(30,2)
44
Fig.2. 4: j-Shaped beta distribution
Fig.2. 5: inverted j-Shaped beta distribution
0
2
4
6
8
10
12
0
0.04
0.08
0.12
0.16 0.
2
0.24
0.28
0.32
0.36 0.
4
0.44
0.48
0.52
0.56 0.
6
0.64
0.68
0.72
0.76 0.
8
0.84
0.88
0.92
0.96 1
Beta(10,1)
0
2
4
6
8
10
12
0
0.04
0.08
0.12
0.16 0.2
0.24
0.28
0.32
0.36 0.4
0.44
0.48
0.52
0.56 0.
6
0.64
0.68
0.72
0.76 0.8
0.84
0.88
0.92
0.96
1
Beta(1,10)
45
2.5 Applications of the Classical Beta Distribution
Income and Wealth
Thurow (1970) applied the Classical beta distribution to US Census Bureau data of
income distribution for households (families and unrelated individuals) for every year
from 1949 โ 1966, stratified by race. He also studied the impact of various
macroeconomic factors on the parameters of the distribution via regression techniques.
McDonald and Ransom (1979a) employed the Classical beta distribution for
approximating US family incomes for 1960 and 1969 through 1975. When utilizing three
different estimators, it turns out that the distribution is preferable to the gamma and
lognormal distributions but inferior to Singh Maddala. McDonald (1984) estimated beta
distribution of the first kind for 1970, 1975 and 1980 US family incomes.
2.6 Special Cases of the Classical Beta Distribution
2.6.1 Power Distribution Function
This distribution is obtained when ๐ = 1 in the formula of the classical beta distribution.
Thus, from equation (2.1) the pdf becomes
๐ ๐ฅ; ๐ =๐ฅ๐โ1
๐ต(๐, 1)
=๐ฅ๐โ1
ฮ(๐)ฮ(1). ฮ(๐ + 1)
=๐ฅ๐โ1
ฮ(๐)ฮ(1). ๐ฮ(๐)
= ๐๐ฅ๐โ1, ๐ > 0 ________(2.14)
and the CDF is given by
๐น ๐ฅ = ๐(๐ฅ)โ
0
๐๐ฅ
46
= ๐๐ฅ๐โ1โ
0
๐๐ฅ
= ๐๐ฅ๐โ1+1
๐
= ๐ฅ๐ , 0 < ๐ < 1
Equation (2.14) is the power distribution.
Generally letting ๐ฅ = ๐ฆ/๐ผ, with b=1 in equation (2.1) yield the general power
distribution given by
๐ ๐ฆ =๐๐ฆ๐โ1
๐ผ๐, 0 < ๐ < ๐ผ
(Leemis and McQueston, 2008)
2.6.2 Properties of power distribution
i. rth-order moments
Putting b=1 in equation (2.4)
๐ธ ๐๐ =ฮ ๐ + ๐ ฮ ๐ + 1
ฮ a + 1 + r ฮ a
=ฮ(๐ + ๐)๐ฮ(๐)
(a + r)ฮ a + r ฮ(a)
=๐
a + r
ii. Mean
๐ธ ๐ =๐
a + 1
iii. Mode
๐๐๐๐ =๐ โ 1
a + 1 โ 2
=๐ โ 1
a โ 1= 1
47
iv. Variance
๐๐๐ ๐ =๐
(a + 2)(a + 1)2
v. Skewness
๐๐๐๐ค๐๐๐ ๐ =2๐(1 โ ๐)
(a + 1)3(a + 2)(a + 3)
While the standardized skewness is
2๐(1 โ ๐)
(a + 3)
(a + 2)
a
vi. Kurtosis
๐พ๐ข๐๐ก๐๐ ๐๐ =3๐[๐2 1 + 2 โ 2๐ + ๐ + 2 ]
a + 1 4 a + 1 + 1 a + 1 + 2 (a + 1 + 3)
=3๐[3๐2 โ ๐ + 2]
a + 1 4 a + 2 a + 3 (a + 4)
and the standardized kurtosis is expressed as
๐ + 2
๐
3 3๐2 โ ๐ + 2
a + 3 a + 4
2.6.3 Shape of Power distribution
The figure 2.6 below shows a graph of a power distribution with the parameter ๐ = 0.05.
This is an example of Power distribution graph, being used to demonstrate ranking of
popularity. To the right is the long tail, and to the left are the few that dominate. Varying
๐ gives different shapes of the power distribution. Special cases of ๐ = 1 (Uniform
distribution), ๐ =2 (triangular distribution), ๐ < 1 gives inverted J-shaped distributions
and ๐ > 2 (J-shaped distributions).
48
Fig.2. 6: Power distribution
2.6.4 Uniform Distribution Function
Special case when ๐=b=1 in equation (2.1)
๐ ๐ฅ = 1 _______________________(2.15)
Equation (2.15) is a Uniform distribution on the interval [0,1]. The Uniform distribution
function is a special case of the power distribution function when a=1.
2.6.5 Properties of the Uniform distribution
i. rth-order moments
Putting ๐ = 1 and ๐ = 1 in equation 2.4
๐ธ ๐ฅ๐ =ฮ 1 + ๐ ฮ 1 + 1
ฮ 1 + 1 + r ฮ 1
=rฮ ๐ . 1
1 + r r ฮ r . 1
=1
1 + r
0
1
2
3
4
5
6
7
8
9
0
0.0
4
0.0
8
0.1
2
0.1
6
0.2
0.2
4
0.2
8
0.3
2
0.3
6
0.4
0.4
4
0.4
8
0.5
2
0.5
6
0.6
0.6
4
0.6
8
0.7
2
0.7
6
0.8
0.8
4
0.8
8
0.9
2
0.9
6 1
Power Distribution
49
ii. Mean
mean =๐
๐ + ๐,
Letting a=b=1,
=1
1 + 1= 0.5
iii. Mode
Mode is any value in the interval [0,1]
iv. Variance
From the variance of the standard beta distribution given by
Variance =๐๐
(๐ + ๐ + 1)(๐ + ๐)2
a=b=1,
=1
3 2 2
=1
12= 0.0833
v. Skewness
Skewness =2๐๐(b โ a)
a + b 3 a + b + 1 a + b + 2 = 0
Standardized skewness =2(1 โ 1)
(1 + 3)
(1 + 2)
1= 0
vi. Kurtosis
kurtosis =3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
50
=3 3 โ 2 + 3
2 4 3 4 5
=1
16 5 = 0.0125
The standardized kurtosis is given by
๐ + 2
๐
3 3๐2 โ ๐ + 2
a + 3 a + 4
=1 + 2
1
3 3 โ 1 + 2
1 + 3 1 + 4
= 3 3 4
4 5 = 1.8
2.6.6 Shape of Uniform distribution
Figure 2.7 below shows a graphical representation of the Uniform distribution on the
interval (0,1)
Fig.2. 7: Uniform distribution on the interval (0,1)
0
0.2
0.4
0.6
0.8
1
1.2
0
0.0
45
0.0
9
0.1
35
0.1
8
0.2
25
0.27
0.3
15
0.3
6
0.4
05
0.4
5
0.4
95
0.5
4
0.5
85
0.6
3
0.67
5
0.7
2
0.7
65
0.8
1
0.8
55 0.9
0.9
45
0.9
9
Beta(1,1)
51
2.6.7 Arcsine Distribution Function
Special case of the standard beta distribution is the arcsine distribution, when a=b=1/2 in
equation (2.1). In this special case, we have
๐ ๐ฅ;1
2,1
2 =
๐ฅ1/2โ1 1 โ x 1/2โ1
๐ต 12 ,
12
, 0 < ๐ฅ < 1
๐ ๐ฅ =๐ฅโ1/2 1 โ x โ1/2
๐ต 12 ,
12
, 0 < ๐ฅ < 1
๐ ๐ฅ =1
๐ ๐ฅ(1 โ ๐ฅ) , 0 < ๐ฅ < 1 ________(2.16)
and the cumulative distribution function is given by
F x =2
๐arcsin ๐ฅ .
Note: ฮ(1/2) = ๐
Equation (2.16) is an Arcsine distribution
2.6.8 Properties of the arcsine distribution
i. rth-order moments
Putting ๐ =1
2 and ๐ =
1
2 in equation 2.4
๐ธ ๐๐ =ฮ
12 + ๐ ฮ 1
ฮ 1 + r ฮ 1
=ฮ
12 + ๐
rฮ r ฮ 12
52
ii. Mean
From the mean of the beta distribution given by
mean =๐
๐ + ๐,
a = b =1
2
=
12
12 +
12
=1
2= 0.5
iii. Mode
mode =๐ โ 1
๐ + ๐ โ 2=
12 โ 1
12 +
12 โ 2
=
12 โ 1
12 +
12 โ 2
=1
2
iv. Variance
From the variance of the beta distribution given by
Variance =๐๐
(๐ + ๐ + 1)(๐ + ๐)2
a = b =1
2
=
14
2 1 2= 0.125
53
v. Skewness
Skewness =2๐๐(b โ a)
a + b 3 a + b + 1 a + b + 2 = 0
Standardized skewness =2(b โ a)
a + b + 2
๐ + ๐ + 1
๐๐
=2(
12 โ
12)
1 + 2 1 + 1
14
= 0
vi. Kurtosis
kurtosis =3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3
14 [
14 (
12 + 2) โ 2(
14) +
14 (
12 + 2)]
1 4 1 + 1 1 + 2 1 + 3
=
34 [(
58) โ (
48) + (
58)]
2 3 4
=
1832
24 = 0.0234
Standardized kurtosis = 1 + 1
14
3
14
12 + 2 โ 2
14 +
14
12 + 2
1 + 2 1 + 3 = 1.5
54
2.6.9 Shape of Arcsine distribution
Fig.2. 8: Arcsine distribution
2.6.10 Triangular shaped distributions (a=1, b=2)
For a=1 and b=2 we get triangular shaped distribution as follows
f x; 1,2 = 1 โ x
B 1,2
= 2 โ 2๐ฅ ________________________(2.17)
With the cdf given as
๐น ๐ฅ = 2๐ฅ โ ๐ฅ2
2.6.11 Properties of the Triangular shaped distribution (a=1, b=2)
i. rth-order moments
Putting ๐ = 1 and ๐ = 2 in equation 2.4
๐ธ ๐๐ =ฮ 1 + ๐ ฮ 3
ฮ 3 + r ฮ 1
=2rฮ ๐
ฮ r + 3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.0
05
0.0
45
0.0
85
0.1
25
0.1
65
0.2
05
0.2
45
0.2
85
0.3
25
0.3
65
0.4
05
0.4
45
0.4
85
0.5
25
0.5
65
0.6
05
0.6
45
0.6
85
0.7
25
0.7
65
0.8
05
0.8
45
0.8
85
0.9
25
0.9
65
Arcsine Distribution
55
ii. Mean
From the mean of the standard beta distribution given by
mean =๐
๐ + ๐,
a = 1, b = 2
=1
3= 0.3333
iii. Mode
mode =๐ โ 1
๐ + ๐ โ 2=
1 โ 1
1 + 2 โ 2= 0
iv. Variance
From the variance of the standard beta distribution given by
Variance =๐๐
(๐ + ๐ + 1)(๐ + ๐)2
a = 1, b = 2
=2
4 3 2
=1
18= 0.0556
Standard deviation, ๐ = ๐ฃ๐๐ = 0.2358
๐ถ๐ =๐
๐=
0.2358
0.3333= 0.7074
v. Skewness
Skewness =2๐๐(b โ a)
a + b 3 a + b + 1 a + b + 2
=2(2)(1)
3 3 4 5
=1
3 3 5 = 0.0074
56
Standardized skewness =2(b โ a)
a + b + 2
๐ + ๐ + 1
๐๐
a = 1, b = 2
=2
5 2 = 0.5657
vi. Kurtosis
kurtosis =3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3 2 [1(4) โ 4 + 4(3)]
3 4 4 5 6
=1
3 3 5 = 0.0074
Standardized kurtosis = 3 + 1
2
3 1 2 + 2 โ 2 1 2 + 4 1 + 2
3 + 2 3 + 3 = 2.4
57
2.6.12 Shape of triangular distribution (a=1, b=2)
Fig.2. 9: Triangular shaped distribution
2.6.13 Triangular shaped distributions (a=2, b=1)
For a=2 and b=1 we get triangular shaped distribution as follows
๐ ๐ฅ; 2,1 =๐ฅ
๐ต(2,1)
= 2๐ฅ____________________________(2.18)
The CDF of equation 2.18 is given by
๐น(๐ฅ) = ๐ฅ2
2.6.14 Properties of the Triangular shaped distribution (a=2, b=1)
i. rth - order moments
Putting ๐ = 2 and ๐ = 1 in equation 2.4
๐ธ ๐๐ =ฮ 2 + ๐ ฮ 3
ฮ 3 + r ฮ 2
0
0.5
1
1.5
2
2.5
0
0.04
5
0.0
9
0.13
5
0.1
8
0.22
5
0.2
7
0.31
5
0.3
6
0.40
5
0.4
5
0.49
5
0.5
4
0.58
5
0.6
3
0.67
5
0.7
2
0.76
5
0.8
1
0.85
5
0.9
0.94
5
0.9
9
Beta(1,2)
58
=2ฮ 2 + ๐
ฮ r + 3
ii. Mean
From the mean of the beta distribution given by
mean =๐
๐ + ๐,
a=2, b=1,
=2
3= 0.6667
iii. Mode
mode =๐ โ 1
๐ + ๐ โ 2=
2 โ 1
2 + 1 โ 2= 1
iv. Variance
From the variance of the beta distribution given by
Variance =๐๐
(๐ + ๐ + 1)(๐ + ๐)2
a=2, b=1,
=2
4 3 2
=1
18= 0.0556
v. Skewness
Skewness =2๐๐(b โ a)
a + b 3 a + b + 1 a + b + 2
=2(2)(โ1)
3 3 4 5
=โ1
3 3 5 = โ0.0074
59
Standardized skewness =2(b โ a)
a + b + 2
๐ + ๐ + 1
๐๐
a=2, b=1
=โ2
5 2 = โ0.5657
vi. Kurtosis
kurtosis =3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3 2 [4(3) โ 4 + 1(4)]
3 4 4 5 6
=1
3 3 5 = 0.0074
Standardized kurtosis = 3 + 1
2
3 4 1 + 2 โ 2 2 1 + 1 2 + 2
3 + 2 3 + 3 = 2.4
2.6.15 Shape of triangular distribution (a=2, b=1)
Fig.2. 10: Trangular shaped distribution
0
0.5
1
1.5
2
2.5
0
0.0
4
0.0
85
0.1
3
0.1
75
0.2
2
0.2
65
0.31
0.35
5
0.4
0.4
45
0.4
9
0.5
35
0.5
8
0.6
25
0.6
7
0.7
15
0.7
6
0.8
05
0.8
5
0.8
95
0.9
4
0.9
85
Beta(2,1)
60
2.6.16 Parabolic shaped distribution
For a=b=2 we obtain a distribution of parabolic shape,
๐ ๐ฅ; 2,2 =๐ฅ(1 โ ๐ฅ)
๐ต(2,2)
= 6๐ฅ(1 โ ๐ฅ)____________________(2.19)
The cdf of equation 2.19 is given by
๐น(๐ฅ) = 3๐ฅ2 โ 2๐ฅ3
2.6.17 Properties of parabolic shaped distribution
i. rth-order moments
Putting ๐ = 2 and ๐ = 2 in equation 2.4
๐ธ ๐๐ =ฮ 2 + ๐ ฮ 4
ฮ 4 + r ฮ 2
=6
r + 3 (r + 2)
ii. Mean
From the mean of the Classical beta distribution given by
E(X) =๐
๐ + ๐,
a = b = 2,
=2
2 + 2= 0.5
iii. Mode
From the mode of the Classical beta distribution given by
Mode =๐ โ 1
(๐ + ๐ โ 2)
a = b = 2, = 0.5
61
iv. Variance
From the variance of Classical beta distribution given by
Variance =๐๐
(๐ + ๐ + 1)(๐ + ๐)2
a = b = 2,
=4
5 4 2
=1
20= 0.05
standard deviation,๐ = 0.05 = 0.2236
๐ถ๐ =0.2236
0.5= 0.4472
v. Skewness
From the skewness of the Classical beta distribution given by
Skewness =2๐๐(๐ โ ๐)
๐ + ๐ ๐ + ๐ + 1 (๐ + ๐ + 2)
a=b=2,
= 0
Standardized skewness =2(b โ a)
a + b + 2
๐ + ๐ + 1
๐๐= 0
vi. Kurtosis
kurtosis =3๐๐[a2(b + 2) โ 2ab + b2(a + 2)]
a + b 4 a + b + 1 a + b + 2 a + b + 3
=3 4 [4(4) โ 8 + 4(4)]
4 4 5 6 7
=3
4 2 5 7 = 0.0054
62
Standardized kurtosis = 4 + 1
4
3 4 2 + 2 โ 2 2 2 + 4 2 + 2
4 + 2 4 + 3 = 2.1429
2.6.18 Shape of Parabolic distribution
The figure 2.11 below shows the shape of the parabolic distribution.
Fig.2. 11: Parabolic distribution
2.7 Transformation of Special Case of the Classical Beta Distribution
2.7.1 Wigner Semicircle Distribution Function
Letting a=b=3/2 in equation (2.1) we obtain the following distribution function
๐ ๐ฅ;3
2,3
2 =
๐ฅ3/2โ1 1 โ ๐ฅ 3/2โ1
๐ต 32 ,
32
, 0 < ๐ฅ < 1
๐ ๐ฅ =2. ๐ฅ1/2 1 โ ๐ฅ 1/2
12 ๐
2 , 0 < ๐ฅ < 1
๐ ๐ฅ =8 ๐ฅ(1 โ ๐ฅ)
๐
Let
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.60
0.04
0.08
5
0.13
0.17
5
0.22
0.26
5
0.31
0.35
5
0.4
0.44
5
0.49
0.53
5
0.58
0.62
5
0.67
0.71
5
0.76
0.80
5
0.85
0.89
5
0.94
0.98
5
Beta(2,2)
63
๐ฅ =๐ฆ + ๐
2๐ , |๐ฝ| =
1
2๐
Limits changes as ๐ฅ = 0 โ ๐ฆ > โ๐ and ๐ฅ = 1 โ ๐ฆ = ๐
๐ ๐ฆ =8 ๐ฆ + ๐
2๐ 1 โ๐ฆ + ๐
2๐
๐.
1
2๐
=4 ๐ฆ + ๐
2๐ โ ๐ฆ + ๐
2๐ 2
๐ ๐
=4 2๐ ๐ฆ + 2๐ 2 โ ๐ฆ2 โ 2๐ฆ๐ โ ๐ 2
4๐ 2
๐ ๐
=2 ๐ 2 โ ๐ฆ2
๐ 2๐, โ๐ < ๐ฆ < ๐ ________(2.20)
Equation (2.20) is the Wigner semicircle distribution.
2.7.2 Properties of Wigner Semicircle
i. rth-order moments
The rth-order moments for positive integers r, the 2r
thmoments are given by
๐ธ ๐2๐ = ๐
2
2๐
๐ถ๐
Where ๐ถ๐ is the rth catalan number given by ๐ถ๐ =
1
๐+1 2๐
๐ so that the moments are the
catalan numbers if R=2. The odd numbers are zero.
ii. Mean
Mean of Wigner semi-circle, the first order moment, odd moment is zero.
iii. Mode
Mode of Wigner semi-circle is given by
๐
๐๐ฆ๐(๐ฆ) = 0
2
๐ 2๐
๐
๐๐ฆ ๐ 2 โ ๐ฆ2 = 0
๐ฆ = 0
64
iv. Variance
The variance of Wigner semi-circle is given by
Var Y = ๐ธ ๐2 โ ๐ธ(๐)
= ๐
2
2 1
2. 2
=๐ 2
4
v. Skewness
The Skewness of Wigner semi-circle is zero, since it is the third order moment which is
an odd moment.
vi. Kurtosis
The Kurtosis of Wigner semi-circle is given by
Kurtosis = ๐ธ ๐4 โ 4๐ธ ๐ ๐ธ ๐3 + 6 ๐ธ ๐ 2๐ธ ๐2 โ 3 ๐ธ ๐ 4
= ๐ธ ๐4
= ๐
2
2.2 1
2 + 1
2.2
2
=๐ 4
8
Standardized Kurtosis =๐ 4
8.
4
๐ 2 2
= 2
65
2.7.3 Shape of Wigner semicircle distribution
The figure 2.12 below shows the shape of the Wigner semicircle distribution.
Fig.2. 12: Wigner semicircle distribution
0
0.2
0.4
0.6
0.8
1
1.2
1.40
0.04
0.08
5
0.13
0.17
5
0.22
0.26
5
0.31
0.35
5
0.4
0.44
5
0.49
0.53
5
0.58
0.62
5
0.67
0.71
5
0.76
0.80
5
0.85
0.89
5
0.94
0.98
5
Beta(1.5,1.5)
3 Chapter III: Type II Two Parameter Inverted
Beta Distribution
3.1 Construction of Beta Distribution of the Second Kind
The beta distribution of the second kind is derived by using the transformation
๐1: ๐ฅ =y
1 + y
in equation 2.1 to give a new distribution as follows:
๐(๐ฆ; ๐, ๐) = ๐(๐ฅ; ๐, ๐)|๐ฝ|
The limits changes as follows ๐ฅ + ๐ฅ๐ฆ = ๐ฆ โน ๐ฆ 1 โ ๐ฅ = ๐ฅ, ๐ฆ =๐ฅ
1โ๐ฅ
when x = 0, y = 0 and when x = 1, y = โ โน 0 < ๐ฆ < โ
Where
|๐ฝ| =1
(1 + y)2
Therefore
๐ ๐ฆ; ๐, ๐ =๐ฅ๐โ1 1 โ ๐ฅ ๐โ1
๐ต ๐, ๐ . |๐ฝ|
=(
๐ฆ1 + ๐ฆ)๐โ1 1 โ
๐ฆ1 + ๐ฆ
๐โ1
๐ต ๐, ๐ .
1
(1 + y)2
=๐ฆ๐โ1
๐ต ๐, ๐ (1 + y)a+b , 0 < ๐ฆ < โ, ๐, ๐ > 0_______(3.1)
67
Equation (3.1) is a beta distribution of the second kind. It is also known as the standard
form of a Pearson Type VI distribution, sometimes called beta prime distribution or
Inverted beta distribution.
Another method of deriving beta distribution of the second kind is by using the
transformation
๐ฅ =1
1 + y in equation (2.1)
๐๐ฅ
๐๐ฆ=
โ1
1 + y 2
The limits changes as follows
๐ฅ + ๐ฅ๐ฆ = 1 โน ๐ฆ =1 โ x
x, when x = 0, y = โ and when x = 1, y = 0 โน 0 < ๐ฆ < โ
๐(๐ฆ; ๐, ๐) = ๐(๐ฅ; ๐, ๐)|๐ฝ|
=
1
1 + ๐ฆ ๐โ1
1 โ 1
1 + ๐ฆ
๐โ1
๐ต ๐, ๐ .
1
1 + y 2
=(
11 + ๐ฆ)๐โ1
๐ฆ1 + ๐ฆ
๐โ1
๐ต ๐, ๐ .
1
1 โ y 2
=๐ฆ๐โ1
๐ต ๐, ๐ 1 + ๐ฆ a+b
3.2 Properties of Beta Distribution of the Second Kind
3.2.1 rth-order moments
To obtain rth-order moments of the beta distribution of the second kind, we evaluate
๐ธ ๐๐ = ๐ฆ๐๐ ๐ฆ; ๐, ๐ โ
0
๐๐ฆ
68
= ๐ฆ๐ ๐ฆ๐โ1
๐ต(๐, ๐)(1 + y)a+b
โ
0
๐๐ฆ
=1
๐ต(๐, ๐)
๐ฆ๐+๐โ1
(1 + y)a+b
โ
0
๐๐ฆ
To integrate the integral in above expression, substitute
๐ฆ =๐ฅ
1 โ ๐ฅ
๐๐ฆ
๐๐ฅ =
1
(1 โ ๐ฅ)2
๐ฆ 1 โ ๐ฅ = ๐ฅ โน 1 + ๐ฅ =1
(1โ๐ฆ) , and limits of integration become 0 to 1. Then
substituting these terms in the above integral, we have
๐ฆ๐+๐โ1
(1 + y)a+b
โ
0
๐๐ฆ = (๐ฅ
1 โ ๐ฅ)๐+๐โ1 1 โ ๐ฅ ๐+๐(1 โ ๐ฅ)โ2
1
0
๐๐ฅ
= ๐ฅ๐+๐โ1(1 โ ๐ฅ)โ๐โ๐+1+๐+๐โ2
1
0
๐๐ฅ
= ๐ฅ(๐+๐)โ1(1 โ ๐ฅ)(๐โ๐)โ1
1
0
๐๐ฅ
= ๐ต ๐ + ๐, ๐ โ ๐ , ๐ < ๐
Consequently, the moments of beta distribution of the second kind are given by
๐ธ ๐๐ =๐ต ๐ + ๐, ๐ โ ๐
๐ต ๐, ๐
=ฮ ๐ + ๐ ฮ ๐ โ ๐
ฮ ๐ ฮ ๐ ๐ < ๐ ___________________(3.2)
Using recursive relationship ฮ a + 1 = aฮ(a), the first four moments about zero can be
written as follows
69
3.2.2 Mean
The mean can be obtained by substituting r=1 in equation 3.2
๐1 โก ฮผ = E Y =๐ต ๐ + ๐, ๐ โ ๐
๐ต ๐, ๐
=ฮ a + 1 ฮ(b โ 1)
ฮ(a + b).ฮ a + b
ฮ(a)ฮ(b)
=aฮ a ฮ b โ 1
ฮ a + b .ฮ a + b
ฮ(a)ฮ(b)
=aฮ b โ 1
ฮ(b)
=a b โ 2 !
๐ โ 1 !
=๐ ๐ โ 2 !
๐ โ 1 ๐ โ 2 !
=๐
๐ โ 1 , ๐ > 1(๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)_______(3.3)
If b โค 1, the mean is infinite i. e undefined
๐2 โก E Y2 =๐ต ๐ + 2, ๐ โ 2
๐ต ๐, ๐
=ฮ a + 2 ฮ(b โ 2)
ฮ(a + b).ฮ a + b
ฮ(a)ฮ(b)
=(a + 1)aฮ a ฮ b โ 2
ฮ(a)ฮ(b)
=a(a + 1) b โ 3 !
b โ 1 b โ 2 (b โ 3)!
=a(a + 1)
๐ โ 1 ๐ โ 2
= ๐1
(a + 1)
๐ โ 2 , ๐ > 2 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)
70
๐3 โก E Y3 =๐ต ๐ + 3, ๐ โ 3
๐ต ๐, ๐
=ฮ a + 3 ฮ(b โ 3)
ฮ(a + b).ฮ a + b
ฮ(a)ฮ(b)
= a + 2 a + 1 aฮ a b โ 4 !
ฮ(a) b โ 1 b โ 2 b โ 3 (b โ 4)!
=a a + 1 (a + 2)
b โ 1 b โ 2 (b โ 3)
=๐2(a + 2)
(b โ 3) , ๐ > 3 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)
๐4 โก E Y4 =๐ต ๐ + 4, ๐ โ 4
๐ต ๐, ๐
=ฮ a + 4 ฮ(b โ 4)
ฮ(a + b).ฮ a + b
ฮ(a)ฮ(b)
=(a + 3)(a + 2)(a + 1)aฮ a b โ 5 !
ฮ a b โ 1 b โ 2 b โ 3 b โ 4 (b โ 5)!
=a a + 1 a + 2 (a + 3)
b โ 1 b โ 2 b โ 3 (b โ 4)
=๐3(a + 3)
(b โ 4) , ๐ > 4 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)
3.2.3 Mode
Mode is obtained by solving
๐
๐๐ฆ๐ ๐ฆ = 0
=๐
๐๐ฆ
๐ฆ๐โ1
๐ต ๐, ๐ 1 + ๐ฆ a+b = 0
71
=1
๐ต ๐, ๐ ๐ โ 1 ๐ฆ๐โ2 . 1 + ๐ฆ a+b โ a + b 1 + ๐ฆ a+bโ1 . ๐ฆ๐โ1
1 + ๐ฆ 2a+2b = 0
= ๐ โ 1 ๐ฆ๐โ2 . 1 + ๐ฆ a+b โ a + b 1 + ๐ฆ a+bโ1 . ๐ฆ๐โ1 = 0
๐ โ 1 ๐ฆ๐โ2. 1 + ๐ฆ a+b = a + b 1 + ๐ฆ a+bโ1 . ๐ฆ๐โ1
๐ โ 1 ๐ฆ๐โ1 . ๐ฆโ1 . 1 + ๐ฆ a+b = a + b 1 + ๐ฆ a+b 1 + ๐ฆ โ1. ๐ฆ๐โ1
๐ โ 1 ๐ฆโ1 = (a + b) 1 + ๐ฆ โ1
1 + ๐ฆ ๐ โ 1 = ๐ฆ(a + b)
๐ โ 1 + ๐ฆ๐ โ ๐ฆ = ๐ฆ๐ + yb
๐ฆ(๐ + 1) = ๐ โ 1
๐ฆ =๐ โ 1
๐ + 1, if b โฅ 1, 0 otherwise___________________(3.4)
3.2.4 Variance
Since variance is given by ๐ธ ๐2 โ ๐ธ ๐ 2, we have
๐๐๐(๐) = a(a + 1)
๐ โ 1 ๐ โ 2 โ
๐2
๐ โ 1 2
= ๐ โ 1 ๐2 + ๐ โ ๐2(๐ โ 2)
๐ โ 1 2(๐ โ 2)
=๐2๐ + ๐๐ โ ๐2 โ ๐ โ ๐2๐ + 2๐2
๐ โ 1 2(๐ โ 2)
=๐๐ + ๐2 โ ๐
๐ โ 1 2(๐ โ 2)
72
=๐(๐ + ๐ โ 1)
๐ โ 1 2(๐ โ 2), ๐ > 2 (๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)________(3.5)
3.2.5 Skewness
Skewness can be expressed as
Skewness = ๐ข3 = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
= m3 โ 3๐2๐1+2๐13,
=๐ ๐ + 1 (๐ + 2)
๐ โ 1 ๐ โ 2 (๐ โ 3)โ 3
a(a + 1)
๐ โ 1 ๐ โ 2
๐
๐ โ 1 + 2(
๐
๐ โ 1 )3
= ๐2 + ๐ (๐ + 2)
๐ โ 1 ๐ โ 2 (๐ โ 3)โ
3a3 + 3a2
๐ โ 1 2 ๐ โ 2 +
2a3
๐ โ 1 3
= ๐3 + 3๐2 + 2๐
๐ โ 1 ๐ โ 2 (๐ โ 3)โ
3a3 + 3a2
๐ โ 1 2 ๐ โ 2 +
2a3
๐ โ 1 3
= ๐ โ 1 2 ๐3 + 3๐2 + 2๐ โ ๐ โ 1 ๐ โ 3 3a3 + 3a2 + 2a3 ๐ โ 2 (๐ โ 3)
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
=(๐2 โ 2๐ + 1) ๐3 + 3๐2 + 2๐ โ (๐2 โ 4๐ + 3) 3a3 + 3a2 + 2a3(๐2 โ 5๐ + 6)
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
=
๐3๐2 + 3๐2๐2 + 2๐๐2 โ 2๐3๐ โ 6๐2๐ โ 4๐๐ + ๐3 + 3๐2 + 2๐ โ 3a3๐2 โ 3a2๐2 + 12a3b+12๐2๐ โ 9๐3 โ 9๐2 + 2p3๐2 โ 10a3๐ + 12a3
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
=2๐๐2 + 6๐2๐ โ 4๐๐ + 4๐3 โ 6๐2 + 2๐
๐ โ 1 3 ๐ โ 2 ๐ โ 3
=2๐[๐2 + 3๐๐ โ 2๐ + 2๐2 โ 3๐ + 1]
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
=2๐[2๐2 + 3๐๐ โ 3๐ + ๐2 โ 2๐ + 1]
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
73
=2๐[2๐2 + 3๐ ๐ โ 1 + (๐ โ 1)2]
๐ โ 1 3 ๐ โ 2 (๐ โ 3), ๐ > 3_____________(3.6)
(๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)
Since skewness is positive, it can be concluded that the beta distribution of the second
kind has usually along tail to the right.
3.2.6 Kurtosis
Kurtosis can be expressed as
Kurtosis = ๐ข4 = ๐ธ ๐4 โ 4๐ธ ๐ ๐ธ ๐3 + 6{๐ธ ๐ }2๐ธ ๐2 โ 3 ๐ธ ๐ 4
= m4 โ 4๐1๐3+6๐12๐2 โ 3๐1
4,
=a a + 1 a + 2 (a + 3)
b โ 1 b โ 2 b โ 3 (b โ 4)โ 4
๐
๐ โ 1
a a + 1 (a + 2)
b โ 1 b โ 2 (b โ 3)
+ 6 ๐
๐ โ 1
2 a(a + 1)
๐ โ 1 ๐ โ 2 โ 3(
๐
๐ โ 1 )4
=(a2 + a)(a2 + 5a + 6)
b โ 1 b โ 2 b โ 3 (b โ 4)โ
4(a3 + a2)(a + 2)
b โ 1 2 b โ 2 (b โ 3)+
6(a4 + a3)
b โ 1 3 ๐ โ 2
โ3a4
b โ 1 4
=(a4 + 5a3 + 6a2 + a3 + 5a2 + 6a)
b โ 1 b โ 2 b โ 3 (b โ 4)โ
(4a4 + 8a3 + 4a3 + 8a2)
b โ 1 2 b โ 2 (b โ 3)+
(6a4 + 6a3)
b โ 1 3 ๐ โ 2
โ3a4
b โ 1 4
= b โ 1 3 a4 + 5a3 + 6a2 + a3 + 5a2 + 6a โ b โ 1 2(b โ 4)(4a4 + 8a3 + 4a3 + 8a2)
b โ 1 4 b โ 2 b โ 3 (b โ 4)
+ b โ 1 b โ 3 b โ 4 6a4 + 6a3 โ b โ 1 b โ 2 b โ 3 (b โ 4)3a4
b โ 1 4 b โ 2 b โ 3 (b โ 4)
74
=(b โ 1) b2 โ 2b + 1 a4 + 6a3 + 11a2 + 6a โ b2 โ 2b + 1 (b โ 4)(4a4 + 12a3
b โ 1 4 b โ 2 b โ 3 (b โ 4)
+8a2) b2 โ 4b + 3 b โ 4 6a4 + 6a3 โ (b โ 2)(b2 โ 7b + 12)3a4
b โ 1 4 b โ 2 b โ 3 b โ 4
=(b3 โ 3b2 + 3b โ 1) a4 + 6a3 + 11a2 + 6a โ b3 โ 6b2 + 9b โ 4 (4a4 + 12a3
b โ 1 4 b โ 2 b โ 3 (b โ 4)
+8a2) b3 โ 8b2 + 19b โ 12 6a4 + 6a3 โ (b โ 2)(3a4b2 โ 21a4b + 36a4)
b โ 1 4 b โ 2 b โ 3 b โ 4
= [(a4b3 + 6a3b3 + 11a2b3 + 6ab3)+(โ3a4b2โ18a3b2โ33a2b2 โ 18ab2)+(3a4b +
18a3b + 33a2b + 18ab ) + โa4 โ 6a3 โ 11a2 โ 6a + โ4a4b3 โ 12a3b3 โ 8a2b3 +
(24a4b2 + 72a3b2 + 48a2b2)+(โ36a4b โ 108a3b โ 72a2b) + 16a4 + 48a3 +
32a2+6a4b3โ48a4b2+114a4bโ72a4+6a3b3โ48a3b2+114a3bโ72a3+โ3a4b3+27
a4b2โ78a4b+72a4]/bโ14bโ2bโ3bโ4
=
3a2b3 + 6ab3+6a3b2+15a2b2 โ 18ab2+3a4b + 24a3b โ 39a2b + 18ab + 15a4 โ 30a3
+21a2 โ 6a b โ 1 4 b โ 2 b โ 3 b โ 4
=
3a(ab3 + 2b3+2a2b2 + 5ab2 โ 6b2+a3b + 8a2b โ 13ab + 6b + 5a3 โ 10a2
+7a โ 2)
b โ 1 4 b โ 2 b โ 3 b โ 4 _______(3.7)
(๐ถ๐๐ก๐๐๐๐๐๐ ๐๐ก ๐๐. , 2010)
75
3.2.7 Shapes of Beta distribution of the second kind
Fig.3. 1: Beta distribution of the second kind
3.3 Application of Beta Distribution of the Second Kind
Vartia and Vartia (1980) fitted a four parameter shifted beta distribution of the second
kind under the name of scaled and shifted F distribution to the 1967 distribution of taxed
income in Finland. Using maximum likelihood and method of momentโs estimators, they
found that their model fits systematically better than the two- and three-parameter
lognormal distributions. McDonald (1984) estimated the beta distribution of the second
kind and both beta of the first and second kind distributions for 1970, 1975 and 1980 US
family incomes.
3.4 Special Cases of Beta Distribution of the Second Kind
3.4.1 Lomax (Pareto II) distribution function
Special case when a=1 in equation (3.1) gives the following density function
๐ ๐ฅ; ๐ =1
๐ต 1, ๐ 1 + x 1+b , 0 < ๐ฅ < โ, ๐ > 0
0
2
4
6
8
10
12
14
16
0
0.05
5
0.11
0.16
5
0.22
0.27
5
0.33
0.38
5
0.44
0.49
5
0.55
0.60
5
0.66
0.71
5
0.77
0.82
5
0.88
0.93
5
0.99
B(0.5,0.5)
B(1,1)
B(1,2)
B(2,1)
B(2,2)
B(10,1)
B(1,10)
B(2,30)
76
๐ต 1, ๐ =1
๐
=๐
(1 + x)1+b, ๐ฅ > 0 , ๐ > 0___________________(3.8)
Equation (3.8) is a Lomax distribution.
3.4.2 Properties of Lomax distribution
i. rth-order moments
Putting ๐ = 1 in equation 3.2
๐ธ ๐๐ =ฮ 1 + ๐ ฮ ๐ โ ๐
ฮ 1 ฮ ๐
=ฮ 1 + ๐ ฮ ๐ โ ๐
ฮ ๐
ii. Mean
E(X) =ฮ 1 + 1 ฮ ๐ โ 1
ฮ ๐
= b โ 2 !
b โ 1 b โ 2 !
=1
๐ โ 1
iii. Mode
Mode =๐ โ 1
2
iv. Variance
๐๐๐(๐) =๐
(๐ โ 2)(๐ โ 1)2
77
v. Skewness
Skewness =2 2 + 3 ๐ โ 1 + ๐ โ 1 2
๐ โ 1 3 ๐ โ 2 ๐ โ 3
=2๐ ๐ + 1
๐ โ 1 3 ๐ โ 2 ๐ โ 3
vi. Kurtosis
kurtosis =3(3b3 + b2 + 2b)
b โ 1 4 b โ 2 b โ 3 b โ 4
3.4.3 Shape of Lomax distribution
The figure 3.2 below shows the shape of the Lomax distribution with parameter b=10.
The graph is equivalent to the graph of beta distribution of the second kind with
parameters a=1 and b=10.
Fig.3. 2: Lomax distribution
3.4.4 Log-Logistic (Fisk) distribution function
Special case of beta distribution of the second kind when a=b=1 in equation (3.1)
๐ ๐ฅ =1
(1 + ๐ฅ)2 , ๐ฅ > 0 _____________________(3.9)
Equation 3.9 is the log-logistic distribution, also known as Fisk distribution.
0
1
2
3
4
5
6
7
8
9
10
00.
035
0.0
70.
105
0.1
40.
175
0.2
10.
245
0.2
80.
315
0.3
50.
385
0.4
20.
455
0.4
90.
525
0.5
60.
595
0.6
30.
665
0.7
0.73
50
.77
0.80
50
.84
0.87
50
.91
0.94
50
.98
Lomax distribution
78
3.4.5 Properties of Log-logistic distribution
i. rth-order moments
Putting ๐ = 1 and ๐ = 1 in equation 3.2
๐ธ ๐๐ =ฮ 1 + ๐ ฮ 1 โ ๐
ฮ 1 ฮ 1
= ฮ 1 + ๐ ฮ 1 โ ๐
=rฯ
sin rฯ
ii. Mean
E(X) = ฮ 1 + ๐ ฮ 1 โ ๐
= ฮ 1 + 1 ฮ โ1 + 1
=ฯ
sin ฯ
iii. Mode
Mode =1 โ 1
1 + 1= 0
iv. Variance
Var X =2ฯ
sin 2ฯโ
ฯ
sin ฯ
2
v. Skewness
Skewness = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2{๐ธ(๐)}3
=3ฯ
sin 3ฯโ 3
2ฯ
sin 2ฯ
ฯ
sin ฯ + 2
ฯ
sin ฯ
3
79
vi. Kurtosis
kurtosis = ๐ธ ๐4 โ 4๐ธ ๐ ๐ธ(๐3) + 6{๐ธ(๐)}2๐ธ ๐2 โ 3{๐ธ(๐)}4
=4ฯ
sin 4ฯโ 4
ฯ
sin ฯ
3ฯ
sin 3ฯ+ 6
ฯ
sin ฯ
2 2ฯ
sin 2ฯโ 3
ฯ
sin ฯ
4
3.4.6 Shape of Log-logistic distribution
Fig.3. 3: Log-logistic distribution
3.5 Limiting Distribution
Gamma distribution
The two parameter beta distribution in equation 3.1 can be shown to approach the gamma
distribution as bโโ with scale parameter a changing with the parameter b as a= ๐ฮฒ. The
resultant probability density function is given by
๐(x; a, ฮฒ, b) = ๐ฅ๐ฮฒโ1
๐ฮฒ ฮ ๐ฮฒ ๐
โ ๐ฅ๐๐ฝ
0 โค ๐ฅ _______________(3.10)
Equation (3.10) is the gamma distribution function which is covered in details under the
generalized beta distribution of the second kind in chapter VI.
0
0.2
0.4
0.6
0.8
1
1.20
0.0
45
0.09
0.1
35
0.18
0.2
25
0.27
0.3
15
0.36
0.4
05
0.45
0.4
95
0.54
0.5
85
0.63
0.6
75
0.72
0.7
65
0.81
0.8
55
0.9
0.9
45
0.99
Log-logistic distribution
80
4 Chapter IV: The Three Parameter Beta
Distributions
4.1 Introduction
This chapter highlights some of the three parameter beta distribution functions used by
various authors in literature. These three parameter beta distributions are special cases of
the more generalized four parameter beta distributions.
4.2 Case I: Libby and Novick Three Parameter Beta Distribution
Libby and Novick (1982) used the following approach in deriving the three parameter
beta distribution
Let ๐0 = ๐0 and ๐1 =๐1
๐0+๐1
โน ๐0 = ๐0 and ๐1(๐0 + ๐1) = ๐1
๐1๐0 + ๐1๐1 = ๐1
๐1๐0 = (1 โ ๐1)๐1
โด ๐1 =๐0๐1
1 โ ๐1
๐ฝ =
d๐ฅ0
d๐ฆ0
d๐ฅ0
d๐ฆ1
d๐ฅ1
d๐ฆ0
d๐ฅ1
d๐ฆ1
= 1 0๐1
1 โ ๐1
๐0
1 โ ๐1 2 =
๐0
1 โ ๐1 2
๐ ๐ฆ0,๐ฆ1 = ๐(๐ฅ0, ๐ฅ1) ๐ฝ
= ๐(๐ฅ0) ๐(๐ฅ1) ๐0
1 โ ๐1 2
=๐ข๐
ฮ(a)๐โ๐ข๐ฅ0๐ฅ0
๐โ1 ๐ฃ๐
ฮ(b)๐โ๐ฃ๐ฅ1๐ฅ1
๐โ1 .๐0
1 โ ๐1 2
=๐ข๐๐ฃ๐
ฮ(a)ฮ(b)๐โ๐ข๐ฅ0โ๐ฃ๐ฅ1๐ฅ0
๐โ1๐ฅ1๐โ1
๐0
1 โ ๐1 2
๐ ๐ฆ0,๐ฆ1 =๐ข๐๐ฃ๐
ฮ a ฮ b ๐โ๐ข๐ฅ0โ๐ฃ๐ฅ1๐ฅ0
๐โ1๐ฅ1๐โ1
๐0
1 โ ๐1 2
=๐ข๐๐ฃ๐
ฮ(a)ฮ(b)๐
โ๐ข๐ฆ0โ๐ฃ ๐0๐11โ๐1
๐ฆ0
๐โ1 ๐0๐1
1 โ ๐1
๐โ1
๐0
1 โ ๐1 2
81
โด ๐ ๐ฆ0,๐ฆ1 =๐ข๐๐ฃ๐
ฮ(a)ฮ(b)๐
โ๐ฆ0 ๐ข+ ๐ฃ๐1
1โ๐1
๐ฆ0
๐+๐โ1๐ฆ1๐โ1
1 โ ๐1 ๐+1
=๐ข๐๐ฃ๐
ฮ(a)ฮ(b)๐
โ๐ข๐ฆ0 1+ ๐1๐11โ๐1
๐ฆ0
๐+๐โ1๐ฆ1๐โ1
1 โ ๐1 ๐+1
๐ ๐ฆ1 =๐ข๐๐ฃ๐
ฮ(a)ฮ(b)
๐ฆ1๐โ1
1 โ ๐1 ๐+1 ๐
โ๐ฆ0 ๐ข+ ๐ฃ๐1
1โ๐1 โ
0
๐ฆ0๐+๐โ1 ๐๐ฆ0
๐ฅ = ๐๐ฆ0 โน ๐ฆ0 =๐ฅ
๐
๐๐ฆ0 =๐๐ฅ
๐
๐โ๐ฅโ
0
๐ฅ
๐ ๐+๐โ1
๐๐ฅ
๐
๐ ๐ฆ1 =๐ข๐๐ฃ๐
ฮ(a)ฮ(b)
๐ฆ1๐โ1
1 โ ๐1 ๐+1 ๐โ๐๐ฆ0
โ
0
๐ฆ0๐+๐โ1 ๐๐ฆ0
๐โ๐ฅโ
0
๐ฅ๐+๐โ1
๐๐+๐๐๐ฅ
ฮ(a + b)
ฮ a ฮ b
๐ข๐๐ฃ๐
u +v๐ฆ1
1 โ ๐ฆ1
a+b
๐ฆ1๐โ1
1 โ ๐1 ๐+1
1
B(a, b)
๐ข๐๐ฃ๐
uab 1 +ฮป1๐ฆ1
1 โ ๐ฆ1
a+b
๐ฆ1๐โ1
1 โ ๐1 ๐+1
vu
b
B a, b
๐ฆ1
1 โ ๐ฆ1
๐โ1
1
1 โ ๐1
2
1 + ฮป1 ๐ฆ1
1 โ ๐ฆ1
a+b, 0 < ๐ฆ1 < 1
ฮป1b
B a, b
๐ฆ1
1 โ ๐ฆ1
๐โ1
1
1 โ ๐1
2
1 + ฮป1 ๐ฆ1
1 โ ๐ฆ1
a+b
๐ ๐ฆ1 =ฮป1
b
B a, b
๐ฆ1
1 โ ๐ฆ1
๐โ1
1
1 โ ๐1
2
1 + ฮป1 ๐ฆ1
1 โ ๐ฆ1
a+b
82
=ฮป1
b
B a, b
๐ฆ1๐โ1
(1 โ ๐1)b+1 1 + ฮป1 ๐ฆ1
1 โ ๐ฆ1
a+b
=ฮป1
b
B a, b
๐ฆ1๐โ1
(1 โ ๐1)b+1 1 โ ๐ฆ1 + ฮป1๐ฆ1
1 โ ๐ฆ1
a+b
=ฮป1
b
B a, b
๐ฆ1๐โ1 1 โ ๐1
aโ1
1 โ (1 โ ฮป1)๐ฆ1 a+b_________________________(4.1)
Equation (4.1) was first used by Libby and Novick (1982) for utility function fitting and
by Chen and Novick (1984) as prior in some binomial sampling model. The equation
becomes standard beta distribution when c=1.
An alternative method of deriving the three parameter beta distribution introduced by
Libby and Novick (1982) is by using transformation as follows:
From the Classical beta distribution in equation (2.1), a third parameter c is introduced
using the transformation
x =๐๐ฆ
1 โ ๐ฆ + ๐๐ฆ
๐๐ฅ
๐๐ฆ =
๐(1 โ ๐ฆ + ๐๐ฆ โ [ โ1 + ๐ ๐๐ฆ]
(1 โ ๐ฆ + ๐๐ฆ)2
= ๐ โ ๐๐ฆ + ๐๐ฆ2 โ [โ๐๐ฆ + ๐๐ฆ2]
(1 โ ๐ฆ + ๐๐ฆ)2
=๐
(1 โ ๐ฆ + ๐๐ฆ)2
We get
83
f y; c, a, b =(
๐๐ฆ1 โ ๐ฆ + ๐๐ฆ)aโ1 1 โ
๐๐ฆ1 โ ๐ฆ + ๐๐ฆ
bโ1
B a, b .
๐
(1 โ ๐ฆ + ๐๐ฆ)2
=1
B a, b
๐aโ1๐ฆaโ1
(1 โ ๐ฆ + ๐๐ฆ)๐โ1
(1 โ ๐ฆ + ๐๐ฆ โ ๐๐ฆ)bโ1
(1 โ ๐ฆ + ๐๐ฆ)๐โ1.
๐
(1 โ ๐ฆ + ๐๐ฆ)2
=1
B a, b
๐aโ1+1๐ฆaโ1(1 โ ๐ฆ)bโ1
(1 โ ๐ฆ + ๐๐ฆ)๐โ1+๐โ1+2
=1
B a, b
๐a๐ฆaโ1(1 โ ๐ฆ)bโ1
(1 โ (1 โ ๐)๐ฆ)๐+๐
for 0 < ๐ฆ < 1, ๐ > 0, ๐ > 0, ๐ > 0
4.3 Case II: McDonaldโs Three Parameter Beta I Distribution
The general three parameter beta (a,b,c) can be derived from the standard beta
distribution with two parameters in equation (2.1) by using the transformation
๐ฅ =๐ฆ
๐
๐๐ฅ
๐๐ฆ =
1
๐
f y; a, b, c =
yc
aโ1
1 โyc
bโ1
B a, b .
1
๐
=
yc
aโ1
1 โ yc
bโ1
c B a, b
84
=yaโ1 1 โ
yc
bโ1
ca B a, b , 0 < ๐ฆ < ๐ , ๐ > 0, ๐ > 0, ๐ > 0__________(4.2)
The three parameter beta distribution in equation 4.2 is referred to as the beta of the first
kind in McDonald, 1995. The Classical beta distribution corresponds to equation 4.2
when c=1.
4.4 Gamma Distribution as a Limiting Distribution
We show that the three parameter generalized beta distribution in equation 4.2
approaches the gamma distribution as bโโ as follows:
Let the scale factor b changes with b as c=ฮฒ ๐ + ๐ . Substituting this in equation 4.2
gives
๐(๐ฅ; ๐, b, ๐) = ๐ฅ๐โ1 1 โ
๐ฅฮฒ a + b
๐โ1
ฮฒ a + b ๐๐ต ๐, ๐
Collecting the terms yields
= ๐ฅ๐โ1
ฮ(๐)ฮฒ๐
ฮ(๐ + ๐)
ฮ(๐) a + b ๐ 1 โ
๐ฅ
ฮฒ a + b
๐โ1
For large values of b, the gamma function can be approximated by Stirlingโs formula
ฮ(๐) = ๐โ๐๐๐โ12 2๐
โน ฮ(๐ + ๐)
ฮ(๐) a + b ๐ =
๐โ๐โ๐(๐ + ๐)๐+๐โ12 2๐
๐โ๐๐๐โ12 2๐ (๐ + ๐)๐
85
โด lim ๐โโ
๐โ๐(๐ + ๐)๐โ12 2๐
๐๐โ12 2๐
= 1
Similarly, lim ๐โโ
1 โ๐ฅ
ฮฒ(a + b)
๐โ1
= ๐โ
๐ฅ๐ฝ
โด ๐(x; a, ฮฒ) =๐ฅ๐โ1
ฮฒ๐
ฮ ๐ ๐
โ ๐ฅ๐ฝ
0 โค ๐ฅ _______________(4.3)
Equation 4.3 is the gamma distribution function.
4.5 Case III
4.5.1 3 Parameter Beta Distribution Derived from the Beta Distribution of
the Second Kind
From the Classical beta distribution in equation 2.1, let
๐ฅ =cy
1 + y
๐๐ฅ
๐๐ฆ =
c
1 + y 2
f(๐ฆ; ๐, ๐, ๐) = ๐(๐ฅ; ๐, ๐) ๐ฝ
=๐ฅ๐โ1 1 โ ๐ฅ ๐โ1
๐ต ๐, ๐ . ๐ฝ
=
๐๐ฆ1 + ๐ฆ
๐โ1
1 โ๐๐ฆ
1 + ๐ฆ ๐โ1
๐ต ๐, ๐ .
c
(1 + y)2
=๐๐๐ฆ๐โ1 1 + 1 โ ๐ ๐ฆ ๐โ1
๐ต ๐, ๐ (1 + y)a+b , 0 < ๐ฆ < 1 ________(4.4)
The three parameter beta distribution in equation 4.4 reduces to the beta distribution of
the second kind when ๐ = 1. When ๐ = 2 it reduces to beta type 3 given by
86
๐ ๐ฆ; ๐, ๐ =2๐๐ฆ๐โ1 1 โ ๐ฆ ๐โ1
๐ต ๐, ๐ (1 + y)a+b , 0 < ๐ฆ < 1______________(4.5)
which is defined on a finite interval and may serve as an alternative to the Classical beta
distribution for many practical applications. The beta type 3 was introduced and studied
by Cardeลo et.al (2004)
4.5.2 Case IV
An alternative form of the probability density function of the three parameter beta
distribution of the second kind can be obtained by substituting ๐ฆ = x/c , ๐ > 0 in
equation 3.1,
Then we obtain
f(๐ฅ; ๐, ๐, ๐) = ๐(๐ฆ; ๐, ๐) ๐ฝ
= ๐ฅ๐
๐โ1
๐๐ต ๐, ๐ 1 + ๐ฅ๐
a+b
=๐ฅ๐โ1
๐๐๐ต ๐, ๐ 1 + ๐ฅ๐
a+b , 0 โค ๐ฅ < โ, ๐ > 0, ๐ > 0________(4.6)
The three parameter beta distribution in equation (4.6) was used as the beta of the second
kind by Chotikapanich et.al (2010). It is referred to as beta of the second kind in
McDonald and Xu, (1995). Equation 4.6 becomes the beta distribution of the second kind
when c=1.
4.5.3 Case V
A three parameter probability density function of beta distribution of the second kind can
be obtained by using the transformation ๐ฆ = ๐๐ฅ , ๐ > 0 in equation 3.1,
Then we obtain
f(๐ฅ; ๐, ๐, ๐) = ๐(๐ฆ; ๐, ๐) ๐ฝ
87
= ๐๐ฅ ๐โ1
๐ต ๐, ๐ 1 + ๐๐ฅ a+b
. ๐
=๐๐๐ฅ๐โ1
๐ต ๐, ๐ 1 + ๐๐ฅ a+b
, 0 โค ๐ฅ < โ, ________________(4.7)
๐ > 0, ๐ > 0, ๐ > 0
Equation 4.7 becomes the beta distribution of the second kind when c=1.
4.5.4 Case VI
A three parameter probability density function of the Classical beta distribution can be
obtained by using the transformation ๐ฅ = ๐ฆ๐ ๐๐ฅ
๐๐ฆ = ๐๐ฆ๐โ1in equation 2.1,
๐ ๐ฆ; ๐, ๐, ๐ =๐ฆ๐๐โ๐ 1 โ ๐ฆ๐ ๐โ1
๐ต ๐, ๐ . ๐๐ฆ๐โ1
=๐๐ฆ๐๐โ1 1 โ ๐ฆ๐ ๐โ1
๐ต ๐, ๐ , 0 < ๐ฆ < 1, ๐ > 0, ๐ > 0, ๐ > 0____(4.8)
Special case when c=1 in equation 4.8 gives the Classical beta distribution given in
equation 2.1.
Special case when a=1 in equation 4.8 gives
๐ ๐ฆ; ๐, ๐ =๐๐ฆ๐โ1 1 โ ๐ฆ๐ ๐โ1
๐ต 1, ๐
= ๐๐๐ฆ๐โ1 1 โ ๐ฆ๐ ๐โ1 0 < ๐ฆ < 1, ๐ > 0, ๐ > 0, ๐ > 0_____(4.9)
Equation 4.9 is a Kumaraswamy distribution.
Special case when c=1 and b=1 in equation 4.8 gives
๐ ๐ฆ; ๐, ๐ = ๐๐ฅ๐โ1 _______________(4.10)
Equation 4.10 is a power distribution function. When a=1 in equation 4.10 we get a
Uniform distribution function.
88
5 Chapter V: The Four Parameter Beta
Distributions
5.1 Introduction
This chapter provides for the construction, properties and special cases of four parameter
generalized beta distribution of the first kind and second kind introduced by McDonald
(1984) as an income distribution, whereas the beta distribution of the first kind was used
for the same purpose more than a decade earlier by Thurow (1970). It also provides for
the construction and properties of the five parameter generalized beta distributions which
includes both generalized beta distribution of the first kind and second kind and many
other distributions.
5.2 Generalized Four Parameter Beta Distribution of the First Kind
The generalized beta distribution of the first kind was introduced by McDonald (1984). It
can be obtained from the Classical beta distribution in equation 2.1 by using the
transformation
๐ฅ = ๐ฆ
๐
๐
in equation (2.1)
๐๐ฅ
๐๐ฆ =
๐
๐๐ ๐ฆ๐โ1
The limits changes as follows ๐ฅ๐๐ = ๐ฆ๐ , when ๐ฅ = 0, ๐ฆ๐ = 0 and when
๐ฆ = 1, ๐ฆ๐ = ๐๐
The new distribution is
๐(๐ฆ; ๐, ๐, ๐, ๐) = ๐(๐ฅ; ๐, ๐) ๐ฝ
= ๐ฆ๐
๐๐โ๐
1 โ ๐ฆ๐
๐
๐โ1
๐ต(๐, ๐).
๐
๐๐๐ฆ๐โ1
89
= ๐
๐ฆ๐
๐
๐โ1
1 โ ๐ฆ๐
๐
๐โ1
๐ต ๐, ๐ .
1
๐๐๐ฆ๐โ1
= ๐ ๐ฆ๐๐โ1 1 โ
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ , 0 โค ๐ฆ๐ โค ๐๐ , ______(5.1)
The CDF of 5.1 is given by
๐น ๐ฆ =๐ต ๐, ๐,
๐ฆ๐
๐
๐ต ๐, ๐ , 0 < ๐ฆ < ๐
where all the four parameters ๐, ๐, ๐ and ๐ are positive. ๐ is a scale parameter and ๐, ๐, ๐
are shape parameters.
The probability distribution function in equation 5.1 is a four parameter generalized beta
distribution of the first kind. If p=q=1, the generalized beta distribution of the first kind in
equation 5.1 becomes the two parameter Classical beta distribution in equation 2.1 over
the interval (0,1). If p=1, in equation 4.1, we obtain a three parameter beta distribution in
equation 4.2. When p=-1 and b=1 in equation 5.1, one obtain the inverse beta distribution
of the first kind (Pareto type 1) with density function given by
๐(๐ฆ; ๐, ๐) =๐ฆโ๐โ1
๐โ๐๐ต(๐, 1)
=๐๐๐
๐ฆ1+๐ , for ๐ < ๐ฆ
The generalized beta distribution of the first kind is supported on a bounded domain and
is useful in the modeling of size phenomena, particularly the distribution of income.
90
5.2.1 Properties of generalized beta distribution of the first kind
i. rth-order moments
To obtain the rth
-order moments of generalized beta distribution of the first kind, we
evaluate
๐ธ ๐๐ = ๐ฆ๐๐ ๐ฆ; ๐, ๐
๐
0
๐๐ฆ
= ๐ ๐ฆ๐+๐๐โ1 1 โ
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐
๐
0
๐๐ฆ
=1
๐๐๐๐ต ๐, ๐
๐ ๐ฆ๐
๐๐
+๐ โ1 1 โ
๐ฆ๐
๐
๐โ1
๐๐
๐๐
+๐ ๐ต
๐๐ + ๐, ๐
๐
0
๐๐+๐๐๐ต ๐
๐+ ๐, ๐ ๐๐ฆ
=๐๐+๐๐๐ต
๐๐ + ๐, ๐
๐๐๐๐ต ๐, ๐
=๐๐๐ต
๐๐ + ๐, ๐
๐ต ๐, ๐
=๐๐ฮ
๐๐ + ๐ ฮ ๐ ฮ a + b
ฮ ๐๐ + ๐ + ๐ ฮ a ฮ b
=๐๐ฮ a + b ฮ ๐ +
๐๐
ฮ ๐ + ๐ +๐๐ ฮ a
, ๐ +๐
๐> 0_________________(5.2)
ii. Mean
The mean is given by substituting r=1 in equation 5.2 as
๐ธ ๐ = ๐๐ต ๐ +
1๐ , ๐
๐ต ๐, ๐
91
=๐ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
___________(5.3)
iii. Mode
The mode of generalized beta distribution of the first kind is given by
๐๐๐๐ =๐
๐๐ฆ๐ ๐ฆ = 0
๐
๐๐๐๐ต ๐, ๐
๐
๐๐ฆ ๐ฆ๐๐โ1 1 โ
๐ฆ
๐
๐
๐โ1
= 0
๐๐ โ 1 ๐ฆ๐๐โ2 1 โ ๐ฆ
๐
๐
๐โ1
+ ๐ฆ๐๐โ1(๐ โ 1) 1 โ ๐ฆ
๐
๐
๐โ2 โ๐
๐๐๐ฆ๐โ1 = 0
(๐๐ โ 1) โ๐๐ฆ๐(๐ โ 1)
๐๐ 1 โ ๐ฆ๐
๐
= 0
๐๐ โ 1 =๐๐ฆ๐(๐ โ 1)
๐๐ โ ๐ฆ๐
(๐๐ โ 1)(๐๐ โ ๐ฆ๐) = ๐๐ฆ๐(๐ โ 1)
๐ฆ = ๐๐ ๐๐ โ 1
๐๐ + ๐๐ โ 1 โ ๐
1๐
iv. Variance
Variance of generalized beta distribution of the first kind can be obtained from the rth
-
order moments as follows
Variance = ๐ธ ๐2 โ ๐ธ ๐ 2
=๐2ฮ a +
2p ฮ b ฮ a + b
ฮ a + b +2p
โ ๐ฮ a +
1p ฮ b ฮ a + b
ฮ a + b +1p
2
92
= ๐2
ฮ a +
2p ฮ b ฮ a + b
ฮ a + b +2p
โ ฮ a +
1p ฮ b ฮ a + b
ฮ a + b +1p
2
= ๐2 ฮ a +
2p ฮ b ฮ a + b
ฮ a + b +2p
โฮ2 a +
1p ฮ2 b ฮ2 a + b
ฮ2 a + b +1p
v. Skewness
Skewness can be given by
๐๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐ ๐ธ(๐2) +2 {๐ธ ๐ }3
=๐3ฮ a + b ฮ a +
3๐
ฮ a + b +3๐ ฮ a
โ 3๐ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
๐2ฮ a + b ฮ a +2๐
ฮ a + b +2๐ ฮ a
+ 2 ๐ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
3
= ๐3
ฮ a + b ฮ a +
3๐
ฮ a + b +3๐ ฮ a
โ 3ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
ฮ a + b ฮ a +2๐
ฮ a + b +2๐ ฮ a
+ 2 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
3
vi. Kurtosis
Kurtosis can be given by
Kurtosis = ๐ธ ๐4 โ 4๐ธ ๐ ๐ธ ๐3 + 6{๐ธ ๐ }2๐ธ ๐2 โ 3 ๐ธ ๐ 4
93
=q4ฮ a + b ฮ a +
4๐
ฮ a + b +4๐ ฮ a
โ 4qฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
q3ฮ a + b ฮ a +3๐
ฮ a + b +3๐ ฮ a
+ 6 qฮ a + b ฮ q +
1๐
ฮ a + b +1๐ ฮ a
2
q2ฮ a + b ฮ a +2๐
ฮ q + b +2๐ ฮ a
โ 3 qฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
4
= q4
ฮ a + b ฮ a +
4๐
ฮ a + b +4๐ ฮ a
โ 4ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
ฮ a + b ฮ a +3๐
ฮ a + b +3๐ ฮ a
+ 6 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
2
ฮ a + b ฮ a +2๐
ฮ a + b +2๐ ฮ a
โ 3 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
4
5.3 Applications of Generalized Beta Distribution of the First Kind
McDonald (1984) estimated both the generalized beta of the first and second kind and
both beta of the first and second kind distributions for 1970, 1975 and 1980 US family
incomes and found out that the performance of the generalized beta of the first kind is
comparable to the generalized gamma and beta of the second kind distribution.
5.4 Special Cases of Generalized Beta Distribution of the First Kind
The four parameter probability density function of the generalized beta distribution of the
first kind is very flexible and includes the following distributions:
94
5.4.1 Pareto type 1 distribution function
Special case of the generalized beta distribution of the first kind when p=-1 and b=1 in
equation 5.1 gives
๐(๐ฆ; ๐, ๐) =๐ฆโ๐โ1
๐โ๐๐ต(๐, 1)(1 โ (
๐ฆ
๐)โ1)1โ1
=๐๐
๐ต(๐, 1)๐ฆ๐+1
=๐๐๐
๐ฆ๐+1 , 0 < ๐ โค ๐ฆ______________________(5.4)
The cdf of equation 5.4 is given by
๐น ๐ฆ = 1 โ ๐
๐ฆ
๐
, ๐ฆ โฅ ๐
The probability distribution function in equation 5.4 is the Pareto type 1 distribution also
known as classical Pareto distribution. This special case also include inverse beta of the
first kind (IB1), implying Pareto (y;q,a) =IB1 (y;q,a,b=1) (McDonald 1995).
5.4.2 Properties of Pareto type 1 distribution
i. rth- order moments
Putting p=-1 and b=1 in equation 5.2
๐ธ(๐๐) =๐๐ฮ a + 1 ฮ ๐ โ ๐
ฮ ๐ + 1 โ ๐ ฮ a
=๐๐a
๐ โ ๐, ๐ โ ๐ > 0
95
ii. Mean
๐ธ ๐ = ๐ฮ a + 1 ฮ ๐ โ 1
ฮ ๐ + 1 โ 1 ฮ a
=๐๐ฮ(๐ โ 1)
ฮ(๐)
=๐๐ ๐ โ 2 !
(๐ โ 1)!
=๐๐ ๐ โ 2 !
(๐ โ 1)(๐ โ 2)!
=๐๐
๐ โ 1, ๐ > 1
iii. Mode
Mode of Pareto type 1 distribution can be obtained by substituting b=1 and p = โ1 in the
mode of generalized beta distribution of the first kind
๐ฆ = ๐โ1 โ1๐ โ 1
โ1๐ โ 1.1 โ 1 โ 1
โ11
= ๐ โ๐ โ 1 + 1 โ 1
โ๐ โ 1
= ๐
iv. Variance
Substituting ๐ = โ1 and b = 1 in the expression of variance for the generalized beta
distribution of the second kind we obtain
Var Y = m2 โ m12
=๐2a
๐ โ 2โ
๐a
๐ โ 1
2
๐2
=๐2a a2 โ 2a + 1 โ ๐2๐2(๐ โ 2)
(a โ 2) a โ 1 2
=๐2a
(a โ 2) a โ 1 2
96
v. Skewness
Skewness of Pareto type 1 distribution can be obtained by substituting b=1 and p = โ1
in the skewness of generalized beta distribution of the first kind
๐๐๐๐ค๐๐๐ ๐
= ๐3 ฮ a + 1 ฮ a โ 3
ฮ a + 1 โ 3 ฮ a โ 3
ฮ a + 1 ฮ a โ 1
ฮ a + 1 โ 1 ฮ a
ฮ a + 1 ฮ a โ 2
ฮ a + 1 โ 2 ฮ a
+ 2 ฮ a + 1 ฮ a โ 1
ฮ a + 1 โ 1 ฮ a
3
๐๐๐๐ค๐๐๐ ๐ = ๐3 a
(a โ 3)โ
3a2
a โ 1 (a โ 2)+ 2
a
a โ 1
3
= ๐3 a a โ 1 3 a โ 2 โ 3a2 a โ 1 2(a โ 3) + 2a3 a โ 2 (a โ 3)
a โ 1 3 a โ 2 (a โ 3)
= ๐3
(a5 โ 5a4 + 9a3 โ 7a2 + 2a) + (โ3a5 + 15a4 โ 21a3 + 9a2)
+(2a5 โ 10a4 + 12a3)
a โ 1 3 a โ 2 (a โ 3)
= 2๐๐3 1 + a
a โ 1 3 a โ 2 (a โ 3)
๐ ๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐ ๐๐๐ค๐๐๐ ๐ = 2๐๐3 1 + a
a โ 1 3 a โ 2 (a โ 3) โ
a โ 2 a โ 1 2
๐2a
32
= 2๐๐3 1 + a
a โ 1 3 a โ 2 (a โ 3) โ
a โ 2 a โ 2 a โ 1 3
๐3a a
=2(1 + a)
a โ 3
a โ 2
a , ๐ > 3
vi. Kurtosis
Kurtosis of Pareto type 1 distribution can be obtained by substituting b=1 and p = โ1 in
the Kurtosis of generalized beta distribution of the first kind
97
๐พ๐ข๐๐ก๐๐ ๐๐
= q4 ฮ a + 1 ฮ a โ 4
ฮ a + 1 โ 4 ฮ a โ 4
ฮ a + 1 ฮ a โ 1
ฮ a + 1 โ 1 ฮ a
ฮ a + 1 ฮ a โ 3
ฮ a + 1 โ 3 ฮ a
+ 6 ฮ a + 1 ฮ a โ 1
ฮ a + 1 โ 1 ฮ a
2ฮ a + 1 ฮ a โ 2
ฮ a + 1 โ 2 ฮ a โ 3
ฮ a + 1 ฮ a โ 1
ฮ a + 1 โ 1 ฮ a
4
= q4 a
a โ 4 โ
4a2
(a โ 1)(a โ 3)+
6a3
(a โ 1)2(a โ 2)โ
3a4
(a โ 1)4
= q4
a a โ 1 4 a โ 2 a โ 3 โ 4a2 a โ 1 3 a โ 2 a โ 4 +
6a3 a โ 1 2 a โ 3 a โ 4 โ 3a4 a โ 2 a โ 3 a โ 4
a โ 1 4 a โ 2 a โ 3 a โ 4
= q4
(a7โ9a6+32a5โ58a4+57a3 โ 29a2 + 6a) +
โ4a7 + 36a6 โ 116a5 + 172a4 โ 120a3 + 32a2 +
6a7 โ 54a6 + 162a5 โ 186a4 + 72a3 +
(โ3a7 + 27a6 โ 78a5 + 72a4)
a โ 1 4 a โ 2 a โ 3 a โ 4
= 3aq4 3a2 + a + 2
a โ 1 4 a โ 2 a โ 3 a โ 4
๐ ๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐พ๐ข๐๐ก๐๐ ๐๐
= 3aq4 3a2 + a + 2
a โ 1 4 a โ 2 a โ 3 a โ 4 โ
a โ 2 a โ 1 2
๐2a
2
= 3 3a2 + a + 2
a โ 3 a โ 4 โ
a โ 2
a
=3(3a3 โ 5a2 โ 4)
a a โ 3 a โ 4
5.4.3 Stoppa (generalized Pareto type 1) distribution function
Special case when a=1 and p=-p in equation 5.1 gives the following density function
98
f y; p, q, b = p๐๐๐๐ฆโ๐โ1 1 โ ๐ฆ
๐
โ๐
bโ1
, y โฅ q > 0____________(5.5)
The CDF of equation 5.5 is given by
F y = 1 โ ๐ฆ
๐
โ๐
b
, y โฅ q > 0
Equation 5.5 is a Stoppa distribution also known as the generalized Pareto type 1
distribution.
5.4.4 Properties of Stoppa distribution
i. rth- order moments
Putting a=1 and p = โp in equation 5.2
๐ธ ๐๐ =qrฮ 1 + ๐ ฮ 1 โ
๐๐
ฮ 1 + ๐ โ๐๐ ฮ 1
=qr๐ฮ ๐ ฮ 1 โ
๐๐
ฮ 1 + ๐ โ๐๐
ii. Mean
The mean is given by substituting r=1 in the expression for the rth-order moments
E(Y) =๐bฮ ๐ ฮ 1 โ
1๐
ฮ 1 + ๐ โ1๐
iii. Mode
Putting a=1 and p = โp in the mode of generalized beta distribution of the first kind
๐ฆ = ๐โ๐ โ๐ โ 1
โ๐ โ ๐๐ โ 1 + ๐
โ1๐
= ๐ 1 + ๐๐
1 + ๐
1๐
, ๐ > 1
99
Compared to the classical Pareto distribution, the Stoppa distribution is more flexible
since it has an additional shape parameter b that allows for unimodal (for ๐ > 1) and
zero-modal (for ๐ โค 1) densities.
iv. Variance
Putting a=1 and p=-p in the variance of the generalized beta distribution of the first kind
Var Y = q2 ฮ 1 โ
2p ฮ b ฮ 1 + b
ฮ 1 + b โ2p
โฮ2 1 โ
1p ฮ2 b ฮ2 1 + b
ฮ2 1 + b โ1p
v. Skewness
Putting a=1 and p = โp in the skewness of generalized beta distribution of the first kind
Skewness
= ๐3
ฮ 1 + b ฮ 1 โ
3๐
ฮ 1 + b โ3๐ ฮ 1
โ 3ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
ฮ 1 + b ฮ 1 โ2๐
ฮ 1 + b โ2๐ ฮ 1
+ 2 ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
3
Skewness
= ๐3
bฮ b ฮ 1 โ
3๐
b โ3๐ ฮ b โ
3๐
โ 3bฮ b ฮ 1 โ
1๐
b โ1๐ ฮ b โ
1๐
bฮ b ฮ 1 โ2๐
b โ2๐ ฮ b โ
2๐
+ 2 bฮ b ฮ 1 โ
1๐
b โ1๐ ฮ b โ
1๐
3
vi. Kurtosis
Putting a=1 and p = โp in the Kurtosis of generalized beta distribution of the first kind
100
๐พ๐ข๐๐ก๐๐ ๐๐
= q4
ฮ 1 + b ฮ 1 โ
4๐
ฮ 1 + b โ4๐ ฮ 1
โ 4ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
ฮ 1 + b ฮ 1 โ3๐
ฮ 1 + b โ3๐ ฮ 1
+ 6 ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
2
ฮ 1 + b ฮ 1 โ2๐
ฮ 1 + b โ2๐ ฮ 1
โ 3 ฮ 1 + b ฮ 1 +
1๐
ฮ 1 + b +1๐ ฮ 1
4
= q4
bฮ b ฮ 1 โ
4๐
ฮ 1 + b โ4๐
โ 4bฮ b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
bฮ b ฮ 1 โ3๐
ฮ 1 + b โ3๐
+ 6 ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐
2
ฮ 1 + b ฮ 1 โ2๐
ฮ 1 + b โ2๐
โ 3 ฮ 1 + b ฮ 1 +
1๐
ฮ 1 + b +1๐
4
5.4.5 Kumaraswamy distributions
Special case when a=1 and q=1 in equation 5.1 gives the following density function
f(y; p, b) =p๐ฆ๐โ1(1 โ ๐ฆ๐)bโ1
B(1, b)
= pb๐ฆ๐โ1(1 โ ๐ฆ๐ )bโ1 , b > 0____________(5.6)
Equation 5.6 is a Kumaraswamy distribution. On the other hand, if b=1 in equation 5.6
then we get the power distribution ๐ y; p = pzpโ1 as in equation 2.12 or equation 4.10.
101
This implies that the power distribution is a special case of Kumaraswamy distribution
with parameters (a,1). If p=b=1 in equation 5.6, then the Kumaraswamy distribution
becomes a uniform distribution.
5.4.6 Properties of Kumaraswamy distribution
i. rth-order moments
The rth-order moments are obtained by substituting a=1 and q=1 in equation 5.2
๐ธ ๐๐ =๐ต
๐๐ + 1, ๐
๐ต 1, ๐
=ฮ
๐๐ + 1 bฮ ๐
ฮ ๐๐ + 1 + ๐
=bฮ
๐๐ + 1 ฮ(๐)
ฮ(1 +๐๐ + ๐)
ii. Mean
The mean is given by substituting r=1 in the expression for the rth-order moments
E(Y) =bฮ
1๐ + 1 ฮ(๐)
ฮ(1 +1๐ + ๐)
iii. Mode
The mode of the Kumaraswamy distribution occurs at
d
dyf y = 0
d
dy pbypโ1 1 โ yp bโ1 = 0
pb p โ 1 ypโ2 1 โ yp bโ1 + b โ 1 1 โ yp bโ2 . โpypโ1 . ypโ1 = 0
p โ 1 ypโ1 . yโ1 1 โ yp bโ1 โ b โ 1 1 โ yp bโ1(1 โ yp)โ1pypโ1ypyโ1 = 0
p โ 1 โ b โ 1 (1 โ yp)โ1pyp = 0
102
p โ 1 = b โ 1 (1 โ yp)โ1pyp = b โ 1 pyp
(1 โ yp )
p โ 1 (1 โ yp) = b โ 1 pyp
p โ pyp โ 1 + yp = pbyp โ pyp
yp โp + 1 โ pb + p = 1 โ p
yp = 1 โ ๐
1 โ ๐๐
๐ฆ = 1 โ ๐
1 โ ๐๐
1๐
for p โฅ 1, b โฅ 1, (p, b) โ (1,1)
iv. Variance
Substituting a=1 and q=1in the variance of the generalized beta distribution of the first
kind we get
Var (Y) =ฮ 1 +
2p ฮ b ฮ 1 + b
ฮ 1 + b +2p
โฮ2 1 +
1p ฮ2 b ฮ2 1 + b
ฮ2 1 + b +1p
= b! b โ 1 !
2p !
2p + b !
โ b! b โ 1 !
1p !
1p + b !
2
v. Skewness
Putting a=1 and q = 1 in the skewness of generalized beta distribution of the first kind
and is given by
๐๐๐๐ค๐๐๐ ๐
=bฮ b ฮ a +
3๐
ฮ 1 + b +3๐
โ 3bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
bฮ b ฮ 1 +2๐
ฮ 1 + b +2๐
+ 2 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
3
103
vi. Kurtosis
Putting a=1 and q = 1 in the Kurtosis of generalized beta distribution of the first kind and
is given by
๐พ๐ข๐๐ก๐๐ ๐๐
=bฮ b ฮ 1 +
4๐
ฮ 1 + b +4๐
โ 4bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
bฮ b ฮ 1 +3๐
ฮ 1 + b +3๐
+ 6 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
2
bฮ b ฮ 1 +2๐
ฮ 1 + b +2๐
โ 3 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
4
5.4.7 Generalized Power distribution Function
Special case when p=1 and b=1 in equation (5.1) gives the following density
๐ ๐ฆ; ๐, ๐ =๐ฆ๐โ1
๐๐๐ต(๐, 1)
=๐๐ฆ๐โ1
๐๐, 0 < ๐ฆ < ๐ ________(5.7)
Equation (5.7) is a generalized power distribution function.
5.4.8 Properties of power distribution
i. rth-order moments
Putting ๐ = 1 and ๐ = 1 in equation 5.2
๐ธ ๐๐ =๐๐ฮ a + 1 ฮ ๐ + ๐
ฮ ๐ + 1 + ๐ ฮ a
=๐๐aฮ a ฮ ๐ + ๐
a + r ฮ ๐ + ๐ ฮ a
=๐๐a
a + r , ๐ + ๐ > 0
104
ii. Mean
๐ธ ๐ =๐๐
a + 1
iii. Variance
Putting ๐ = 1 and b = 1 in the expression of variance for the generalized beta
distribution of the second kind we obtain
Var Y = ๐2 ฮ a + 2 ฮ 1 ฮ a + 1
ฮ a + 1 + 2 โ
ฮ2 a + 1 ฮ2 1 ฮ2 a + 1
ฮ2 a + 1 + 2
= ๐2 aฮ a
a + 2 โ
aฮ a aฮ a
(a + 1)2 a + 2 2
= ๐2 (a + 1)2 a + 2 aฮ a โ a2ฮ2 a
(a + 1)2 a + 2 2
iv. Skewness
Putting p=1 and b = 1 in the skewness of generalized beta distribution of the first kind
and is given by
Skewness
= ๐3 ฮ a + 1 ฮ a + 3
ฮ a + 1 + 3 ฮ a โ 3
ฮ a + 1 ฮ a + 1
ฮ a + 1 + 1 ฮ a
ฮ a + 1 ฮ a + 2
ฮ a + 1 + 2 ฮ a
+ 2 ฮ a + 1 ฮ a + 1
ฮ a + 1 + 1 ฮ a
3
= ๐3 a
a + 3โ
3a2
(a + 1) a + 2 +
2a3
(a + 1)3
= ๐3 a a + 1 3 a + 2 โ 3a2(a + 1)2 a + 3 + 2a3(a + 2) a + 3
(a + 1)3(a + 2) a + 3
= ๐3 (a5+5a4+9a3+7a2 + 2a) + โ3a5 โ 15a4 โ 21a3 โ 9a2 + 2a5 + 10a4 + 12a3
(a + 1)3(a + 2) a + 3
=2๐๐3(1 โ a)
(a + 1)3(a + 2) a + 3
Standardized Skewness
=2๐๐3(1 โ a)
(a + 1)3(a + 2) a + 3 .
(a + 1)2 a + 2 2
๐2(a + 1)2 a + 2 aฮ a โ a2ฮ2 a
32
105
=2๐(โa3 โ 3a2 + 4)
a + 3
1
(a + 1)2 a + 2 aฮ a โ a2ฮ2 a
32
v. Kurtosis
Putting p=1 and b = 1 in the Kurtosis of generalized beta distribution of the first kind
and is given by
๐พ๐ข๐๐ก๐๐ ๐๐
= q4 ฮ a + 1 ฮ a + 4
ฮ a + 1 + 4 ฮ a โ 4
ฮ a + 1 ฮ a + 1
ฮ a + 1 + 1 ฮ a
ฮ a + 1 ฮ a + 3
ฮ a + 1 + 3 ฮ a
+ 6 ฮ a + 1 ฮ a + 1
ฮ a + 1 + 1 ฮ a
2ฮ a + 1 ฮ a + 2
ฮ a + 1 + 2 ฮ a โ 3
ฮ a + 1 ฮ a + 1
ฮ a + 1 + 1 ฮ a
4
= q4 a
a + 4โ
4a2
(a + 1)(a + 3)+
6a3
a + 1 2(a + 2)โ
3a4
a + 1 4
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐พ๐ข๐๐ก๐๐ ๐๐
= (a + 1)2 a + 2 2
๐2(a + 1)2 a + 2 aฮ a โ a2ฮ2 a
2
. q4 a
a + 4โ
4a2
(a + 1)(a + 3)
+6a3
a + 1 2(a + 2)โ
3a4
a + 1 4
5.4.9 Generalized Gamma Distribution Function
We show that the generalized beta distribution of the first kind approaches the
generalized gamma distribution as bโโ as follows:
Let the scale factor q changes with b as q=ฮฒ(a + b)1
๐ . Substituting this in equation 5.1 we
get
๐(๐ฆ; ๐, ฮฒ, ๐) =
๐ ๐ฆ๐๐โ1(1 โ (๐ฆ
ฮฒ(a + b)1๐
)๐)๐โ1
(ฮฒ(a + b)1๐)๐๐๐ต ๐, ๐
106
Collecting the terms yields
= ๐ ๐ฆ๐๐โ1
ฮ(๐)ฮฒ๐๐
ฮ(๐ + ๐)
ฮ(๐) a + b ๐ 1 โ
๐ฆ๐
ฮฒp(a + b)
๐โ1
For large values of b, the gamma function can be approximated by Stirlingโs formula
ฮ(๐) = ๐โ๐๐๐โ12 2๐
โน ฮ(๐ + ๐)
ฮ(๐) a + b ๐ =
๐โ๐โ๐(๐ + ๐)๐+๐โ12 2๐
๐โ๐๐๐โ12 2๐ (๐ + ๐)๐
โด lim ๐โโ
๐โ๐(๐ + ๐)๐โ12 2๐
๐๐โ12 2๐
= 1
Similarly, lim ๐โโ
1 โ๐ฆ๐
ฮฒp(a + b)
๐โ1
= ๐โ
๐ฆ๐ฝ
๐
โด ๐(y; p, ฮฒ, a) = ๐ ๐ฆ๐๐โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
๐
0 โค ๐ฆ _______________(5.8)
Equation 5.8 is the generalized Gamma (GG) distribution function. See the relationship
of generalized gamma to other distribution under the GG derived from generalized beta
of the second kind where it includes lognormal, Weibull, Gamma, Exponential, Normal
and Pareto distributions as special or limiting cases.
5.4.10 Unit Gamma Distribution Function
Special case of generalized beta distribution of the second kind when ๐ = ๐ฟ/๐ and
getting the limit as ๐ โ 0 in equation 5.1 gives
107
๐ y; q, ๐ฟ, b = lim ๐โ0
๐ ๐ฆ
๐ ๐ฟ๐ โ1
(1 โ (๐ฆ๐)๐)๐โ1
(qp)๐ฟ๐ B(
๐ฟ๐
, b)
=๐ฆ๐ฟโ1
q๐ฟlim
๐โ0
๐ (1 โ (๐ฆ๐)๐)๐โ1
B(๐ฟ๐ , b)
=๐ฆ๐ฟโ1
q๐ฟlim
๐โ0
๐ (1 โ (๐ฆ๐)๐)๐โ1
ฮ ๐ฟ๐ ฮ b
. ฮ(๐ฟ
๐+ b)
=๐ฆ๐ฟโ1
q๐ฟฮ b lim
๐โ0
๐ (1 โ (
๐ฆ๐)๐)๐โ1
ฮ ๐ฟ๐
. ฮ ๐ฟ
๐+ b
=๐ฆ๐ฟโ1
q๐ฟฮ b ๐ฟ๐ ln
๐ฆ
๐
๐โ1
, 0 < ๐ฆ < ๐ _______________(5.9)
Equation 5.9 is the Unit gamma distribution. The unit gamma with q=1 is mentioned in
Patil et al (1984) and has been used as a mixing distribution for the parameter p in the
binomial, Grassia (1977).
5.5 Generalized Four Parameter Beta Distribution of the Second Kind
The generalized beta distribution of the second kind can be derived from the beta
distribution of the second kind in equation 3.1 by using the transformation
๐ฆ = ๐ฅ
๐
๐
๐๐ฆ
๐๐ฅ =
๐
๐๐ ๐ฅ๐โ1
The new limits are as follows (๐ฆ๐๐)1
๐ = ๐ฅ
108
When ๐ฆ = 0, ๐ฅ = 0 and when ๐ฆ = โ, ๐ฅ = โ
The new distribution is
๐(๐ฅ; ๐, ๐, ๐, ๐) = ๐(๐ฆ; ๐, ๐) ๐ฝ
= (
๐ฅ๐)๐
๐โ1
๐ต ๐, ๐ 1 + (๐ฅ๐)๐
a+b
๐
๐๐ ๐ฅ๐โ1
= ๐ ๐ฅ๐๐โ1
๐๐๐๐ต(๐, ๐)(1 + (๐ฅ๐)๐)a+b
, 0 < ๐ฅ < โ , ๐, ๐, ๐, ๐ > 0 ______(5.10)
The probability density function in equation (5.10) is a four parameter generalized beta
distribution of the second kind. It is a result of the pioneering work of McDonald (1984)
and Venter (1983) which led to this generalized beta of the second kind (or transformed
beta) distribution. It is also known in the statistical literature as the generalized F (see,
e.g., Kalbfleisch and Prentice, 1980). If p=q=1, then the generalized beta distribution of
the second kind in equation 5.10 reduces to beta distribution of the second kind in
equation 3.1. If p=1 in equation 5.10, one obtain the three parameter beta distribution in
equation 4.6.
5.5.1 Properties of generalized beta distribution of the second kind
i. rth-order moments
To obtain the rth
-order moments of generalized beta distribution of the second kind, we
evaluate
๐ธ ๐๐ = ๐ฆ๐๐ ๐ฆ; ๐, ๐
โ
0
๐๐ฆ
= ๐ ๐ฆ๐+๐๐โ1
๐๐๐๐ต ๐, ๐ 1 + (๐ฆ๐)๐
๐+๐
โ
0
๐๐ฆ
109
=1
๐ต ๐, ๐
๐ ๐ฆ๐ ๐+
๐๐ โ1
๐๐๐ 1 + ๐ฆ๐
๐
๐+๐
โ
0
๐๐ฆ
=1
๐๐๐๐ต ๐, ๐
๐ ๐ฆ๐ ๐+
๐๐ โ1
1 โ (๐ฆ๐)๐
๐โ1
๐๐ ๐+
๐๐ 1 + (
๐ฆ๐)๐
๐+๐๐ +๐โ
๐๐
โ
0
๐๐ ๐+
๐๐ ๐๐ฆ
=๐
๐ ๐+๐๐
๐๐๐๐ต ๐, ๐ ๐ต ๐ +
๐
๐, ๐ โ
๐
๐
=๐๐๐+๐โ๐๐
๐ต ๐, ๐ ๐ต ๐ +
๐
๐, ๐ โ
๐
๐
=๐๐๐ต ๐ +
๐๐ , ๐ โ
๐๐
๐ต ๐, ๐
=๐๐ฮ ๐ +
๐๐ ฮ ๐ โ
๐๐
ฮ(๐)ฮ(๐), โ๐๐ < ๐ < ๐๐_______________________(5.11)
ii. Mean
The mean of the generalized beta distribution of the second kind is found by substituting
r=1 in equation 5.11
๐1 โก ๐ธ ๐ = ๐๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
=๐ฮ ๐ +
1๐ ฮ ๐ โ
1๐
ฮ ๐ + ๐ .ฮ ๐ + ๐
ฮ ๐ ฮ ๐
=๐ฮ ๐ +
1๐ ฮ ๐ โ
1๐
ฮ ๐ ฮ ๐ , ๐, ๐ > 1
110
๐2 โก ๐ธ ๐2 =๐๐๐ต ๐ +
๐๐ , ๐ โ
๐๐
๐ต ๐, ๐
=๐2๐ต ๐ +
2๐ , ๐ โ
2๐
๐ต ๐, ๐
iii. Mode
The mode of the generalized beta distribution of the second kind occurs at
๐
๐๐ฆ๐ ๐ฆ; ๐, ๐, ๐, ๐ = 0
๐
๐๐ฆ
๐ ๐ฆ๐๐โ1
๐๐๐๐ต ๐, ๐ (1 + (๐ฆ๐)๐)a+b
= 0
๐
๐๐๐๐ต ๐, ๐
๐
๐๐ฆ
๐ฆ๐๐โ1
(1 + (๐ฆ๐
)๐)a+b = 0
๐๐ โ 1 ๐ฆ๐๐โ1โ1 (1 + (๐ฆ๐)๐)a+b โ a + b (1 + (
๐ฆ๐)๐)a+bโ1
pypโ1
qp ๐ฆ๐๐โ1
1 + (๐ฆ๐)๐
a+b
2
= 0
๐๐ โ 1 ๐ฆ๐๐โ1โ1 (1 + (๐ฆ
๐)๐)a+b โ a + b (1 + (
๐ฆ
๐)๐)a+bโ1
pypโ1
qp ๐ฆ๐๐โ1
= 0
๐๐ โ 1 ๐ฆ๐๐โ1๐ฆโ1 (1 + (๐ฆ
๐)๐)a+b
โ a + b (1 + (๐ฆ
๐)๐)a+b (1 + (
๐ฆ
๐)๐)โ1
pypyโ1
qp ๐ฆ๐๐โ1 = 0
111
๐๐ โ 1 = a + b 1
(1 + (๐ฆ๐)๐)
pyp
qp
(qp + qp (๐ฆ
๐)๐)(๐๐ โ 1) = apyp + bpyp
qp๐๐ โ qp + ๐ฆ๐๐๐ โ ๐ฆ๐ = apyp + bpyp
๐ฆ๐(๐๐ + ๐๐ โ ๐๐ + 1) = qp (๐๐ โ 1)
๐ฆ๐ = ๐๐(๐๐ โ 1)
(๐๐ + 1)
๐ฆ = ๐ ๐๐ โ 1
๐๐ + 1
1๐
if ๐๐ โฅ 1
iv. Variance
The variance of the generalized beta distribution of the second kind is given by
Var ๐ = ๐ธ ๐2 โ ๐ธ ๐ 2
= ๐2
๐ต ๐ +
2๐ , ๐ โ
2๐
๐ต ๐, ๐ โ
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
Coefficient of Variation
๐ถ๐ =
๐2
๐ต ๐ +
2๐ , ๐ โ
2๐
๐ต ๐, ๐ โ ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
๐๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
=
๐ต ๐ +
2๐
, ๐ โ2๐ ๐ต2 ๐, ๐
๐ต ๐, ๐ ๐ต2 ๐ +1๐ , ๐ โ
1๐
โ ๐ต ๐ +
1๐
, ๐ โ1๐
๐ต ๐, ๐
2
๐ต2 ๐, ๐
๐ต2 ๐ +1๐ , ๐ โ
1๐
112
= ๐ต ๐ +
2๐ , ๐ โ
2๐ ๐ต ๐, ๐
๐ต2 ๐ +1๐
, ๐ โ1๐
โ 1
v. Skewness
Skewness of the generalized beta distribution of the second kind can be given by
๐๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐ ๐ธ(๐2) +2 {๐ธ ๐ }3
=๐3๐ต ๐ +
3๐ , ๐ โ
3๐
๐ต ๐, ๐ โ 3
๐๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
๐2๐ต ๐ +2๐ , ๐ โ
2๐
๐ต ๐, ๐
+ 2 ๐๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
3
= ๐3
๐ต ๐ +
3๐
, ๐ โ3๐
๐ต ๐, ๐ โ 3
๐ต ๐ +1๐
, ๐ โ1๐
๐ต ๐, ๐
๐ต ๐ +2๐
, ๐ โ2๐
๐ต ๐, ๐
+ 2 ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
3
Standardized Skewness =๐ธ ๐3 โ 3๐ธ ๐ ๐ธ(๐2) + 2 {๐ธ ๐ }3
๐3
vi. Kurtosis
Kurtosis of generalized beta distribution of the second kind can be given by
๐พ๐ข๐๐ก๐๐ ๐๐ = ๐ธ ๐4 โ 4๐ธ(๐)๐ธ ๐3 + 6๐ธ{ ๐ }2๐ธ(๐2) โ3 {๐ธ ๐ }4
113
=๐4๐ต ๐ +
4๐ , ๐ โ
4๐
๐ต ๐, ๐ โ 4
๐๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
๐3๐ต ๐ +3๐ , ๐ โ
3๐
๐ต ๐, ๐
+ 6 ๐๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
๐2๐ต ๐ +2๐ , ๐ โ
2๐
๐ต ๐, ๐
โ 3 ๐๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
4
= ๐4
๐ต ๐ +
4๐ , ๐ โ
4๐
๐ต ๐, ๐ โ 4
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
๐ต ๐ +3๐ , ๐ โ
3๐
๐ต ๐, ๐
+ 6 ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
๐ต ๐ +2๐ , ๐ โ
2๐
๐ต ๐, ๐ โ 3
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
4
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐พ๐ข๐๐ก๐๐ ๐๐ =๐ธ ๐4 โ 4๐ธ(๐)๐ธ ๐3 + 6๐ธ{ ๐ }2๐ธ(๐2) โ 3 {๐ธ ๐ }4
๐4
5.6 Special Case of the Generalized Beta Distribution of the Second
Kind
The generalized beta distribution of the second kind nests many distributions as special or
limiting cases, below is a consideration of some of the distributions and their properties
5.6.1 Singh-Maddala (Burr 12) Distribution
Special case of the generalized beta distribution of the second kind when a=1 in equation
5.10 gives the following density function
๐ ๐ฆ; ๐, ๐, ๐ = ๐ ๐ฆ๐โ1
๐๐๐ต 1, ๐ (1 + (๐ฆ๐)๐)1+b
114
= ๐ ๐๐ฆ๐โ1
๐๐ 1 + (๐ฆ๐)๐ 1+b
, ๐ฆ > 0_______________________(5.12)
The CDF of equation 2.1 is given by
F y = 1 โ 1 + (๐ฆ
๐)๐ โb
Where all three parameters p, q, b are positive. q is a scale parameter and p, b are shape
parameters; b only affects the right tail, whereas p affects both tails (Kleiber and Kotz,
2003). Singh Maddala distribution is known under a variety of names: Usually called
Burr Type 12 distribution, or simply the Burr distribution. The Singh-Maddala
distribution includes the Weibull and Fisk distributions as special cases.
5.6.2 Properties of Singh-Maddala distribution
i. rth-order moments
The rth-order moments of the Singh-Maddala distribution can be obtained from the
expression of the rth โorder moments of the generalized beta distribution of the second
kind by substituting a=1 in equation 5.11. Thus
๐ธ ๐๐ =๐๐๐ต 1 +
๐๐ , ๐ โ
๐๐
๐ต 1, ๐
=๐๐ฮ 1 +
๐๐ ฮ ๐ โ
๐๐
ฮ b , โ๐ < ๐ < ๐๐
ii. Mean
The mean is given by
๐ธ ๐ =๐ฮ 1 +
1๐ ฮ ๐ โ
1๐
ฮ b
iii. Mode
The mode of the Singh-Maddala distribution is given by
115
Mode = ๐ ๐ โ 1
๐๐ + 1
1๐
if ๐ โฅ 1
and at zero elsewhere. Thus the mode is decreasing with b, reflecting the fact that the
right tail becomes lighter as b increases
iv. Variance
Var Y = ๐2 =๐2๐ต 1 +
2๐ , ๐ โ
2๐
๐ต 1, ๐ โ
๐๐ต 1 +1๐ , ๐ โ
1๐
๐ต 1, ๐
2
=๐2bฮ 1 +
2๐ ฮ b โ
2p
bฮ(๐)โ
๐๐ฮ(1 +1๐)ฮ ๐ โ
1๐
bฮ ๐
2
= ๐2 ฮ(b)ฮ 1 +
2๐ ฮ ๐ โ
2๐ โ ฮ2(1 +
1๐)ฮ2 ๐ โ
1๐
ฮ2(๐)
Hence the coefficient of variation is given by
๐ถ๐ =๐
๐
๐ถ๐ =
๐2 ฮ b ฮ 1 +
2๐ ฮ ๐ โ
2๐ โ ฮ2 1 +
1๐ ฮ2 ๐ โ
1๐
ฮ2 ๐
๐ฮ(1 +1๐
)ฮ(๐ โ1๐
)
ฮ(b)
=
๐ฮ ๐ ฮ b ฮ 1 +
2๐ ฮ ๐ โ
2๐ โ ฮ2 1 +
1๐ ฮ2 ๐ โ
1๐
๐ฮ(1 +1๐)ฮ(๐ โ
1๐)
ฮ(b)
= ฮ b ฮ 1 +
2๐ ฮ ๐ โ
2๐ โ ฮ2 1 +
1๐ ฮ2 ๐ โ
1๐
ฮ(1 +1๐)ฮ(๐ โ
1๐)
116
= ฮ b ฮ 1 +
2๐ ฮ ๐ โ
2๐
ฮ2 1 +1๐ ฮ2 ๐ โ
1๐
โฮ2 1 +
1๐ ฮ2 ๐ โ
1๐
ฮ2 1 +1๐ ฮ2 ๐ โ
1๐
ฮ b ฮ 1 +
2๐ ฮ ๐ โ
2๐
ฮ2 1 +1๐ ฮ2 ๐ โ
1๐
โ 1
v. Skewness
Putting a=1in the Skewness of generalized beta distribution of the second kind
๐๐๐๐ค๐๐๐ ๐
= ๐3
๐ต 1 +
3๐ , ๐ โ
3๐
๐ต 1, ๐ โ 3
๐ต 1 +1๐ , ๐ โ
1๐
๐ต 1, ๐
๐ต 1 +2๐ , ๐ โ
2๐
๐ต 1, ๐
+ 2 ๐ต 1 +
1๐ , ๐ โ
1๐
๐ต 1, ๐๐
3
= ๐3 ๐๐ต 1 +3
๐, ๐ โ
3
๐ โ 3๐๐ต 1 +
1
๐, ๐ โ
1
๐ ๐๐ต 1 +
2
๐, ๐ โ
2
๐
+ 2 ๐๐ต 1 +1
๐, ๐ โ
1
๐
3
vi. Kurtosis
Kurtosis of Singh-Maddala distribution can be obtained from the Kurtosis of generalized
beta distribution of the second kind by letting a=1
๐พ๐ข๐๐ก๐๐ ๐๐ = ๐ธ ๐4 โ 4๐ธ(๐)๐ธ ๐3 + 6๐ธ{ ๐ }2๐ธ(๐2) โ3 {๐ธ ๐ }4
117
= ๐4
๐ต 1 +
4๐ , ๐ โ
4๐
๐ต 1, ๐ โ 4
๐ต 1 +1๐ , ๐ โ
1๐
๐ต 1, ๐
๐ต 1 +3๐ , ๐ โ
3๐
๐ต 1, ๐
+ 6 ๐ต 1 +
1๐ , ๐ โ
1๐
๐ต 1, ๐
2
๐ต 1 +2๐ , ๐ โ
2๐
๐ต 1, ๐ โ 3
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต 1, ๐
4
= ๐4 ๐๐ต 1 +4
๐, ๐ โ
4
๐ โ 4๐2๐ต 1 +
1
๐, ๐ โ
1
๐ ๐ต 1 +
3
๐, ๐ โ
3
๐
+ 6 ๐๐ต 1 +1
๐, ๐ โ
1
๐
2
๐๐ต 1 +2
๐, ๐ โ
2
๐
โ 3 ๐๐ต ๐ +1
๐, ๐ โ
1
๐
4
5.6.3 Shapes of Singh-Maddala distribution
0
1
2
3
4
5
6
7
8
0
0.0
4
0.0
8
0.1
2
0.16 0.
2
0.2
4
0.2
8
0.3
2
0.36 0.
4
0.4
4
0.4
8
0.5
2
0.5
6
0.6
0.6
4
0.6
8
0.7
2
0.7
6
0.8
0.8
4
0.8
8
0.9
2
0.9
6 1(a,q,b)=(1,1,1)
(a,q,b)=(1,2,1)
(a,q,b)=(1,3,1)
(a,q,b)=(2,1,1)
(a,q,b)=(3,1,1)
(a,q,b)=(0.5,2,1)
118
5.6.4 Generalized Lomax (Pareto type II) Distribution
Special case of the generalized beta distribution of the second kind when p=1, ๐ =1
ฮป and
๐ = 1 in equation 5.10 gives
๐ y; ฮป, b = 1 ๐ฆ1(1)โ1
(1๐)1.1๐ต(1, ๐)(1 + (
๐ฆ1๐
)1)1+b
=๐๐
(1 + ๐๐ฆ)1+b, ๐ฆ > 0, __________(5.13)
The CDF is given by
F y = 1 โ (1 + ๐๐ฆ)โb
Equation 5.13 is the general expression of Lomax distribution which becomes equation
3.8 when ๐ = 1.
5.6.5 Properties of Generalized Lomax distribution
i. rth-order moments
The rth
-order moments of the generalized Lomax distribution can be obtained from the
expression of the rth
โorder moments of the generalized beta distribution of the second
kind by substituting ๐ = 1, ๐ =1
๐, ๐ = 1 in equation 5.11.
๐ธ ๐๐ =
1๐
๐
ฮ(1 + ๐)ฮ(๐ โ ๐)
ฮ(b)
ii. Mean
The mean is given by
๐ธ ๐ =
1๐ ฮ 1 + 1 ฮ ๐ โ 1
ฮ b
119
=
1๐
๐ โ 2 !
b โ 1 (b โ 2)!=
1
๐ b โ 1 , ๐ > 1
iii. Mode
The mode of generalized Lomax distribution is given by
Mode =1
๐
1 โ 1
๐ + 1 = 0
iv. Variance
The variance of generalized Lomax distribution can be obtained from the variance of
generalized beta distribution of the second kind by substituting ๐ = 1, ๐ =1
๐ and ๐ = 1
Var Y =
1๐
2
๐ต 1 + 2, ๐ โ 2
๐ต 1, ๐ โ
๐๐ต 1 + 1, ๐ โ 1
๐ต 1, ๐
2
=2
1๐
2
b โ 3 !
b โ 1 b โ 2 b โ 3 !โ
1๐ ๐ โ 2 !
b โ 1 b โ 2 !
2
= 1
๐
2 1
b โ 1
2
b โ 2 โ
1
b โ 1
= 1
๐
2 1
b โ 1
2 ๐ โ 1 โ (๐ โ 2)
b โ 2 (b โ 1)
=b
๐2 b โ 1 2 b โ 2 , ๐ > 2
Hence the coefficient of variation is given by
๐ถ๐ =๐
๐
๐ถ๐ =
b๐2 b โ 1 2 b โ 2
1๐ b โ 1
120
= ๐2 b โ 1 2b
๐2 b โ 1 2 b โ 2
= b
b โ 2
v. Skewness
Skewness of generalized Lomax distribution is given by
๐ ๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
=
1๐
3
ฮ(4)ฮ(๐ โ 3)
ฮ(b)โ 3
1๐
2
ฮ(3)ฮ(๐ โ 2)
ฮ(b)
1
๐ b โ 1 + 2
1
๐3 b โ 1 3
=1
๐3
6
b โ 1 b โ 2 b โ 3 โ
6
b โ 1 2 b โ 2 +
2
b โ 1 3
=1
๐3
6 b2 โ 2b + 1 + 6 โb2 + 4b โ 3 + 2 b2 โ 5b + 6
b โ 1 3 b โ 2 b โ 3
=1
๐3
2b b + 1
b โ 1 3 b โ 2 b โ 3
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐ ๐๐๐ค๐๐๐ ๐ =2b(b + 1)
๐3 b โ 1 3 b โ 2 (b โ 3).
1
b
๐2 b โ 1 2 b โ 2
3
=2b(b + 1)
๐3 b โ 1 3 b โ 2 (b โ 3). ๐3 b โ 1 3
b โ 2
b
3
=2(b + 1)
(b โ 3)
b โ 2
b
vi. Kurtosis
Kurtosis of generalized Lomax distribution can be obtained from the kurtosis of
generalized beta distribution of the second kind by substituting ๐ = 1, ๐ =1
๐ and ๐ = 1
121
๐พ๐ข๐๐ก๐๐ ๐๐
= 1
๐
4
๐ต 1 + 4, ๐ โ 4
๐ต 1, ๐ โ 4
๐ต 1 + 1, ๐ โ 1
๐ต 1, ๐
๐ต 1 + 3, ๐ โ 3
๐ต 1, ๐
+ 6 ๐ต 1 + 1, ๐ โ 1
๐ต 1, ๐
2๐ต 1 + 2, ๐ โ 2
๐ต 1, ๐ โ 3
๐ต 1 + 1, ๐ โ 1
๐ต 1, ๐
4
= 1
๐
4
๐๐ต 5, ๐ โ 4 โ 4๐๐ต 2, ๐ โ 1 ๐๐ต 4, ๐ โ 3 + 6 ๐๐ต 2, ๐ โ 1 2๐๐ต 3, ๐ โ 2
โ 3 ๐๐ต 2, ๐ โ 1 4
= 1
๐
4
24
b โ 1 b โ 2 b โ 3 (b โ 4)โ
4
(b โ 1)
6
b โ 1 b โ 2 (b โ 3)
+ 6 1
(b โ 1)
2 2
b โ 1 (b โ 2)โ 3
1
(b โ 1)
4
= 1
๐
4
24
b โ 1 b โ 2 b โ 3 (b โ 4)โ
24
(b โ 1)2 b โ 2 (b โ 3) +
12
(b โ 1)3(b โ 2)
โ3
(b โ 1)4
5.6.6 Inverse Lomax Distribution
Special case of the generalized beta distribution of the second kind when b=1and p=1 in
equation 5.10 gives
๐ y; q, a =๐๐ฆ๐โ1
๐๐(1 +๐ฆ๐)1+a
, ๐ฆ > 0, ๐ > 0, ๐ > 0__________________(5.14)
Equation 5.14 is the inverse Lomax distribution
122
5.6.7 Properties of Inverse Lomax distribution
i. rth-order moments
The rth-order moments of Inverse Lomax distribution can be obtained from the expression
of the rth
โorder moments of the generalized beta distribution of the second kind by
substituting ๐ = 1, ๐ = 1 in equation 5.11.
๐ธ ๐๐ = ๐ ๐ฮ(๐ + ๐)ฮ(1 โ ๐)
ฮ(๐)ฮ(1)
ii. Mean
The mean of inverse Lomax distribution is given by
๐ธ ๐ = ๐ ฮ(๐ + 1)ฮ(1 โ 1)
ฮ(๐)ฮ(1)
iii. Mode
The mode of inverse Lomax distribution is given by
๐๐๐๐ = ๐ ๐ โ 1
2
iv. Variance
The variance of inverse Lomax distribution can be obtained from the variance of
generalized beta distribution of the second kind by substituting ๐ = 1, and ๐ = 1
Var Y = ๐2 ๐๐ต 3, ๐ โ 2 โ ๐๐ต 2, ๐ โ 1 2
Var(Y) = ๐2 ๐๐ต ๐ + 2,1 โ 2 โ ๐๐ต ๐ + 1,1 โ 1 2
= ๐2 ๐๐ต ๐ + 2, โ1 โ ๐๐ต ๐ + 1,0 2
v. Skewness
The skewness of inverse Lomax distribution can be obtained from the skewness of
generalized beta distribution of the second kind by substituting ๐ = 1, and ๐ = 1
= ๐3 ๐ต ๐ + 3,1 โ 3
๐ต ๐, 1 โ 3
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
๐ต ๐ + 2,1 โ 2
๐ต ๐, 1 + 2
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
3
123
vi. Kurtosis
The Kurtosis of inverse Lomax distribution can be obtained from the kurtosis of
generalized beta distribution of the second kind by substituting ๐ = 1, and ๐ = 1
๐พ๐ข๐๐ก๐๐ ๐๐
= ๐4 ๐ต ๐ + 4,1 โ 4
๐ต ๐, 1 โ 4
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
๐ต ๐ + 3,1 โ 3
๐ต ๐, 1
+ 6 ๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
2๐ต ๐ + 2,1 โ 2
๐ต ๐, 1 โ 3
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
4
5.6.8 Dagum (Burr type 3) Distribution
Special case of the generalized beta distribution of the second kind with b=1 in equation
5.10 gives the following density function
๐ y; p, q, a = ๐ ๐๐ฆ๐๐โ1
๐๐๐ (1 + (๐ฆ๐
)๐)1+a, ๐ฆ > 0, ๐ > 0, ๐ > 0,
๐ > 0____________(5.15)
Equation 5.15 is the Dagum distribution. The Dagum distribution is also known as Burr
Type 3 distribution in the statistic literature (Kleiber and Kotz, 2003)
5.6.9 Properties of Dagum distribution
i. rth-order moments
The rth-order moments of the Dagum distribution can be obtained from the expression of
the generalized beta distribution of the second kind by substituting b=1 in equation 5.11.
Thus
๐ธ ๐๐ =๐๐๐ต ๐ +
๐๐
, 1 โ๐๐
๐ต ๐, 1
=๐๐ฮ(๐ +
๐๐)ฮ(1 โ
๐๐)
ฮ(a), โ๐๐ < ๐ < ๐
124
ii. Mean
The mean is given by
๐ธ ๐ =๐ฮ(๐ +
1๐)ฮ(1 โ
1๐)
ฮ(a)
iii. Mode
The mode of the Dagum distribution is given by
Mode = ๐ ๐๐ โ 1
๐ + 1
1๐
if ๐๐ โฅ 1
and at zero elsewhere.
iv. Variance
Var Y = ๐2 =๐2๐ต ๐ +
2๐ , 1 โ
2๐
๐ต ๐, 1 โ
๐๐ต ๐ +1๐ , 1 โ
1๐
๐ต ๐, 1
2
=๐2aฮ ๐ +
2๐ ฮ 1 โ
2p
aฮ(๐)โ
๐๐ฮ(๐ +1๐)ฮ 1 โ
1๐
aฮ ๐
2
= ๐2 ฮ(a)ฮ ๐ +
2๐ ฮ 1 โ
2๐ โ ฮ2(๐ +
1๐)ฮ2 1 โ
1๐
ฮ2(๐)
Hence the coefficient of variation is
๐ถ๐ = ฮ(a)ฮ ๐ +
2๐ ฮ 1 โ
2๐
ฮ2(๐ +1๐)ฮ2 1 โ
1๐
โ 1
125
5.6.10 Para-logistic distribution
Special case of the generalized beta distribution of the second kind with a=1 and b=p in
equation 5.10 (Klugman, Panjer and Willmot, 1998)
๐ y; p, q = ๐ ๐ฆ๐โ1
๐๐๐ต 1, ๐ (1 + (๐ฆ๐)๐)1+p
=๐2๐ฆ๐โ1
๐๐(1 + (๐ฆ๐)๐)1+p
, ๐ฆ > 0__________________(5.16)
5.6.11 General Fisk (Log-Logistic) Distribution Function
Special case of the generalized beta distribution of the second kind when a=b=1 and
๐ =1
๐ in equation 5.10
๐ y; p, ๐ = ๐ ๐ง๐ .1โ1
1๐
๐ .1
๐ต(1,1)(1 + (๐ฆ1๐
)๐)1+1
=๐ ๐ (๐๐ฆ)๐โ1
(1 + (๐ฆ๐)๐)2 , ๐ฆ > 0 ___________________(5.17)
Equation 5.17 is the general form of the log logistic distribution. The Fisk (Log-logistic)
distribution can also be obtained from Singh-Maddala distribution by letting b=1 and q= 1
๐
or from Dagum distribution by letting a=1 and ๐ =1
๐
126
5.6.12 Properties of Fisk (log-logistic) distribution
i. rth-order moments
The rth-order moments is obtained from the r
thโorder moments of the generalized beta
distribution of the second kind by substituting a=b=1 in equation 5.11. Thus we get
๐ธ ๐๐ = ๐๐๐ต 1 +๐
๐, 1 โ
๐
๐
= ๐๐ฮ 1 +๐
๐ ฮ 1 โ
๐
๐ , โ๐ < ๐ < ๐
ii. Mean
The mean is given by
๐ธ ๐ = ๐ฮ 1 +1
๐ ฮ 1 โ
1
๐
= ๐1
๐ฮ
1
๐ ฮ 1 โ
1
๐
iii. Mode
The mode of the Fisk distribution is given by
Mode = ๐ ๐ โ 1
๐ + 1
1๐
if ๐ > 1
and at zero for ๐ โค 1.
iv. Variance
Var Y = ๐2 = ๐2๐ต 1 +2
๐, 1 โ
2
๐ โ ๐๐ต 1 +
1
๐, 1 โ
1
๐
2
127
= ๐2ฮ 1 +2
๐ ฮ 1 โ
2
p โ ๐ฮ(1 +
1
๐)ฮ 1 โ
1
๐
2
= ๐2 ฮ 1 +2
๐ ฮ 1 โ
2
๐ โ ฮ2(1 +
1
๐)ฮ2 1 โ
1
๐
5.6.13 Fisher (F) Distribution Function
Special case of the generalized beta distribution of the second kind when ๐ =๐ข
2, ๐ =
๐ฃ
2, ๐ = 1 and ๐ =
๐ฃ
๐ข in equation 5.10
๐ y; ๐ข, ๐ฃ = 1 ๐ฆ1.
๐ข2โ1
๐ฃ๐ข
1.๐ข2๐ต
๐ข2 ,
๐ฃ2 1 +
๐ฆ๐ฃ๐ข
1
๐ข2
+๐ฃ2
=ฮ
๐ข + ๐ฃ2
๐ข๐ฃ
๐ข2๐ฆ
๐ข2โ1
ฮ ๐ข2 ฮ
๐ฃ2 1 +
๐ฆ๐ข๐ฃ
๐ข+๐ฃ2
___________________(5.18)
Equation 5.18 is the Fisher (F) distribution.
5.6.14 Properties of F distribution
i. rth-order moments
The rth-order moments is obtained from the r
thโorder moments of the generalized beta
distribution of the second kind by substituting ๐ =๐ข
2, ๐ =
๐ฃ
2, ๐ = 1 and ๐ =
๐ฃ
๐ข in equation
5.11. Thus we get
๐ธ ๐๐ = ๐ฃ๐ข
๐
๐ต ๐ข2
+ ๐,๐ฃ2
โ ๐
๐ต ๐ข2 ,
๐ฃ2
= ๐ฃ๐ข
๐
ฮ ๐ข2 + ๐ ฮ
๐ฃ2 โ ๐
ฮ ๐ข2 ฮ
๐ฃ2
, โ๐ข
2< ๐ < 1
128
ii. Mean
The mean of F distribution is given by
๐ธ ๐ = ๐ฃ๐ข ฮ
๐ข2 + 1 ฮ
๐ฃ2 โ 1
ฮ ๐ข2 ฮ
๐ฃ2
= ๐ฃ๐ข
๐ข2 ฮ
๐ข2
๐ฃ2 โ 2 !
ฮ ๐ข2
๐ฃ2 โ 1
๐ฃ2 โ 2 !
= ๐ฃ2
๐ฃ2 โ 1
=๐ฃ
๐ฃ โ 2, ๐ฃ > 2
iii. Mode
The mode of F distribution is given by
Mode =๐ฃ
๐ข
๐ข2 โ 1
๐ฃ2 + 1
=๐ฃ
๐ข ๐ข โ 2
๐ฃ + 2
= ๐ข โ 2
๐ข
๐ฃ
๐ฃ + 2 , ๐ข > 2
iv. Variance
The variance for F distribution can be given by
Var Y =(๐ฃ๐ข)2๐ต
๐ข2 + 2,
๐ฃ2 โ 2
๐ต(๐ข2 ,
๐ฃ2)
โ ๐ฃ
๐ฃ โ 2
2
129
=(๐ฃ๐ข)2ฮ
๐ข2 + 2 ฮ
๐ฃ2 โ 2 ฮ
u + v2
ฮ u + v
2 ฮ ๐ข2 ฮ
๐ฃ2
โ ๐ฃ
๐ฃ โ 2
2
=(๐ฃ๐ข)2
๐ข2 + 1
๐ข2 ฮ
๐ข2
๐ฃ2 โ 3 !
ฮ ๐ข2
๐ฃ2 โ 1
๐ฃ2 โ 2
๐ฃ2 โ 3 !
โ ๐ฃ
๐ฃ โ 2
2
=(๐ฃ๐ข
)2 ๐ข2
+ 1 ๐ข2
๐ฃ2 โ 1
๐ฃ2 โ 2
โ ๐ฃ
๐ฃ โ 2
2
= ๐ฃ2
1๐ข
๐ข + 22
12
๐ฃ โ 2
2
๐ฃ โ 42
โ
1
๐ฃ โ 2
2
= ๐ฃ2 ๐ข + 2
๐ข ๐ฃ โ 2 ๐ฃ โ 4 โ
1
๐ฃ โ 2
2
= ๐ฃ2 ๐ฃ๐ข + 2๐ฃ โ 2๐ข โ 4 โ ๐ข๐ฃ + 4๐ข
๐ข ๐ฃ โ 2 2 ๐ฃ โ 4
= 2๐ฃ2 ๐ข + ๐ฃ โ 2
๐ข ๐ฃ โ 2 2 ๐ฃ โ 4 , ๐ฃ > 4
5.6.15 Half Studentโs t distribution
Special case of the generalized beta distribution of the second kind when ๐ =1
2, ๐ =
๐
2, ๐ = 2 and ๐ = ๐ in equation 5.10
๐ y; ๐ = 2 ๐ฆ2.
12โ1
๐ 2.
12๐ต
12 ,
๐2 1 +
๐ง
๐
2
12
+๐2
=2ฮ
1 + ๐2
๐ฯฮ ๐2 1 +
๐ฆ2
๐
1+๐2
, โโ < ๐ฆ < โ ___________________(5.19)
Equation 5.19 is the Half studentโs t distribution.
130
5.6.16 Logistic Distribution Function
Letting ๐ฆ = ๐๐ฅ and ๐๐ฆ
๐๐ฅ = ๐๐ฅ
In equation 5.10 for the generalized beta distribution of the second kind, we get the
following density function
๐ x; p, ๐ = ๐ ๐๐๐๐ฅ โ๐ฅ
๐๐๐๐ต(๐, ๐)(1 + (๐๐ฅ
๐ )๐)a+b . ๐๐ฅ
= ๐ ๐๐๐๐ฅ
๐๐๐๐๐๐ (๐)๐ต(๐, ๐)(1 + (๐๐ฅ๐โlog (๐))๐)a+b
= ๐ ๐๐๐ ๐ฅโ๐๐๐ ๐
๐ต(๐, ๐) 1 + ๐๐ ๐ฅโlog q a+b, โโ < ๐ฅ < โ _________________(5.20)
Equation 5.20 is the generalized logistic distribution. McDonald and Xu (1995) referred
to equation 5.20 as the density of an exponential generalized distribution. The standard
logistic distribution can be obtained by setting p = q = a = b = 1 as
๐(๐ฅ) =๐๐ฅ
1 + ๐๐ฅ 2
5.6.17 Generalized Gamma Distribution Function
We show that the generalized beta distribution of the second kind approaches the
generalized gamma distribution as bโโ as follows:
Let q=ฮฒb1
๐ . Substituting this expression in equation 5.10 yields
๐(๐ฆ; ๐, ฮฒ, ๐, ๐) = ๐ ๐ฆ๐๐โ1
(ฮฒb1๐)๐๐๐ต ๐, ๐ (1 + (
๐ฆ
ฮฒb1๐
)๐)๐+๐
= ๐ ๐ฆ๐๐โ1
๐๐ฮฒ๐๐ ๐ต ๐, ๐ (1 +
1๐ (
๐ฆฮฒ
)๐)๐+๐
131
Grouping the terms gives
๐(๐ฆ; ๐, ฮฒ, ๐, ๐) = ๐ ๐ฆ๐๐โ1
ฮ(๐)ฮฒ๐๐
ฮ(๐ + ๐
ฮ ๐ b๐
1
(1 +1๐ (
๐ฆฮฒ
)๐)
๐+๐
For large values of b, the gamma function can be approximated by Stirlingโs formula
thus
ฮ(๐ + ๐)
ฮ(๐)b๐ =
๐โ๐โ๐(๐ + ๐)๐+๐โ12 2๐
๐โ๐๐๐โ12 2๐๐๐
limbโโ
ฮ(๐ + ๐)
ฮ(๐)b๐ = lim
bโโ
๐โ๐(๐ + ๐)๐โ12 2๐
๐๐โ12 2๐
= 1
Similarly, limbโโ
1
(1 +1๐ (
๐ฆฮฒ
)๐)๐+๐ = ๐
โ ๐ฆ๐ฝ
๐
taking the limit of ๐ ๐ฆ; ๐, ฮฒ, ๐, ๐ as ๐ โ โ yields
๐ y; p, ฮฒ, a = ๐ ๐ฆ๐๐โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
๐
, 0 โค ๐ฆ ______________________(5.21)
Equation 5.21 is the generalized gamma (GG) distribution function equivalent to the one
derived in equation 5.8.
5.6.18 Properties of Generalized Gamma distribution
i. rth-order moments
To obtain the rth-order moments of the generalized gamma distribution, we evaluate
132
๐ธ ๐๐ = ๐ฆ๐๐(๐ฆ; ๐, ฮฒ, a)๐๐ฆโ
0
= ๐ฆ๐ ๐ ๐ง๐๐โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
๐
๐๐ฆโ
0
= ๐ ๐ฆ(๐๐+๐)โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
๐
๐๐ฆโ
0
= ๐ ๐ฆ
๐(๐+๐๐
)โ1. ฮ ๐ +
๐๐
ฮฒ๐(๐+
๐๐
). ฮฒโ๐ . ฮ ๐ ฮ ๐ +
๐๐
๐โ
๐ง๐ฝ
๐
๐๐ฆโ
0
=ฮฒ
๐ฮ ๐ +
๐๐
ฮ ๐
๐ ๐ฆ๐(๐+
๐๐
)โ1
ฮฒ๐ (๐+
๐๐
)ฮ ๐ +
๐๐
๐โ
๐ฆ๐ฝ
๐
๐๐ฆโ
0
The expression under the integral sign integrates to 1
โด ๐ธ ๐๐ =ฮฒ
๐ฮ ๐ +
๐๐
ฮ ๐ , โpa < ๐ < โ________________________(5.22)
ii. Mean
The mean is given by
๐ธ ๐ =ฮฒฮ ๐ +
1๐
ฮ ๐
iii. Mode
The mode of the generalized gamma distribution occurs at
d
dy๐(๐ฆ) = 0
d
dy ๐ ๐ฆ๐๐โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
๐
= 0
๐๐ โ 1 ๐ฆ๐๐โ1. ๐ฆโ1๐โ
๐ฆ๐ฝ
๐
โ ๐๐ฆ๐โ1
๐ฝ๐. ๐ฆ๐๐โ1๐
โ ๐ฆ๐ฝ
๐
= 0
๐๐ โ 1 = ๐๐ฆ๐
๐ฝ๐
133
๐ฆ = ๐ฝ ๐ โ1
๐
1๐
and at zero for ๐ โค 1.
iv. Variance
Var Y =๐ฝ2ฮ ๐ +
2๐
ฮ(a)โ
๐ฝฮ ๐ +1๐
ฮ(a)
2
=๐ฝ2ฮ(a)ฮ ๐ +
2๐ โ ๐ฝ2ฮ2 ๐ +
1๐
ฮ2(a)
=๐ฝ2
ฮ2 a ฮ a ฮ ๐ +
2
๐ โ ฮ2 ๐ +
1
๐
v. Skewness
Skewness of the generalized gamma distribution can be expressed as follows
Skewness = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
=ฮฒ
3ฮ ๐ +
3๐
ฮ ๐ โ
3ฮฒ3ฮ ๐ +
2๐ ฮ ๐ +
1๐
ฮ2 ๐ +
2ฮฒ3ฮ3 ๐ +
1๐
ฮ3 ๐
=ฮฒ
3ฮ2 ๐ ฮ ๐ +
3๐ โ 3ฮฒ
3ฮ a ฮ ๐ +2๐ ฮ ๐ +
1๐ + 2ฮฒ
3ฮ3 ๐ +
1๐
ฮ3 ๐
vi. Kurtosis
Kurtosis of the generalized gamma distribution can be expressed as follows
Kurtosis = ๐ธ ๐4 โ 4๐ธ ๐3 ๐ธ ๐ + 6๐ธ ๐2 {๐ธ ๐ }2โ3 ๐ธ ๐ 4
=ฮฒ
4ฮ ๐ +
4๐
ฮ ๐ โ
4ฮฒ3ฮ ๐ +
3๐ ฮฒฮ ๐ +
1๐
ฮ ๐ ฮ(a)+
6ฮฒ2ฮ ๐ +
2๐ ฮฒ
2ฮ2 ๐ +
1๐
ฮ ๐ ฮ2 ๐
โ 3ฮฒ
4ฮ4 ๐ +
1๐
ฮ4 ๐
134
= ฮฒ4
ฮ ๐ +4๐
ฮ ๐ โ
4ฮ ๐ +3๐ ฮ ๐ +
1๐
ฮ2 ๐ +
6ฮ ๐ +2๐ ฮ2 ๐ +
1๐
ฮ3 ๐ โ 3
ฮ4 ๐ +1๐
ฮ4 ๐
5.6.19 Gamma Distribution Function
Special case of the generalized gamma distribution derived in equation 5.21 when p=1
gives
๐ y; ฮฒ, a =๐ฆ๐โ1
ฮฒ๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
, 0 โค ๐ฆ _________________(5.23)
Equation 5.23 is the gamma distribution.
5.6.20 Properties of Gamma distribution
i. rth-order moments
The rth-order moments of the Gamma distribution can be obtained from the expression of
the moments of the generalized Gamma distribution by substituting p=1 in equation 5.22.
Thus
๐ธ ๐๐ =ฮฒ
๐ฮ ๐ +
๐๐
ฮ ๐
=ฮฒ
๐ฮ ๐ + ๐
ฮ ๐
ii. Mean
The mean is given by
๐ธ ๐ =ฮฒฮ ๐ + 1
ฮ ๐ = ฮฒa
135
iii. Mode
The mode of the Gamma distribution occurs at
Mode = ๐ฝ ๐ โ1
๐
1๐
= ฮฒ a โ 1 , a > 1
iv. Variance
The variance of gamma distribution can be obtained from the variance of generalized
gamma distribution by substituting p=1 as follows:
Var Y =๐ฝ2
ฮ2 a ฮ a ฮ ๐ + 2 โ ฮ2 ๐ + 1
=๐ฝ2
ฮ2 a ฮ a a + 1 (a)ฮ ๐ โ (a)ฮ ๐ (a)ฮ ๐
=๐ฝ2
ฮ2 a (a)ฮ2 a a + 1 โ (a)
= a๐ฝ2
v. Skewness
๐๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
=ฮฒ
3ฮ ๐ + 3
ฮ ๐ โ 3
ฮฒ2ฮ ๐ + 2
ฮ ๐ . ๐ฮฒ + 2 ๐ฮฒ 3
=ฮฒ
3 a + 2 a + 1 (a)ฮ ๐
ฮ ๐ โ
3๐ฮฒ3 a + 1 (a)ฮ ๐
ฮ ๐ + 2 ๐ฮฒ 3
= ฮฒ3 a + 2 a + 1 (a) โ 3๐ฮฒ
3 a + 1 (a) + 2 ๐ฮฒ 3
= a3ฮฒ3 + 3a2ฮฒ
3 + 2๐ฮฒ3 โ 3a3ฮฒ
3 โ 3a2ฮฒ3 + 2a3ฮฒ
3
= 2๐ฮฒ3
The standardized skewness is given by
136
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐ ๐๐๐ค๐๐๐ ๐ =2๐ฮฒ
3
(a๐ฝ2)32
=2
๐
vi. Kurtosis
Kurtosis is given by
๐พ๐ข๐๐ก๐๐ ๐๐ = ๐ธ ๐4 โ 4๐ธ ๐3 ๐ธ ๐ + 6๐ธ ๐2 ๐ธ ๐ 2 โ 3 ๐ธ ๐ 4
= ฮฒ
4ฮ ๐ + 4
ฮ ๐ โ 4
ฮฒ3ฮ ๐ + 3
ฮ ๐ . ๐ฮฒ + 6
ฮฒ2ฮ ๐ + 2
ฮ ๐ . (๐ฮฒ)2 โ 3 ๐ฮฒ 4
= ฮฒ4 ๐ + 3 ๐ + 2 ๐ + 1 (๐) โ 4๐ฮฒ
4 ๐ + 2 ๐ + 1 (๐) + 6๐2ฮฒ4(๐ + 1)(๐)
โ 3๐4ฮฒ4
= ๐4ฮฒ4
+ 6๐3ฮฒ4
+ 11๐2ฮฒ4
+ 6๐ฮฒ4 โ 4๐4ฮฒ
4โ 12๐3ฮฒ
4โ 8๐2ฮฒ
4+ 6๐4ฮฒ
4+ 6๐3ฮฒ
4
โ 3๐4ฮฒ4
= 3๐2ฮฒ4
+ 6๐ฮฒ4
The standardized kurtosis is given by
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐๐ข๐ก๐๐ ๐๐ =3๐2ฮฒ
4+ 6๐ฮฒ
4
๐2ฮฒ4
= 3 +6
๐
5.6.21 Inverse Gamma distribution
Special case of the generalized gamma distribution derived in equation 5.21 when
๐ = โ1 gives
๐ y; ฮฒ, a =๐ฆโ๐โ1
ฮฒโ๐
ฮ ๐ ๐
โ ๐ฆ๐ฝ
โ1
, 0 โค ๐ฆ
=ฮฒ
๐
๐ฆ1+๐ฮ ๐ ๐
โ ๐ฝ๐ฆ , 0 โค ๐ฆ_________________(5.24)
Equation 5.24 is the inverse gamma distribution
137
5.6.22 Properties of Inverse Gamma distribution
i. rth-order moments
The rth-order moments of the Inverse Gamma distribution can be obtained from the
expression of the moments of the generalized Gamma distribution by substituting
๐ = โ1 in equation 5.22.
๐ธ ๐๐ =ฮฒ
๐ฮ ๐ โ ๐
ฮ ๐
ii. Mean
The mean is given by
๐ธ ๐ =ฮฒฮ ๐ โ 1
ฮ ๐
=ฮฒ ๐ โ 2 !
a โ 1 a โ 2 !
=ฮฒ
a โ 1
iii. Mode
The mode of the inverse Gamma distribution occurs at
Mode = ๐ฝ ๐ + 1 โ1
=ฮฒ
a + 1 , a > 1
iv. Variance
The variance of inverse gamma distribution can be obtained from the variance of
generalized gamma distribution by substituting ๐ = โ1 as follows:
Var ๐ =๐ฝ2
ฮ2 a ฮ a ฮ ๐ โ 2 โ ฮ2 ๐ โ 1
=๐ฝ2( ๐ โ 1 ! ๐ โ 3 ! โ ๐ โ 2 ! ๐ โ 2 !
a โ 1 ! a โ 1 !
138
= ๐ฝ2 1
a โ 1 a โ 2 โ
1
a โ 1 a โ 1
=๐ฝ2
(๐ โ 1)2 a โ 2 , ๐ > 2
v. Skewness
Skewness of inverse gamma distribution can be as follows
๐๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2 ๐ธ ๐ 3
=ฮฒ
3ฮ ๐ โ 3
ฮ ๐ โ 3
ฮฒ2ฮ ๐ โ 2
ฮ ๐ .
ฮฒ
a โ 1+ 2
ฮฒ
a โ 1
3
=ฮฒ
3 a โ 4 !
a โ 1 a โ 2 a โ 3 a โ 4 !โ
3ฮฒ3 a โ 3 !
a โ 1 2 a โ 2 a โ 3 ! +
2๐ฝ3
(๐ โ 1)3
=ฮฒ
3
a โ 1 a โ 2 a โ 3 โ
3ฮฒ3
a โ 1 2 a โ 2 +
2๐ฝ3
(๐ โ 1)3
= ฮฒ3
๐2 โ 2๐ + 1 โ 3๐2 โ 12๐ + 9 + 2 ๐2 โ 5๐ + 6
a โ 1 3 a โ 2 a โ 3
= ฮฒ3
4
a โ 1 3 a โ 2 a โ 3
=4ฮฒ
3
a โ 1 3 a โ 2 a โ 3
The standardized skewness is given by
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐ ๐๐๐ค๐๐๐ ๐ =
4ฮฒ
3
a โ 1 3 a โ 2 a โ 3
(๐ฝ2
(๐ โ 1)2 a โ 2 )32
=4ฮฒ
3
a โ 1 3 a โ 2 a โ 3
a โ 1 3 a โ 2 a โ 2
ฮฒ3
139
=4 (a โ 2)
a โ 3
vi. Kurtosis
Kurtosis is given by
๐พ๐ข๐๐ก๐๐ ๐๐ = ๐ธ ๐4 โ 4๐ธ ๐3 ๐ธ ๐ + 6๐ธ ๐2 ๐ธ ๐ 2 โ 3 ๐ธ ๐ 4
= ฮฒ
4ฮ ๐ โ 4
ฮ ๐ โ 4
ฮฒ3ฮ ๐ โ 3
ฮ ๐ .
ฮฒ
a โ 1+ 6
ฮฒ2ฮ ๐ โ 2
ฮ ๐ .
ฮฒ
a โ 1
2
โ 3 ฮฒ
a โ 1
4
=ฮฒ
4
๐ โ 1 ๐ โ 2 ๐ โ 3 ๐ โ 4 โ
4ฮฒ4
(๐ โ 1)2 ๐ โ 2 ๐ โ 3 +
6ฮฒ4
๐ โ 1 3(๐ โ 2)
โ 3 ฮฒ
a โ 1
4
= ฮฒ4
๐ โ 1 3 โ 4 ๐ โ 1 2 ๐ โ 4 +6 ๐ โ 1 ๐ โ 3 ๐ โ 4 โ 3 ๐ โ 2 ๐ โ 3 ๐ โ 4
๐ โ 1 4 ๐ โ 2 ๐ โ 3 ๐ โ 4
= ฮฒ4
๐3 โ 3๐2 + 3๐ โ 1 โ 4๐3 + 24๐2 โ 36๐ + 16 + 6๐3 โ 48๐2 + 114๐โ72โ3๐3 + 27๐2 โ 78๐ + 72 ๐ โ 1 4 ๐ โ 2 ๐ โ 3 ๐ โ 4
= ฮฒ4
3๐ + 15
๐ โ 1 4 ๐ โ 2 ๐ โ 3 ๐ โ 4
The standardized kurtosis is given by
๐๐ก๐๐๐๐๐๐๐๐ง๐๐ ๐๐ข๐๐ก๐๐ ๐๐ = ฮฒ4
3๐ + 15
๐ โ 1 4 ๐ โ 2 ๐ โ 3 ๐ โ 4 .
(๐ โ 1)4 a โ 2 2
๐ฝ4
=3(๐2 + 3๐ โ 10)
๐ โ 3 ๐ โ 4
=3(๐ โ 2)(๐ + 5)
๐ โ 3 ๐ โ 4 , ๐ > 4
140
5.6.23 Weibull Distribution Function
Special case of the generalized gamma distribution derived in equation 5.21 when a=1
๐ y; p, ฮฒ = ๐ ๐ฆ๐โ1
ฮฒ๐ ๐
โ ๐ฆ๐ฝ
๐
, 0 โค ๐ฆ
= ๐
ฮฒ๐ ๐ฆ๐โ1 ๐
โ ๐ฆ๐ฝ
๐
_________________(5.25)
The cdf of the distribution in equation 5.25 is given by
๐น ๐ฆ = 1 โ ๐โ
๐ฆ๐ฝ
๐
Equation 5.25 is the Weibull distribution with ๐ > 0 as the shape parameter and ๐ฝ > 0 as
the scale parameter.
5.6.24 Properties of Weibull distribution
i. rth-order moments
The rth-order moments of the Weibull distribution can be obtained from the expression of
the moments of the generalized Gamma distribution by substituting a=1 in equation 5.22.
Thus
๐ธ ๐ =ฮฒ
๐ฮ 1 +
๐๐
ฮ 1
= ฮฒ๐ฮ 1 +
๐
๐
ii. Mean
The mean is given by
๐ธ ๐ = ฮฒฮ 1 +1
๐ =
ฮฒ
pฮ
1
p
iii. Mode
The mode occurs at
141
๐
๐๐ฆ๐ ๐ฆ = 0
๐
๐๐ง ๐
ฮฒ๐ ๐ฆ๐โ1 ๐
โ ๐ฆ๐ฝ
๐
= 0
๐
ฮฒ๐ ๐ โ 1 ๐ฆ๐โ2 ๐
โ ๐ฆ๐ฝ
๐
โ๐๐ฆ๐โ1
๐ฝ๐๐
โ ๐ฆ๐ฝ
๐
. ๐ฆ๐โ1 = 0
(๐ โ 1)๐ฆโ1 =๐
๐ฝ๐. ๐ฆ๐โ1
๐ฝ๐(๐ โ 1) = ๐๐ฆ๐
๐ฆ = ๐ฝ ๐ โ 1
๐
1๐
, ๐ > 0
iv. Variance
Var(๐) = ๐ธ ๐2 โ (๐ธ(๐))2
= ฮฒ2ฮ 1 +
2
๐ โ ฮฒ
2ฮ2 1 +
1
๐
= ฮฒ2ฮ 1 +
2
๐ โ ฮผ2
5.6.25 Exponential distribution Function
Special case of the generalized gamma distribution in equation 5.21 with p=a=1 gives
๐(y; p = 1, ฮฒ, a = 1) = 1 ๐ง1โ1
ฮฒ1ฮ(1)
๐โ(
๐ฆ๐ฝ
)1
0 โค ๐ฆ
=1
ฮฒ ๐
โ(๐ฆ๐ฝ
) , 0 โค ๐ฆ_________________(5.26)
and the cdf is given by
๐น ๐ = 1 โ ๐โ(
๐ฆ๐ฝ
)
142
Equation 5.26 is the exponential distribution. This exponential distribution can be
expressed as a special case of gamma distribution in equation 5.23 with a=1 and also a
special case of Weibull distribution in equation 5.25 with p=1.
5.6.26 Properties of exponential distribution
i. rth-order moments
The rth
-order moments for the exponential distribution can be obtained from the
expression of moments of the generalized gamma distribution in equation 5.22 by
substituting p = a =1
๐ธ ๐๐ =ฮฒ
๐ฮ 1 +
๐1
ฮ 1
= ฮฒ๐ฮ 1 + ๐
= ฮฒ๐rฮ ๐
ii. Mean
The mean is given by
๐ธ ๐ = ฮฒ1(1)
= ฮฒ
iii. Mode
The mode of the exponential distribution occurs at
๐
๐๐ฆ๐ ๐ฆ = 0
=๐
๐๐ฆ
1
ฮฒ๐
โ๐ฆฮฒ = 0
=1
ฮฒ โ
1
ฮฒ๐
โ๐ฆฮฒ = 0
143
โ๐ฆ
ฮฒ= 0
๐ฆ = 0
iv. Variance
From the rth-order moments given by
E(Yr) = ฮฒ๐ r
the variance is given by ฯ2 = 2ฮฒ2 โ (ฮฒ)2
= ฮฒ2
5.6.27 Chi-Squared distribution function
Special case of the generalized gamma distribution in equation 5.21 when p=1, a =n
2 and
ฮฒ = 2 gives
๐ y; n = 1 ๐ฆ
n2โ1
2n2ฮ
n2
๐โ ๐ฆ2
1
, 0 โค ๐ฆ
=1
2n2ฮ
n2 ๐ฆ
n2โ1๐โ
๐ฆ2 , 0 โค ๐ฆ _________________(5.27)
Equation 5.27 is the chi-squared distribution. Chi-squared distribution is a special case of
the gamma distribution in equation 5.24 when ๐ =๐
2 and ๐ฝ = 2
5.6.28 Properties of Chi-squared distribution
i. rth-order moments
The rth
-order moments for the Chi-squared distribution can be obtained from the
expression of moments of the generalized gamma distribution in equation 5.22 by
substituting ๐ = 1, a =n
2 and ฮฒ = 2
๐ธ ๐๐ =2๐ฮ
n2 +
๐1
ฮ n2
144
=2๐ฮ
n2 + ๐
ฮ n2
ii. Mean
The mean is given by
๐ธ ๐ =2ฮ
n2 + 1
ฮ n2
=2(
n2)ฮ
n2
ฮ n2
= n
iii. Mode
The mode of the Chi-squared distribution can be obtained from the mode of generalized
gamma distribution by substituting ๐ = 1, a =n
2 and ฮฒ = 2 and is given by
๐ฆ = 2 n
2โ 1
= n โ 2
iv. Variance
The variance of Chi-squared distribution can be obtained from the variance of
generalized gamma distribution by substituting ๐ = 1, a =n
2 and ฮฒ = 2
Var(๐) =4
ฮ2 n2
ฮ n
2 ฮ
n
2+ 2 โ ฮ2
n
2+ 1
= 4 n
2
n
2+ 1 โ
n
2
2
= 2n
145
5.6.29 Rayleigh distribution function
Special case of the generalized gamma distribution in equation 5.21 when p=2, a = 1 and
ฮฒ = ฮฑ 2 gives
๐ y; ฮฑ = 2 ๐ฆ2โ1
(ฮฑ 2)2ฮ 1 ๐
โ ๐ฆ
ฮฑ 2
2
, 0 โค ๐ฆ
=๐ฆ
ฮฑ2 ๐
โ ๐ฆ
ฮฑ 2
2
, 0 โค ๐ฆ_________________(5.28)
The cdf is given by
๐น y = 1 โ ๐โ
๐ฆ2
2ฮฑ2 , 0 โค ๐ฆ
Equation 5.28 is the Rayleigh distribution. Rayleigh distribution is a special case of
Weibull distribution with p=2 and ฮฒ=ฮฑ 2 in equation 5.25.
5.6.30 Properties of Rayleigh distribution
i. rth-order moments
The rth
-order moments for the Rayleigh distribution can be obtained from the expression
of moments of the generalized gamma distribution in equation 5.22 by substituting p=2,
a=1 and ฮฒ = ฮฑ 2
๐ธ ๐๐ = ฮฑ 2
๐ฮ 1 +
๐2
ฮ 1
= ฮฑ 2 ๐ฮ 1 +
๐
2
ii. Mean
The mean is given by
146
๐ธ ๐ = ฮฑ 2 ฮ 1 +1
2
=ฮฑ
2 2ฯ = ฮฑ
ฯ
2
iii. Mode
The mode of the Rayleigh distribution occurs at
๐
๐๐ฆ๐ ๐ฆ = 0
=๐
๐๐ฆ
๐ฆ
ฮฑ2๐
โ๐ฆ2
2ฮฑ2 = 0
=1
ฮฑ2 ๐
โ๐ฆ2
2ฮฑ2 โ ๐ฆ2๐ฆ
2ฮฑ2๐
โ๐ฆ2
2ฮฑ2 = 0
1 โ๐ฆ2
ฮฑ2= 0
๐ฆ = ฮฑ
iv. Variance
From the rth-order moments given by
E(Yr) = ฮฑ 2 ๐ฮ 1 +
๐
2
the variance is given by ฯ2 = ฮฑ 2 2ฮ 1 +
2
2 โ ฮฑ 2 ฮ 1 +
1
2 2
=4ฮฑ2 โ ฮฑ2๐
2
= ฮฑ2 4 โ ๐
2
5.6.31 Maxwell distribution function
Special case of the generalized gamma distribution in equation 5.21 when p=2, a =3
2 and
ฮฒ = ฮฑ 2 gives
147
๐ y; ฮฑ = 2 ๐ฆ2(
32
)โ1
(ฮฑ 2)2(32
)ฮ
32
๐โ
๐ฆ
ฮฑ 2
2
, 0 โค ๐ฆ
=2๐ฆ2
ฮฑ 2 3ฮ
12 + 1
๐โ๐ฆ2/(ฮฑ 2)2
=2๐ฆ2
2 2 ฮฑ3 12 ฮ
12
๐โ
๐ฆ2
2โ2
=2๐ฆ2
2 ฮฑ3 ฯ ๐
โ ๐ฆ2
2โ2
=๐ฆ2
ฯ2 ฮฑ3
๐โ
๐ฆ2
2โ2
= 2
ฯ
๐ฆ2๐โ
๐ฆ2
2โ2
ฮฑ3 , 0 โค ๐ฆ_________________(5.29)
Equation 5.29 is the Maxwell distribution.
5.6.32 Properties of Maxwell distribution
i. rth-order moments
The rth
-order moments for the Maxwell distribution can be obtained from the expression
of moments of the generalized gamma distribution in equation 5.22 by substituting p=2,
a=3/2 and ฮฒ = ฮฑ 2
๐ธ ๐๐ = ฮฑ 2
๐ฮ
32 +
๐2
12 ฯ
148
ii. Mean
The mean is given by
๐ธ ๐ = ฮฑ 2 ฮ
32 +
12
12 ฯ
=2ฮฑ 2
ฯ
iii. Mode
The mode of the Maxwell distribution occurs at
๐
๐๐ฆ๐ ๐ฆ = 0
=๐
๐๐ฆ ๐ฆ2๐
โ๐ฆ2
2ฮฑ2 = 0
2๐ฆ๐โ
๐ฆ2
2ฮฑ2 โ ๐ฆ22๐ฆ
2ฮฑ2๐
โ๐ฆ2
2ฮฑ2 = 0
2๐ฆ โ๐ฆ3
ฮฑ2= 0
๐ฆ = 2ฮฑ
iv. Variance
From the rth-order moments given by
E(Yr) = ฮฑ 2 ๐ฮ
1
2+
๐
2
the variance is given by
๐๐๐(๐) = ฮฑ 2
2ฮ
32 +
22
12 ฯ
โ 2ฮฑ 2
ฯ
2
149
= 3ฮฑ2 โ8ฮฑ2
ฯ
= ฮฑ2 3ฯ โ 8
ฯ
5.6.33 Half Standard normal function
Special case of the generalized gamma distribution in equation 5.21 when p=2, ๐ =1
2 and
ฮฒ = 2
๐ y = 2 ๐ฆ2(
12
)โ1
2 2(
12
)ฮ
12
๐โ
๐ฆ
2
2
=2
2ฯ ๐โ
๐ฆ2
2 , 0 โค ๐ฆ_________________(5.30)
5.6.34 Properties of Half Standard normal distribution
i. rth-order moments
The rth-order moments of the Half-standard normal distribution can be obtained from the
expression of the moments of the generalized gamma distribution by substituting p=2,
a =1
2 and ฮฒ = 2 in equation 5.22 as follows
๐ธ ๐๐ = 2
๐ฮ
12 +
๐2
ฮ 12
ii. Mean
The mean is given by
๐ธ ๐ = 2
1ฮ
12 +
12
ฮ 12
= 2
ฯ
150
iii. Mode
The mode is given by
๐ฆ = 2 1
2โ
1
2
12
= 0
iv. Variance
Variance is given by
Var Y = 2
2ฮ
12 +
22
ฮ 12
โ2
ฯ
= 1
5.6.35 Half-normal distribution function
Special case of the generalized gamma distribution in equation 5.21 when ๐ = 2, ๐ =1
2
and ฮฒ = ฯ 2 gives
๐ y; ฯ = 2 ๐ฆ2(
12
)โ1
(ฯ 2)2(12
)ฮ
12
๐โ
๐ฆ
ฯ 2
2
, 0 โค ๐ฆ
=2
ฯ 2 ฮ 12
๐โ๐ฆ2/(ฯ 2)2
=2
ฯ 2 ฯ ๐
โ ๐ฆ2
2๐2
= 2
๐ ฯ ๐
โ ๐ฆ2
2๐2
= 2
ฯ
๐โ
๐ฆ2
2๐2
๐ , 0 โค ๐ฆ_________________(5.31)
Equation 5.31 is the Half-normal distribution.
151
5.6.36 Properties of Half-normal distribution
i. rth-order moments
The rth-order moments of the Half-normal distribution can be obtained from the
expression of the moments of the generalized gamma distribution by substituting p=2,
a =1
2 and ฮฒ = ฯ 2 in equation 5.22 as follows
๐ธ ๐๐ = ฯ 2
๐ฮ
12 +
๐2
ฮ 12
ii. Mean
The mean is given by
๐ธ ๐ = ฯ 2 ฮ
12 +
12
ฯ
=ฯ 2
ฯ
iii. Mode
The mode of the Half-normal distribution is given by
๐ฆ = ฯ 2 1
2โ
1
2
12
= 0
iv. Variance
Variance is given by
Var Y = ฯ 2
2ฮ
12 +
22
ฮ 12
โ2ฯ2
ฯ
= ฯ2 1 โ2
ฯ
152
5.6.37 Half Log-normal distribution function
Letting
๐ฆ = (ln x โ ฮผ) and ๐๐ฆ
๐๐ฅ =
1
x in equation 5.21 we get
๐ x; ฮผ, ฯ = ๐ ln ๐ฅ โ ๐ pa โ1
๐ฝpa ฮ ๐ ๐
โ ln ๐ฅ โ๐
๐ฝ ๐
.1
๐ฅ
Substituting p=2, a =1
2 and ฮฒ = ฯ 2 gives
๐ x; ฮผ, ฯ =2
ฯ๐ฅ 2ฯ ๐
โ(ln ๐ฅ โ๐ )2
2ฯ2
๐ x; ฮผ, ฯ =2
ฯ๐ฅ 2ฯ ๐
โ ln ๐ฅ โ๐ 2
2ฯ2 , ๐ฅ > 0_________________(5.32)
Equation 5.32 is the Half Log-normal distribution. The log-normal distribution can be
obtained as a limiting case of the generalized gamma distribution by setting ฮฒ = (ฯ2p2)1
p
and a =pฮผ+1
ฯ2p2 as p โ 0.
5.6.38 Properties of Half Log-normal distribution
i. rth-order moments
The rth-order moments of the Half Log-normal distribution is given by
๐ธ ๐๐ = e๐๐+12๐2๐2
ii. Mean
The mean is given by
๐ธ ๐ = e๐+12๐2
153
iii. Mode
The mode of the Half Log-normal distribution is given by
๐ฆ = e๐ +๐2
iv. Variance
Var Y = e2๐ +2๐2โ e2๐ +๐2
= e2๐ +๐2 e๐2
โ 1
๐ถ๐ =e๐+
12๐2
e๐2โ 1
12
e๐+12๐2
= e๐2โ 1
12
5.7 Four Parameter Generalized Beta Distribution Function
From the beta distribution with two shape parameters a and b in equation
(2.1), ๐(๐ฅ; ๐, ๐) =๐ฅ๐โ1(1โ๐ฅ)๐โ1
๐ต ๐ ,๐ ), supported on the range 0 โ 1. The location and scale
parameters of the distribution can be altered by introducing two further parameters p and
q representing the minimum and maximum values of the distribution respectively using
the following transformation:-
๐ฅ =๐ฆ โ ๐
๐ โ ๐
๐๐ฅ
๐๐ฆ =
1
๐ โ ๐
โด ๐ ๐ฆ; ๐, ๐, ๐, ๐ =1
๐ต ๐, ๐
(๐ฆ โ ๐)๐โ1
(๐ โ ๐)๐โ1 1 โ
๐ฆ โ ๐
๐ โ ๐
๐โ1 1
๐ โ ๐
=1
๐ต ๐, ๐
(๐ฆ โ ๐)๐โ1
(๐ โ ๐)๐โ1 ๐ โ ๐ โ ๐ฆ + ๐
๐ โ ๐
๐โ1 1
๐ โ ๐
154
=1
๐ต ๐, ๐ (๐ฆ โ ๐)๐โ1 ๐ โ ๐ฆ ๐โ1
1
(๐ โ ๐)๐โ1+๐โ1+1
=1
๐ต ๐, ๐ (๐ฆ โ ๐)๐โ1 ๐ โ ๐ฆ ๐โ1
1
(๐ โ ๐)๐โ1+๐โ1+1
=(๐ฆ โ ๐)๐โ1 ๐ โ ๐ฆ ๐โ1
๐ต ๐, ๐ (๐ โ ๐)๐+๐โ1 _____________________(5.33)
The probability distribution function in equation 5.33 is a four parameter beta distribution
function.
5.7.1 Properties of the four parameter generalized beta distribution
i. Mean
๐ธ ๐ = ๐ฆ๐ ๐ฆ; ๐, ๐, ๐, ๐ ๐๐ฆโ
0
= ๐ฆ(๐ฆ โ ๐)๐โ1 ๐ โ ๐ฆ ๐โ1
๐ต ๐, ๐ (๐ โ ๐)๐+๐โ1
โ
0
๐๐ฆ
=1
๐ต ๐, ๐ (๐ โ ๐)๐+๐โ1 ๐ฆ(๐ฆ โ ๐)๐โ1 ๐ โ ๐ฆ ๐โ1
โ
0
๐๐ฆ
=๐๐ + ๐๐
๐ + ๐
ii. Mode
= ๐ โ 1 ๐ + (๐ โ 1)๐
๐ + ๐ โ 2 ๐๐๐ ๐ > 1, ๐ > 1
iii. Variance
=๐๐(๐ โ ๐)2
๐ + ๐ 2(๐ + ๐ + 1)
155
6 Chapter VI: The Generalized Beta Distribution
Based on Transformation Technique
6.1 Introduction
This chapter provides the construction, properties and special cases of the five parameter
generalized beta distributions due to McDonaldโs. It gives an extension of the
McDonaldโs five parameter to a five parameter weighted generalized beta distribution. It
further considers a five parameter exponential generalized beta distribution function. The
five parameter generalized beta distribution nests both the generalized beta distribution of
the first kind and the generalized beta distribution of the second kind. It includes many
distributions as special or limiting cases.
6.2 Five Parameter Generalized Beta Distribution Function Due to
McDonald
McDonald and Xu (1995) introduced the five parameter generalized beta distribution and
showed that it contains the distributions previously mentioned under 4-3-2 parameters, as
special or limiting cases. The probability density function of the five parameter
generalized beta distribution can be obtained from the classical beta distribution with two
shape parameters a and b in equation (2.1), using the following transformation
๐ฅ = ๐ฆ๐
๐๐ + ๐๐ฆ๐
๐๐ฅ
๐๐ฆ =
๐๐ฆ๐โ1 ๐๐ + ๐๐ฆ๐ โ [๐๐๐ฆ๐โ1(๐ฆ๐)]
(๐๐ + ๐๐ฆ๐)2
=๐๐๐๐ฆ๐โ1
(๐๐ + ๐๐ฆ๐)2
โด ๐ ๐ฆ; ๐, ๐, ๐, ๐, ๐ =๐ฆ๐๐โ๐
(๐๐ + ๐๐ฆ๐)๐โ1 1 โ
๐ฆ๐
๐๐ + ๐๐ฆ๐
๐โ1 1
๐ต ๐, ๐
๐๐๐๐ฆ๐โ1
(๐๐ + ๐๐ฆ๐)2
156
=๐ฆ๐๐โ๐
(๐๐ + ๐๐ฆ๐)๐โ1 ๐๐ + ๐๐ฆ๐ โ ๐ฆ๐
๐๐ + ๐๐ฆ๐
๐โ1 1
๐ต ๐, ๐
๐๐๐๐ฆ๐โ1
(๐๐ + ๐๐ฆ๐)2
=๐๐๐๐ฆ๐๐โ๐+๐โ1(๐๐ + ๐๐ฆ๐ โ ๐ฆ๐)๐โ1
๐ต ๐, ๐ (๐๐ + ๐๐ฆ๐)๐โ1+๐โ1+2
=๐๐๐๐ฆ๐๐โ1(๐๐)๐โ1 1 + ๐
๐ฆ๐
๐๐ โ๐ฆ๐
๐๐ ๐โ1
๐ต ๐, ๐ (๐๐)๐+๐ 1 + ๐๐ฆ๐
๐๐ ๐+๐
=๐๐๐+๐๐โ๐๐ฆ๐๐โ1 1 + ๐ โ 1 (
๐ฆ๐)๐
๐โ1
๐ต ๐, ๐ ๐๐๐+๐๐ 1 + ๐(๐ฆ๐)๐
๐+๐
=๐๐๐๐๐ฆ๐๐โ1 1 โ 1 โ ๐
๐ฆ๐
๐
๐โ1
๐๐๐+๐๐ ๐ต ๐, ๐ 1 + ๐(๐ฆ๐) ๐
๐+๐
=๐๐ฆ๐๐โ1 1 โ 1 โ ๐
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ 1 + ๐ ๐ฆ๐
๐
๐+๐ ___________________________________________(6.1)
for 0 < ๐ฆ๐ < ๐๐
1โ๐ , 0 โค ๐ โค 1 and ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0
Equation 6.1 is a five parameter generalized beta distribution. The five parameter
generalized beta distribution encompasses both the four parameter generalized beta
distribution in equation 5.1 and 5.10 as follows:-
i. Letting c=0 in the five parameter generalized beta distribution in equation 6.1 gives a
four parameter generalized beta distribution in equation 5.1
ii. Letting c=1 in the five parameter generalized beta distribution in equation 6.1 gives
the four parameter generalized beta distribution in equation 5.10.
157
However, when fitting this five parameter generalized beta distribution model to 1985
family income, McDonald and Xu (1995) found that the four parameter generalized beta
of the second kind subfamily is selected (in terms of likelihood and several other criteria).
Thus, it appears that the five parameter generalized beta distribution does not provide
additional flexibility.
6.3 Properties of the Five Parameter Generalized Beta Distribution
6.3.1 rthโorder moments
To obtain the rth-order moments for the five parameter generalized beta distribution, we
evaluate
๐ธ ๐๐ = ๐ฆ๐๐ ๐ฆ; ๐, ๐, ๐, ๐, ๐ ๐๐ฆโ
โโ
= ๐ฆ๐๐๐ฆ๐๐โ1(1 โ (1 โ ๐)(
๐ฆ๐)๐)๐โ1
๐๐๐๐ต ๐, ๐ (1 + ๐(๐ฆ๐)๐)๐+๐
โ
โโ
๐๐ฆ
replacing 1 โ 1 โ ๐ (๐ฆ
๐)๐
๐โ1
by its binomial expansion, letting
๐ข = (1 โ ๐)(๐ฆ
๐)๐ and collecting like terms yield
๐ธ ๐๐ = ๐๐ ๐ + ๐ ๐
โ๐1 โ ๐
๐
(1 โ ๐)๐+
๐๐
+๐
โ
๐=0
๐ข๐+
๐๐
+๐(1 โ ๐ข)๐โ1
1
0
๐๐ข
= ๐๐ ๐ + ๐ ๐
โ๐1 โ ๐
๐
1 โ ๐ ๐+
๐๐
+๐
โ
๐=0
๐ต(๐ +๐
๐+ ๐, ๐)
= ๐๐ ๐ + ๐ ๐
โ๐1 โ ๐
๐
ฮ ๐ +๐๐ + ๐ ฮ b
1 โ ๐ ๐+
๐๐
+๐ฮ ๐ +
๐๐ + ๐ + ๐
โ
๐=0
=๐๐๐ต(๐ +
๐๐ , ๐)
1 โ ๐ ๐+
๐๐๐ต(๐, ๐)
๐ + ๐, ๐ +๐
๐;
โ๐
1 โ ๐
๐ + ๐ +๐
๐
158
=๐๐๐ต(๐ +
๐๐ , ๐)
๐ต(๐, ๐)
๐ +๐
๐,
๐
๐; ๐
๐ + ๐ +๐
๐
_______________________________(6.2)
Equation 6.2 is defined for all r if ๐ < 1 or for โ๐ <๐
๐< ๐ if ๐ = 1
6.3.2 Mean
The mean is given by
๐ธ(๐) =๐๐ต(๐ +
1๐ , ๐)
1 โ ๐ ๐+
1๐๐ต(๐, ๐)
๐ +
1
๐,
1
๐; ๐
๐ + ๐ +1
๐
6.3.3 Variance
Var(Y) =q2B(a +
2p , b)
1 โ c a+
2p B(a, b)
2F1
a +
2
p,
2
p; c
a + b +2
p
โ
qB(a +
1p , b)
1 โ c a+
1p B(a, b)
2F1
a +
1
p,
1
p; c
a + b +1
p
2
6.4 Special Cases of the Five Parameter Generalized Beta Distribution
Function
6.4.1 The four parameter generalized beta distribution of the first kind
This is a special case when ๐ = 0 in equation 6.1
๐(๐ฆ; ๐, ๐, ๐, ๐) =๐๐ฆ๐๐โ1 1 โ 1 โ 0
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ 1 + 0 ๐ฆ๐
๐
๐+๐
159
=๐๐ฆ๐๐โ1 1 โ
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐
for 0 < ๐ฆ๐ < ๐๐ , and ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0
6.4.2 The four parameter generalized beta distribution of the second kind
This is a special case when ๐ = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐, ๐, ๐ =๐๐ฆ๐๐โ1 1 โ 1 โ 1
๐ฆ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ 1 + 1 ๐ฆ๐
๐
๐+๐
=๐๐ฆ๐๐โ1
๐๐๐๐ต ๐, ๐ 1 + ๐ฆ๐
๐
๐+๐
for 0 < ๐ฆ๐ < โ , ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0
6.4.3 The four parameter generalized beta distribution
This is a special case when a = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐, ๐, ๐ =๐๐๐ฆ๐โ1 1 โ 1 โ ๐
๐ฆ๐
๐
๐โ1
๐๐ 1 + ๐ ๐ฆ๐
๐
1+๐
for 0 < ๐ฆ๐ < โ , ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0
6.4.4 The four parameter generalized Logistic distribution function
This is a special case when ๐ = 1, y = ๐๐ฅ , and ๐ฝ = ๐๐ฅ in equation 6.1
๐ ๐๐ฅ ; ๐, ๐, ๐, ๐ =๐(๐๐ฅ)๐๐โ1 1 โ 1 โ 1
(๐๐ฅ)๐
๐
๐โ1
. (๐๐ฅ)
๐๐๐๐ต ๐, ๐ 1 + 1 (๐๐ฅ)๐
๐
๐+๐
=๐๐๐๐(๐ฅโ๐๐๐๐ )
๐๐๐๐ต ๐, ๐ 1 + ๐๐(๐ฅโ๐๐๐๐ ) ๐+๐
160
for โโ < ๐ฅ < โ , ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0
6.4.5 The three parameter Inverse Beta distribution of the first kind
This is a special case when ๐ = 0, and p = โ1 in equation 6.1
๐(๐ฆ; ๐, ๐, ๐) =(โ1)๐ฆ๐(โ1)โ1 1 โ 1 โ 0
๐ฆ๐
(โ1)
๐โ1
๐๐(โ1)๐ต ๐, ๐ 1 + 0 ๐ฆ๐
(โ1)
๐+๐
=๐๐ 1 โ
๐๐ฆ
๐โ1
๐ต ๐, ๐ ๐ฆ๐+1
for 0 < ๐ฆ < 1 , and ๐ > 0, ๐ > 0, ๐ > 0
6.4.6 The three parameter Stoppa (generalized pareto type I) distribution
function
This is a special case when ๐ = 0, a = 1 and p = โp in equation 6.1
๐(๐ฆ; ๐, ๐, ๐) =| โ ๐|๐ฆ 1 (โ๐)โ1 1 โ 1 โ 0
๐ฆ๐
โ๐
๐โ1
๐ 1 (โ๐)๐ต 1, ๐ 1 + 0 ๐ฆ๐
โ๐
1+๐
= ๐๐๐๐๐ฆโ๐โ1 1 โ ๐ฆ
๐
โ๐
๐โ1
for ๐ฆ โฅ ๐ > 0 , and ๐ > 0, ๐ > 0
6.4.7 The three parameter Singh-Maddala (Burr12) distribution function
This is a special case when ๐ = 1, and a = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐, ๐ =๐๐ฆ๐(1)โ1 1 โ 1 โ 1
๐ฆ๐
๐
๐โ1
๐๐(1)๐ต 1, ๐ 1 + 1 ๐ฆ๐
๐
(1)+๐
=๐๐๐ฆ๐โ1
๐๐ 1 + ๐ฆ๐
๐
1+๐
161
for 0 < ๐ฆ๐ < โ , ๐ > 0, ๐ > 0, ๐ > 0
6.4.8 The three parameter Dagum (Burr3) distribution function
This is a special case when ๐ = 1, and b = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐, ๐ =๐๐ฆ๐๐โ1 1 โ 1 โ 1
๐ฆ๐
๐
(1)โ1
๐๐๐๐ต ๐, 1 1 + 1 ๐ฆ๐
๐
๐+(1)
=๐๐๐ฆ๐๐โ1
๐๐๐ 1 + ๐ฆ๐
๐
๐+1
for 0 < ๐ฆ๐ < โ , ๐ > 0, ๐ > 0, ๐ > 0
6.4.9 The two parameter Classical Beta distribution
This is a special case when ๐ = 0, p = 1, and q = 1 in equation 6.1
๐(๐ฆ; ๐, ๐) =
(1)๐ฆ๐(1)โ1 1 โ 1 โ 0 ๐ฆ
(1) (1)
๐โ1
(1)๐(1)๐ต ๐, ๐ 1 + 0 ๐ฆ
(1)
(1)
๐+๐
=๐ฆ๐โ1 1 โ ๐ฆ ๐โ1
๐ต ๐, ๐
for 0 < ๐ฆ < 1 , and ๐ > 0, ๐ > 0
6.4.10 The two parameter Pareto type I distribution function
This is a special case when ๐ = 0, p = โ1, and b = 1 in equation 6.1
๐(๐ฆ; ๐, ๐) =(โ1)๐ฆ๐(โ1)โ1 1 โ 1 โ 0
๐ฆ๐
(โ1)
1โ1
๐๐(โ1)๐ต ๐, 1 1 + 0 ๐ฆ๐
(โ1)
๐+1
162
=๐๐๐
๐ฆ๐+1
for 0 < ๐ฆ < 1 , and ๐ > 0, ๐ > 0,
6.4.11 The two parameter Kumaraswamy distribution function
This is a special case when ๐ = 0, a = 1 and q = 1 in equation 6.1
๐(๐ฆ; ๐, ๐) =๐๐ฆ(1)๐โ1 1 โ 1 โ 0
๐ฆ1
๐
๐โ1
1 1 ๐๐ต 1, ๐ 1 + 0 ๐ฆ1
๐
1+๐
= ๐๐๐ฆ๐โ1 1 โ ๐ฆ ๐ ๐โ1
for 0 < ๐ฆ๐ < 1 , and ๐ > 0, ๐ > 0
6.4.12 The two parameter generalized Power distribution function
This is a special case when ๐ = 0, b = 1, and p = 1 in equation 6.1
๐(๐ฆ; ๐, ๐) =(1)๐ฆ๐(1)โ1 1 โ 1 โ 0
๐ฆ๐
(1)
1โ1
๐๐(1)๐ต ๐, 1 1 + 0 ๐ฆ๐
(1)
๐+1
=๐๐ฆ๐โ1
๐๐
for 0 < ๐ฆ < 1 , and ๐ > 0, ๐ > 0,
6.4.13 The two parameter Beta distribution of the second kind
This is a special case when ๐ = 1, p = 1, and q = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐ =
(1)๐ฆ(1)๐โ1 1 โ 1 โ 1 ๐ฆ
(1) (1)
๐โ1
(1)(1)๐๐ต ๐, ๐ 1 + 1 ๐ฆ
(1)
(1)
๐+๐
=๐ฆ๐โ1
๐ต ๐, ๐ 1 + ๐ฆ ๐+๐
for 0 < ๐ฆ < โ , ๐ > 0, ๐ > 0
163
6.4.14 The two parameter Inverse Lomax distribution function
This is a special case when ๐ = 1, b = 1, and p = 1 in equation 6.1
๐ ๐ฆ; ๐, ๐ =(1)๐ฆ(1)๐โ1 1 โ 1 โ 1
๐ฆ๐
(1)
1โ1
๐(1)๐๐ต ๐, 1 1 + 1 ๐ฆ๐
(1)
๐+1
=๐๐ฆ๐โ1
๐๐ 1 + ๐ฆ๐
๐+1
for 0 < ๐ฆ < โ , ๐ > 0, ๐ > 0
6.4.15 The two parameter General Fisk (Log-Logistic) distribution function
This is a special case when ๐ = 1, a = 1, b = 1 and q =1
ฮป in equation 6.1
๐ ๐ฆ; ฮป, ๐ =
๐๐ฆ๐(1)โ1 1 โ 1 โ 1 ๐ฆ1ฮป
๐
(1)โ1
1ฮป ๐(1)
๐ต 1,1 1 + 1 ๐ฆ1ฮป
๐
(1)+(1)
=ฮป๐ ฮป๐ฆ ๐โ1
1 + ฮป๐ฆ ๐ ๐
for 0 < ๐ฆ๐ < โ , ฮป > 0, ๐ > 0, ๐ > 0
6.4.16 The two parameter Fisher (F) distribution function
This is a special case when ๐ = 1, a =u
2, b =
v
2, p = 1 and q =
v
uin equation 6.1
๐ ๐ฆ; ๐ข, ๐ฃ =
(1)๐ฆ 1
๐ข2 โ1 1 โ 1 โ 1
๐ฆvu
(1)
๐ข2 โ1
vu
1 ๐ข2
๐ต ๐ข2 ,
๐ฃ2 1 + 1
๐ฆvu
(1)
๐ข2 +
๐ฃ2
164
=ฮ
๐ข + ๐ฃ2
uv
๐ข2
๐ฆ ๐ข2 โ1
ฮ ๐ข2 ฮ
๐ฃ2 1 +
๐ฆuv
๐ข+๐ฃ2
for 0 < ๐ฆ < โ , u > 0, ๐ฃ > 0
6.4.17 The one parameter Power distribution function
This is a special case when ๐ = 0, a = 1, b = 1 and q = 1 in equation 6.1
๐(๐ฆ; ๐) =๐๐ฆ(1)๐โ1 1 โ 1 โ 0
๐ฆ1
๐
1โ1
1 1 ๐๐ต 1,1 1 + 0 ๐ฆ1
๐
1+1
= ๐๐ฆ๐โ1
for 0 < ๐ฆ < 1 , and ๐ > 0,
6.4.18 The one parameter Wigner semicircle distribution function
This is a special case when ๐ = 0, a =3
2, b =
3
2, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ
32 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
32โ1
1 32 (1)๐ต
32 ,
32 1 + 0
๐ฆ1
(1)
32
+32
=8 ๐ฆ 1 โ ๐ฆ
๐, 0 < ๐ฆ < 1
Letting
๐ฆ =๐ฅ + ๐
2๐ ๐๐๐ |๐ฝ| =
1
2๐
Limits changes as ๐ฆ = 0 โน ๐ฅ > โ๐ and ๐ฆ = 1 โน ๐ฅ = ๐
๐ ๐ฅ =8 ๐ฆ + ๐
2๐ 1 โ๐ฆ + ๐
2๐
๐.
1
2๐
=2 ๐ 2 โ ๐ฆ2
๐ 2๐, โ๐ < ๐ฆ < ๐
165
6.4.19 The one parameter Lomax (pareto type II) distribution function
This is a special case when ๐ = 1, a = 1, p = 1, and q = 1 in equation 6.1
๐ ๐ฆ; ๐ =
(1)๐ฆ(1)(1)โ1 1 โ 1 โ 1 ๐ฆ
(1) (1)
๐โ1
(1)(1)(1)๐ต 1, ๐ 1 + 1 ๐ฆ
(1) (1)
(1)+๐
=๐
1 + ๐ฆ ๐+๐
for 0 < ๐ฆ < โ , ๐ > 0
6.4.20 The one parameter half studentโs (t) distribution function
This is a special case when ๐ = 1, a =1
2, b =
r
2, p = 2 and q = r in equation 6.1
๐ ๐ฆ; ๐ =
(2)๐ฆ 2
12 โ1 1 โ 1 โ 1
๐ฆ
r
(2)
๐2 โ1
r 2
12 ๐ต
12 ,
๐2 1 + 1
๐ฆ
r
(2)
12 +
๐2
=2ฮ
1 + ๐2
rฯ ฮ ๐2 1 +
๐ฆ๐
๐ ๐+๐๐
for 0 < ๐ฆ < โ , ๐ > 0
6.4.21 Uniform distribution function
This is a special case when ๐ = 0, a = 1, b = 1, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ 1 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
1โ1
1 1 (1)๐ต 1,1 1 + 0 ๐ฆ1
(1)
1+1
= 1
166
6.4.22 Arcsine distribution function
This is a special case when ๐ = 0, a =1
2, b =
1
2, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ
12 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
12โ1
1 12 (1)๐ต
12 ,
12 1 + 0
๐ฆ1
(1)
12
+12
=1
๐ ๐ฆ(1 โ ๐ฆ), 0 < ๐ฆ < 1
6.4.23 Triangular shaped distribution functions
i. This is a special case when ๐ = 0, a = 1, b = 2, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ 1 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
2โ1
1 1 (1)๐ต 1,2 1 + 0 ๐ฆ1
(1)
1+2
= 2 1 โ ๐ฆ , 0 < ๐ฆ < 1
ii. This is a special case when ๐ = 0, a = 2, b = 1, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ 2 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
1โ1
1 2 (1)๐ต 2,1 1 + 0 ๐ฆ1
(1)
2+1
= 2๐ฆ, 0 < ๐ฆ < 1
6.4.24 Parabolic shaped distribution functions
This is a special case when ๐ = 0, a = 2, b = 2, p = 1 and q = 1 in equation 6.1
๐(๐ฆ) =(1)๐ฆ 2 (1)โ1 1 โ 1 โ 0
๐ฆ1
(1)
2โ1
1 2 (1)๐ต 2,2 1 + 0 ๐ฆ1
(1)
2+2
= 6๐ฆ 1 โ ๐ฆ , 0 < ๐ฆ < 1
167
6.4.25 Log-Logistic (Fisk) distribution function
This is a special case when ๐ = 1, a = 1, b = 1, p = 1, and q = 1 in equation 6.1
๐ ๐ฆ =
(1)๐ฆ(1)(1)โ1 1 โ 1 โ 1 ๐ฆ
(1) (1)
(1)โ1
(1)(1)(1)๐ต 1,1 1 + 1 ๐ฆ
(1) (1)
(1)+(1)
=1
1 + ๐ฆ ๐
for 0 < ๐ฆ < โ
6.4.26 Logistic distribution function
This is a special case when ๐ = 1, a = 1, b = 1, p = 1, q = 1, y = ๐๐ฅ , and ๐ฝ = ๐๐ฅ
in equation 6.1
๐ ๐ฅ =
(1)(๐๐ฅ)(1)(1)โ1 1 โ 1 โ 1 (๐๐ฅ)(1)
(1)
(1)โ1
. (๐๐ฅ)
(1)(1)(1)๐ต 1,1 1 + 1 (๐๐ฅ)(1)
(1)
(1)+(1)
=๐๐ฅ
1 + ๐๐ฅ 2, for โ โ < ๐ฅ < โ
168
The Five Parameter Beta Distribution and its Special Cases Diagrammatically
5 Parameter
4 Parameter
Generalized beta
Generalized beta
of the 1st kind
Generalized beta
of the 2nd
kind
Inverse beta Stoppa Singh Maddala
(Burr12) Dagum (Burr3)
Pareto
type I
Beta of the
1st kind
Kumaraswamy Generalized
Power
Beta of the
2nd
kind
3 Parameter
2 Parameter
Inverse
Lomax
General
Fisk (Log
logistic)
Fisher F
1 Parameter
Generalized
beta
Power Wigner
Semicircle
Lomax (Pareto
type II) Half studentโs t Uniform
Arcsine Triangular Parabolic Logistic
Fisk (Log
logistic)
169
6.5 Construction of the Five Parameter Weighted Generalized Beta Distribution
Function
Extending the McDonaldโs five parameter model by introducing a sixth parameter, k
gives a six parameter weighted generalized beta distribution as follows.
Let
๐(๐ฆ; ๐, ๐, ๐, ๐, ๐, ๐) =๐๐ฆ๐๐ +๐โ1 1 โ 1 โ ๐
๐ฆ๐
๐
๐โ1
๐๐๐+๐๐ต ๐ +๐๐ , ๐ โ
๐๐ 1 + ๐
๐ฆ๐
๐
๐+๐ ______________________(6.3)
for 0 < ๐ฆ๐ < ๐๐
1โ๐ , 0 โค ๐ โค 1 and ๐ > 0, ๐ > 0, ๐ +
๐
๐> 0, ๐ โ
๐
๐> 0
The Weighted generalized beta distribution of the second kind with polynomial weight
function ๐ค ๐ฆ = ๐ฆ๐ is a special case of equation 6.3 when ๐ =1. Alternatively, it can be
obtained from the generalized beta distribution of the second kind by adding some weight
to the parameters ๐ and ๐.
Let ๐ = ๐ +๐
๐ and ๐ = ๐ โ
๐
๐ in equation 4.1 of generalized beta distribution of the
first kind.
๐ ๐ฆ; ๐, ๐, ๐, ๐, ๐ = ๐ ๐ฆ
๐ ๐+๐๐ โ1
๐๐ ๐+
๐๐ ๐ต ๐ +
๐๐ , ๐ โ
๐๐ 1 +
๐ฆ๐
๐
๐+๐
= ๐ ๐ฆ๐๐+๐โ1
๐๐๐ +๐๐ต ๐ +๐๐ , ๐ โ
๐๐ 1 +
๐ฆ๐
๐
๐+๐ _____________________________________(6.4)
๐ฆ > 0, ๐, ๐, ๐, ๐ > 0, and โ ๐๐ < ๐ < ๐๐
170
Equation 6.4 is a very flexible five parameter weighted generalized beta distribution of
the second kind presented by Yuan et al. (2012).
6.6 Properties of Five Parameter Weighted Generalized Beta
Distribution
6.6.1 rth-order moments
To obtain the rth-order moments for the weighted generalized beta distribution, we
evaluate
๐ธ ๐๐ = ๐ฆ๐๐ ๐ฆ; ๐, ๐, ๐, ๐, ๐ ๐๐ฆโ
โโ
= ๐ฆ๐๐๐ฆ๐๐+๐โ1
๐๐๐ +๐๐ต ๐ +๐๐ , ๐ โ
๐๐ 1 + ๐(
๐ฆ๐)๐
๐+๐
โ
โโ
๐๐ฆ
=๐๐๐ต ๐ +
๐๐ +
๐๐ , ๐ โ
๐๐ โ
๐๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
_____________________________________________(6.5)
6.6.2 Mean
The mean is given by
๐ธ(๐) =๐๐ต ๐ +
๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐
, ๐ โ๐๐
_____________________________________(6.6)
6.6.3 Variance
Var(Y) = ๐2
๐ต ๐ +
๐ + 2๐ , ๐ โ
๐ + 2๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
โ ๐ +
๐ + 1๐ , ๐ โ
๐ โ 1๐
๐ +๐๐ , ๐ โ
๐๐
2
The coefficient of variation is given by
171
๐ถ๐ = ๐ต ๐ +
๐ + 2๐ , ๐ โ
๐ + 2๐ ๐ต ๐ +
๐๐ , ๐ โ
๐๐
๐ต2 ๐ +๐ + 1
๐ , ๐ โ๐ + 1
๐ โ 1
6.6.4 Skewness
The skewness is given by
๐๐๐๐ค๐๐๐ ๐ = ๐ธ ๐3 โ 3๐ธ ๐2 ๐ธ ๐ + 2(๐ธ ๐ )3
=๐3๐ต ๐ +
๐ + 3๐ , ๐ โ
๐ + 3๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
โ 3๐2๐ต ๐ +
๐ + 2๐ , ๐ โ
๐ + 2๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
.
๐๐ต ๐ +๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
+ 2 ๐๐ต ๐ +
๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
2
6.6.5 Kurtosis
The kurtosis is given by
๐๐ข๐๐ก๐๐ ๐๐ = ๐ธ ๐4 โ 4๐ธ ๐3 ๐ธ ๐ + 6๐ธ ๐2 ๐ธ ๐ 2โ 3 ๐ธ ๐
4
๐พ๐ข๐๐ก๐๐ ๐๐
=๐4๐ต ๐ +
๐ + 4๐ , ๐ โ
๐ + 4๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
โ 4๐3๐ต ๐ +
๐ + 3๐ , ๐ โ
๐ + 3๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
.๐๐ต ๐ +
๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
+ 6๐2๐ต ๐ +
๐ + 2๐ , ๐ โ
๐ + 2๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
. ๐๐ต ๐ +
๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
2
โ 3 ๐๐ต ๐ +
๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐๐
4
172
6.7 Special Cases of the Five Parameter Weighted Generalized Beta
Distribution Function
The five parameter weighted generalized beta distribution includes both the generalized
beta distribution of the first and second kinds as special cases, it also includes several
other weighted distributions as special or limiting cases: weighted generalized gamma,
weighted beta of the second kind, weighted Singh-Maddala, weighted Dagum, weighted
gamma, weighted Weibull, and weighted exponential distributions among others as
shown below.
6.7.1 Weighted beta of the second distribution function
This is a special case when p = 1 and q = 1 in equation 6.4
๐ ๐ฆ; ๐, ๐, ๐ =๐ฆ๐+๐โ1
๐ต ๐ + ๐, ๐ โ ๐ 1 + ๐ฆ ๐+๐______________________(6.7)
๐ฆ > 0, ๐, ๐, ๐ > 0, and โ 1 < ๐ < ๐
6.7.2 Weighted Singh-Maddala (Burr12) distribution function
This is a special case when a = 1 in equation 6.4
๐ ๐ฆ; ๐, ๐, ๐, ๐ = ๐ ๐ฆ๐+๐โ1
๐๐+๐๐ต 1 +๐๐ , ๐ โ
๐๐ 1 +
๐ฆ๐
๐
1+๐ ____________(6.8)
๐ฆ > 0, ๐, ๐, ๐ > 0, and โ ๐ < ๐ < ๐๐
6.7.3 Weighted Dagum (Burr3) distribution function
This is a special case when b = 1 in equation 6.4
๐ ๐ฆ; ๐, ๐, ๐, ๐ = ๐ ๐ฆ๐๐+๐โ1
๐๐๐ +๐๐ต ๐ +๐๐ , 1 โ
๐๐ 1 +
๐ฆ๐
๐
๐+1 __________________(6.9)
๐ฆ > 0, ๐, ๐, ๐ > 0, and โ ๐๐ < ๐ < ๐
173
6.7.4 Weighted Generalized Fisk (Log-Logistic) distribution function
This is a special case when , a = 1, b = 1 and q =1
ฮป in equation 6.4
๐ ๐ฆ; ๐, ฮป, ๐ =๐๐ฆ๐+๐โ1
1ฮป
๐+๐
๐ต ๐ +๐๐
, 1 โ๐๐ 1 + ๐๐ฆ๐ 2
___________________________(6.10)
๐ฆ > 0, ๐, ฮป > 0, and โ ๐ < ๐ < ๐
174
The Five Parameter Weighted Beta Distribution and its Special Cases Diagrammatically
Weighted
Generalized Beta2
Weighted Beta of
the 2nd
kind
Weighted
Generalized Fisk
(log-logistic)
Weighted Dagum
(Burr3)
Weighted Singh
Maddala (Burr12)
5 Parameter
4 Parameter
3 Parameter
2 Parameter
175
6.8 Construction of the Five Parameter Exponential Generalized Beta
Distribution Function
This section considers a five parameter exponential generalized beta distribution
function, its construction, properties and special cases. If a random variable ๐ follows the
generalized beta distribution given in equation 6.1 with parameters ๐, ๐, ๐, ๐, ๐ , then the
random variable ๐ = ๐ผ๐ ๐ is said to be distributed as an exponential generalized beta,
with the pdf given by
Let
๐ฆ = ๐๐ง
and
๐๐ฆ
๐๐ง = ๐๐ง
๐(๐ง; ๐, ๐, ๐, ๐, ๐) =๐(๐๐ง)๐๐โ1 1 โ 1 โ ๐
๐๐ง
๐ ๐
๐โ1
. ๐๐ง
๐๐๐๐ต ๐, ๐ 1 + ๐ ๐๐ง
๐ ๐
๐+๐
=๐ ๐๐ง ๐๐ 1 โ 1 โ ๐
๐๐ง
๐ ๐
๐โ1
๐๐๐๐ต ๐, ๐ 1 + ๐ ๐๐ง
๐
๐
๐+๐ ___________________________________________(6.11)
Let ๐ =1
๐ and ๐ = ๐๐ฟ in equation 6.11
๐ ๐ง; ๐, ๐, ๐, ๐, ๐ฟ =
1๐ (๐๐ง)
1๐๐ 1 โ 1 โ ๐
๐๐ง
๐๐ฟ
1๐
๐โ1
(๐๐ฟ)1๐๐๐ต ๐, ๐ 1 + ๐
๐๐ง
๐๐ฟ
1๐
๐+๐
=๐
๐ ๐งโ๐ฟ ๐ 1 โ 1 โ ๐ ๐
๐งโ๐ฟ๐
๐โ1
๐๐ต ๐, ๐ 1 + ๐๐๐งโ๐ฟ๐
๐+๐ ________________________________________________(6.12)
176
for โ โ <๐ง โ ๐ฟ
๐< ๐ผ๐
1
1 โ ๐
Equation 6.12 is the five parameter exponential generalized beta distribution function.
6.9 Properties of Five Parameter Exponential Generalized Beta
Distribution
6.9.1 Moment-generating function
The moment-generating function of the exponential generalized beta distribution variates
can be obtained by using the substitution ๐ข = ๐(๐งโ๐ฟ)/๐ in ๐ธ(๐๐ก๐ง ) and combining terms:
๐ ๐ก = ๐ธ(๐ก๐)
= ๐๐ฟ๐ก ๐ธ(๐๐๐ก ; ๐ = 1, ๐ = 1, ๐, ๐, ๐)
=๐๐ฟ๐ก ๐ต ๐ + ๐๐ก, ๐
๐ต ๐, ๐ 2๐น1
๐ + ๐๐ก, ๐๐ก; ๐๐ + ๐ + ๐๐ก
, ___(6.13)
Equation 6.13 is the moment-generating function of the five parameter exponential
generalized beta distribution. The other moment generating functions related to special
cases can be obtained from equation 6.13 as necessary; for instance, the moment
generating function for the exponential generalized beta distribution of the first kind can
be obtained as a special case when ๐ = 0 and is given by
๐(๐ก) =๐๐ฟ๐ก ๐ต ๐ + ๐๐ก, ๐
๐ต ๐, ๐
While for the exponential generalized beta distribution of the second kind can be
obtained as a special case when ๐ = 1 and is given by
๐ ๐ก =๐๐ฟ๐ก ๐ต ๐ + ๐๐ก, ๐ โ ๐๐ก
๐ต ๐, ๐ .
177
6.9.2 Mean of the exponential generalized beta distribution
The mean of the exponential generalized beta distribution can be obtained by
differentiating the natural logarithm of equation 6.13, the cumulant-generating function,
and substituting ๐ก = 0 to yield
๐ธ ๐ = ๐ฟ + ๐ ๐ ๐ โ ๐ ๐ ๐๐๐
๐ + ๐ 3๐น1
1, 1, ๐ + 1; ๐2, ๐ + ๐ + 1;
The means for the exponential generalized beta of the first kind, exponential generalized
beta of the second kind, and exponential generalized gamma, respectively, can be given
by
๐ธ1 ๐ = ๐ฟ + ๐ ๐ ๐ โ ๐ ๐ + ๐ ,
๐ธ2 ๐ = ๐ฟ + ๐ ๐ ๐ โ ๐ ๐ ,
๐ธ3 ๐ = ๐ฟ + ๐๐ ๐ ,
Where ๐() denotes the digamma function. The higher-order moments for the exponential
generated beta distribution are quite involving, however, relatively simple expressions for
the variance, skewness and Kurtosis for the exponential generated beta distribution of the
first kind, the exponential generated beta distribution of the second kind and the
exponential generated gamma distribution can be easily obtained by differentiating the
desired cumulant-generating functions.
6.10 Special Cases of Exponential Generalized Beta Distribution
Since the five parameter exponential generalized beta and the five parameter generalized
beta distributions are related by the logarithmic transformation, similar โexponentialโ
distributions can be obtained by transforming each of the distributions mentioned under
178
the five parameter generalized beta distribution in section 6.2.3 which corresponds to the
special cases given below:
6.10.1 Exponential generalized beta of the first kind
This is a special case of the exponential generalized beta distribution in equation 6.12
when ๐ = 0
๐ ๐ง; ๐, ๐, ๐, ๐ฟ =๐
๐ ๐งโ๐ฟ ๐ 1โ๐
๐งโ๐ฟ๐
๐โ1
๐๐ต ๐, ๐ __________________________________(6.14)
for โ โ <๐ง โ ๐ฟ
๐< 0
Equation 6.14 is just an alternative representation of the generalized exponential
distribution.
6.10.2 Exponential generalized beta of the second kind
This is a special case of the exponential generalized beta distribution in equation 6.12
when ๐ = 1
๐ ๐ง; ๐, ๐, ๐, ๐ฟ =๐
๐ ๐งโ๐ฟ ๐
๐๐ต ๐, ๐ 1 + ๐๐งโ๐ฟ๐
๐+๐ ____________________________(6.15)
for โ โ <๐ง โ ๐ฟ
๐< โ
Equation 6.15 is just an alternative representation of the generalized logistic distribution.
179
6.10.3 Exponential generalized gamma
The exponential generalized gamma distribution can be derived as a limiting case ๐ โ
โ and ๐ฟ โ= ๐In๐ + ๐ฟ using the same method given in section 5.6.17. It is defined by the
following probability density function:
๐ ๐ง; ๐, ๐, ๐ฟ =๐
๐ ๐งโ๐ฟ ๐ ๐โ๐
๐งโ๐ฟ๐
๐ฮฮ ๐ _______________________________________(6.16)
for โ โ <๐ง โ ๐ฟ
๐< โ
Equation 6.16 is an alternative representation of the generalized Gompertz distribution.
6.10.4 Generalized Gumbell distribution
This is a special case of the exponential generalized beta distribution in equation 6.12
when ๐ = 1 and a = b
๐ ๐ง; ๐, ๐, ๐ฟ =๐
๐ ๐งโ๐ฟ ๐
๐๐ต ๐, ๐ 1 + ๐๐งโ๐ฟ๐
2๐ ____________________________(6.17)
for โ โ <๐ง โ ๐ฟ
๐< โ
6.10.5 Exponential Burr 12
This is a special case of the exponential generalized beta distribution in equation 6.12
when ๐ = 1 and b = 1
๐ ๐ง; ๐, ๐, ๐ฟ =๐๐
๐ ๐งโ๐ฟ ๐
๐ 1 + ๐๐งโ๐ฟ๐
๐+1 ______________________________(6.18)
180
for โ โ <๐ง โ ๐ฟ
๐< โ
6.10.6 Exponential power distribution function
The 3 parameter exponential power distribution function is a special case of the
exponential generalized beta distribution in equation 6.12 when ๐ = 0 and ๐ = 1
๐ ๐ง; ๐, ๐, ๐ฟ =๐๐
๐ ๐งโ๐ฟ ๐
๐____________________________________(6.19)
for โ โ <๐ง โ ๐ฟ
๐< 0
6.10.7 Two parameter exponential distribution function
The 2 parameter exponential distribution function is a special case of the exponential
generalized beta distribution in equation 6.12 when ๐ = 0, ๐ = 1 and ๐ = 1
๐ ๐ง; ๐, ๐ฟ =๐
๐งโ๐ฟ๐
๐___________________________________________(6.20)
for โ โ <๐ง โ ๐ฟ
๐< 0
6.10.8 One parameter exponential distribution function
The 1 parameter exponential distribution function is a special case of the exponential
generalized beta distribution in equation 6.12 when ๐ = 0, ๐ = 1, ๐ = 1 and ๐ฟ = 0
๐ ๐ง; ๐ =๐
๐ง๐
๐____________________________________________(6.21)
for โ โ <๐ง
๐< 0
181
6.10.9 Exponential Fisk (logistic) distribution
The exponential fisk distribution is a special case of the exponential generalized beta
distribution in equation 6.12 when ๐ = 1, a = 1 and b = 1
๐ ๐ง; ๐, ๐ฟ =๐
๐งโ๐ฟ๐
๐ 1 + ๐๐งโ๐ฟ๐
2 ____________________________(6.22)
for โ โ <๐ง โ ๐ฟ
๐< โ
Equation 6.22 of the exponential fisk distribution is also known as the logistic distribution.
6.10.10 Standard logistic distribution
The standard logistic distribution is a special case of the exponential generalized beta
distribution in equation 6.12 when ๐ = 1, a = 1, b = 1 and ๐ฟ = 0
๐ ๐ง; ๐ =๐
๐ง๐
๐ 1 + ๐๐ง๐
2 __________________________________(6.23)
for โ โ <๐ง
๐< โ
6.10.11 Exponential weibull distribution
The exponential weibull distribution is a special case of the exponential generalized
gamma distribution in equation 6.16 when a = 1
๐ ๐ง; ๐, ๐, ๐ฟ =๐
๐งโ๐ฟ๐ ๐โ๐
๐งโ๐ฟ๐
๐ฮฮ ๐ ___________________________(6.24)
for โ โ <๐ง โ ๐ฟ
๐< โ
182
The exponential weibull distribution is the extreme value type 1 distribution or the
Gompertz distribution. Equation 6.24 of the exponential weibull distribution can also be
derived as a limiting case (๐ โ โ with ๐ฟ = ๐ฟโ + ๐In๐) from the exponential burr 12
distribution.
183
The Five Parameter exponential Beta Distribution and its Special Cases Diagrammatically
Exponential
Generalized Beta 5 Parameter
4 Parameter
Exponential
Generalized Beta
of the 1st kind
Exponential
Generalized Beta
of the 2nd
kind
Exponential Burr 12 Exponential
Power
Exponential
Exponential
Exponential Fisk
(Logistic)
Generalized
Gumbell 3 Parameter
2 Parameter
1 Parameter
Standard Logistic
184
7 Chapter VII: Generalized Beta Distributions
Based on Generated Distributions
7.1 Introduction
This chapter looks at the new family of generalized beta distributions. The new family of
generalized beta distributions is based on beta generators and can be classified as follows:
Beta generated distributions
Exponentiated generated distributions
Generalized beta generated distributions
7.2 Beta Generator Approach
The beta generated distribution was first introduced by Eugene et. al (2002) through its
cumulative distribution function (cdf). The cdf of a beta distribution is defined by
๐ ๐ก = ๐ก๐โ1(1 โ t)๐โ1
๐ต ๐, ๐ ๐๐ก
y
0
, ๐ > 0, ๐ > 0
Note that 0 < ๐ฆ < 1 since 0 < ๐ก < 1
Replace ๐ฆ by a cdf say G ๐ฅ , of any distribution, since 0 < G ๐ฅ < 1,
for โโ < ๐ฅ < โ
โด ๐ G ๐ฅ =1
๐ต ๐, ๐ ๐ก๐โ1(1 โ t)๐โ1๐๐ก
G ๐ฅ
0
, ๐ > 0, ๐ > 0
๐ G ๐ฅ = ๐ด ๐๐๐ ๐๐ ๐ ๐๐๐
= ๐ด ๐๐ข๐๐๐ก๐๐๐ ๐๐ ๐ฅ
= ๐น ๐ฅ
๐น ๐ฅ = ๐น G ๐ฅ
โด๐
๐๐ฅ(๐น ๐ฅ =
๐
๐๐ฅ๐น G ๐ฅ
๐ ๐ฅ = {๐น โฒ G ๐ฅ } Gโฒ ๐ฅ
= ๐ G ๐ฅ g ๐ฅ
185
โน ๐(๐ฅ) =๐(๐ฅ)[๐บ(๐ฅ)]๐โ1 1 โ ๐บ ๐ฅ ๐โ1
๐ต ๐, ๐ ________________________(7.1)
for 0 < ๐บ ๐ฅ < 1, โโ < ๐ฅ < โ where ๐ > 0, ๐ > 0, ๐ต ๐, ๐ =ฮ a ฮ b
ฮ a+b is the beta
function.
Equation 7.1 is the beta generator or beta generated distribution. It is also referred to as
the generalized beta-F distribution (Sepanski and Kong, 2007). The equation can be used
to generate a new family of beta distributions usually referred to as beta generated
distributions. Jones (2004) called it generalized ith order statistic because the order
statistic distribution is a special case when ๐ = ๐, ๐ = ๐ โ ๐ + 1 which gives the
following density function
๐ ๐บ ๐ฆ; ๐, ๐ =๐(๐ฆ)[๐บ(๐ฆ)]๐โ1 1 โ ๐บ ๐ฆ ๐โ๐+1 โ1
๐ต ๐, ๐ โ ๐ + 1
=ฮ n + 1
ฮ ๐ ฮ ๐ โ ๐ + 1 ๐(๐ฆ)[๐บ(๐ฆ)]๐โ1 1 โ ๐บ ๐ฆ ๐โ๐
=n!
๐ โ 1 ! ๐ โ ๐ !๐(๐ฆ)[๐บ(๐ฆ)]๐โ1 1 โ ๐บ ๐ฆ ๐โ๐__________________(7.2)
Equation 7.2 is the ith-order statistics generated from the beta distribution.
7.3 Various Beta Generated Distributions
7.3.1 Beta-Normal Distribution - (Eugene et. al (2002))
Eugene et. al (2002) introduced the beta-normal distribution, based on a composition of
the classical beta distribution and the normal distribution. Its importance is more than just
generalize the normal distribution. The beta-normal distribution generalizes the normal
distribution and has flexible shapes, giving it greater applicability. Since then, many
authors generalized other distributions similar to the beta-normal distribution.
The beta-normal distribution is obtained as follows:
186
Let
๐บ ๐ฅ = ฮฆ ๐ฅ โ ๐
๐
be the cumulative density function of normal distribution with parameters ๐ and ๐, and
๐ ๐ฅ = ๐ ๐ฅ โ ๐
๐
be the probability density function of the normal distribution.
By using equation 7.1, the density function of beta-normal distribution is given by
๐ ๐ฅ; ๐, ๐, ๐, ๐ =๐โ1 ฮฆ
๐ฅ โ ๐๐
๐โ1
1 โ ฮฆ ๐ฅ โ ๐
๐ ๐โ1
๐ต ๐, ๐ ๐
๐ฅ โ ๐
๐ ___(7.3)
Where ๐ > 0, ๐ > 0, ๐ > 0, ๐ โ โ ๐๐๐ ๐ฅ โ โ
The parameters ๐ and ๐ are the shape parameters characterizing the skewness, kurtosis
and bimodality of the beta normal distribution. The parameters ๐ and ๐ have the same
role as in normal distribution where, ๐ is a location parameter and ๐ is a scale parameter
that stretches out or shrinks the distribution.
Special case of equation 7.3 when ๐ = 0 and ๐ = 1 gives the standard beta-normal
distribution given by
๐ ๐ฅ; ๐, ๐, =[ฮฆ ๐ฅ ]๐โ1 1 โ ฮฆ ๐ฅ ๐โ1
๐ต ๐, ๐ ๐ ๐ฅ ________________(7.4)
Where ๐ > 0, ๐ > 0, ๐ฅ โ โ
Equation 7.3 is the beta-normal distribution introduced by Eugene et.al who also
discussed some of its properties.
7.3.2 Beta-Exponential distribution โ (Nadarajah and Kotz (2006))
Nadarajah and Kotz (2006) introduced the beta-exponential distribution, based on a
composition of the classical beta distribution and the exponential distribution. They
discussed in their paper the expression for the rth
moment, properties of the hazard
function, results for the distribution of the sum of beta-exponential random variables,
maximum likelihood estimation and some asymptotic results.
187
The beta-exponential distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 โ exp(โฮปx)
be the cdf of exponential distribution with parameter ฮป, and
๐ ๐ฅ = ฮป exp(โฮปx)
be the pdf of exponential distribution
By using equation 7.1, the density of the beta-exponential distribution is given by
๐ ๐ฅ; ๐, ๐, ฮป =ฮป exp(โฮปx)[1 โ exp(โฮปx) ]๐โ1 1 โ 1 โ exp โฮปx ๐โ1
๐ต ๐, ๐
=ฮป exp(โ๐ฮปx)[1 โ exp(โฮปx) ]๐โ1
๐ต ๐, ๐ ____________________(7.5)
๐ > 0, ๐ > 0, ฮป > 0, ๐ฅ >0
Equation 7.5 is the beta-exponential distribution introduced by Nadarajah and Kotz
(2006). They also discussed some of its properties. This beta-exponential distribution
contains the exponentiated-exponential distribution (Gupta and Kundu (1999)) as a
special case for b=1. i.e
๐ ๐ฅ; ๐, ฮป = ๐ฮป exp(โฮปx)[1 โ exp(โฮปx) ]๐โ1
๐ > 0, ฮป > 0, ๐ฅ >0
When ๐ = 1, the beta-exponential distribution coincides with the exponential distribution
with parameters ๐ฮป i.e
๐ ๐ฅ; ๐, ฮป = bฮป exp(โ๐ฮปx)
๐ > 0, ฮป > 0, ๐ฅ >0
188
7.3.3 Beta-Weibull Distribution โ (Famoye et al. (2005))
Famoye et.al (2005) introduced the beta-weibull distribution, based on a composition of
the classical beta distribution and the weibull distribution. Lee et.al (2007) gave some
properties of the hazard function, entropies and an application to censored data. Cordeiro
et.al (2005) derived additional mathematical properties of beta-weibull such as
expression for moments.
The beta-weibull distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 โ eโ
xฮฒ
c
be the cdf of Weibull distribution and
๐ ๐ฅ = ๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
be the pdf of Weibull distribution
By using equation 7.1, the density function of the beta-weibull distribution is given by
๐ ๐ฅ; ๐, ๐, ๐, ๐ฝ =
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
1 โ eโ
xฮฒ
c
๐โ1
1 โ 1 โ eโ
xฮฒ
c
๐โ1
๐ต ๐, ๐
=
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
1 โ eโ
xฮฒ
c
๐โ1
eโ
xฮฒ
c
๐โ1
๐ต ๐, ๐
=
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ๐ ๐ฅ๐ฝ
๐
1 โ eโ
xฮฒ
c
๐โ1
๐ต ๐, ๐ , ๐, ๐, ๐, ฮฒ > 0______(7.6)
189
Equation 7.6 is the four parameter beta-Weibull distribution introduced by Famoye et.al
(2005) and studied by Lee et.al (2007).
7.3.4 Beta-Hyperbolic Secant Distribution โ (Fischer and Vaughan. (2007))
Fisher and Vaughan (2007) introduced the beta-hyperbolic secant distribution, based on a
composition of the classical beta distribution and the hyperbolic secant distribution. They
gave some of its properties like the moments, special and limiting cases and compare it
with other distributions using a real data set.
The beta-hyperbolic secant distribution is obtained as follows:
Let
๐บ ๐ฅ =2
ฯarctan ๐๐ฅ
be the cdf of the hyperbolic secant distribution and
๐ ๐ฅ =1
ฯCosh(x)
be the pdf of the hyperbolic secant distribution
By using equation 7.1, the density function of the beta-hyperbolic secant is given by
๐ ๐ฅ; ๐, ๐, ฯ = 2ฯ arctan ๐๐ฅ
๐โ1
1 โ2ฯ arctan ๐๐ฅ
๐โ1
๐ต ๐, ๐ ฯ cosh x ______________(7.7)
๐ > 0, ๐ > 0 and ๐ฅ โ โ
Equation 7.7 is the beta-hyperbolic secant distribution introduced and studied by. Fischer
and Vaughan (2007).
7.3.5 Beta-Gamma โ (Kong et al. (2007))
Kong et.al (2007) introduced the beta-gamma distribution, based on a composition of the
classical beta distribution and the gamma distribution. They derived in their paper some
190
properties of the limit of the density function and of the hazard function. They gave an
expression for the moments when the shape parameter ๐ is an integer and made an
application of the beta-gamma distribution.
The beta-gamma distribution is obtained as follows:
Let
๐บ ๐ฅ =ฮxฮป ฯ
ฮ ฯ
be the cdf of gamma distribution where
ฮx ฯ = yฯโ1eโyx
0
dy
is the incomplete gamma function and
๐ ๐ฅ = x
ฮป
ฯโ1
eโxฮป
is the pdf of the gamma function.
By using equation 7.1, the density function of the beta-gamma distribution is given by
๐ ๐ฅ; ๐, ๐, ฯ, ฮป =
๐ฅ๐โ1๐โ
๐ฅ๐ฮx
ฮป ฯ ๐โ1 1 โ
ฮxฮป ฯ
ฮ ฯ
๐โ1
๐ต ๐, ๐ ฮ ฯ aฮปฯ___________________(7.8)
๐, ๐, ฯ, ฮป, x > 0
Equation 7.8 is the beta-gamma distribution introduced by Kong et al. (2007). They also
derived some properties of the limit of the density function and hazard function.
7.3.6 Beta-Gumbel Distribution โ (Nadarajah and Kotz (2004))
Nadarajah and Kotz (2004) introduced the beta-Gumbel distribution, based on a
composition of the classical beta distribution and the Gumbel distribution. They
calculated expressions for the rth moment, gave some particular cases, and studied the
density function.
191
The beta-Gumbel distribution is obtained as follows:
Let
๐บ ๐ฅ = exp โ exp โx โ ฮผ
ฯ
be the cdf of Gumbel distribution and
๐ ๐ฅ =1
ฯexp โ
x โ ฮผ
ฯ exp โ exp โ
x โ ฮผ
ฯ
=๐ข
ฯexp โ๐ข
is the pdf of the gumbel distribution. Where
๐ข = exp โx โ ฮผ
ฯ
By using equation 7.1, the density function of the beta-gumbel distribution is given by
๐ ๐ฅ; ๐, ๐, ๐ข, ฯ =1
๐ต(๐, ๐)[exp โ๐ข ]aโ1 [1 โ exp โ๐ข ]bโ1 .
๐ข
ฯexp โ๐ข
= ๐ข
๐๐ต ๐, ๐ eโ๐๐ข 1 โ eโ๐ข bโ1___________________________________________(7.9)
๐, ๐, ๐ข, ฯ, ๐ฅ > 0
Equation 7.9 is the beta-gumbel distribution introduced by Nadarajah and Kotz (2004).
They calculated expressions for the rth moments, gave some particular cases and studied
the density function.
7.3.7 Beta-Frรจchet Distribution (Nadarajah and Gupta (2004))
Nadarajah and Gupta (2004) introduced the beta- Frรจchet distribution, based on a
composition of the classical beta distribution and the Frรจchet distribution. They derived
in their paper some of its properties and hazard function and of the hazard functions.
They also gave an expression for the rth moment. Barreto-Souza et.al (2008) gave another
expression for the moments and derived some additional mathematical properties.
The beta- Frรจchet distribution is obtained as follows:
192
Let
๐บ ๐ฅ = exp โ x
ฯ
โฮป
be the cdf of standard Frรจchet distribution and
๐ ๐ฅ =ฮป๐ฮป
๐ฅ1+ฮปexp โ
x
ฯ โฮป
be the pdf of the Frรจchet distribution.
By using equation 7.1, the density function of the beta-Frรจchet distribution is given by
๐ ๐ฅ; ๐, ๐, ฮผ, ฯ = ฮป๐ฮป exp โa
xฯ
โฮป
1 โ exp โ xฯ
โฮป
bโ1
๐ฅ1+ฮป๐ต ๐, ๐ ______(7.10)
๐, ๐, ฮป, ฯ, x > 0
Equation 7.10 is the beta-Frรจchet distribution introduced by Nadarajah and Gupta (2004).
They derived some of its properties and gave an expression for the rth moment. Barreto-
Souza et.al (2008) gave another expression for the moments and derived some additional
mathematical properties.
7.3.8 Beta-Maxwell Distribution (Amusan (2010))
Amusan (2010) introduced the beta-Maxwell distribution, based on a composition of the
classical beta distribution and the Maxwell distribution. She defined and studied the
three-parameter Maxwell distribution. She also discussed various properties of the
distribution.
The beta-maxwell distribution is obtained as follows:
Let
๐บ ๐ฅ =2ฮณ
32 ,
๐ฅ2
2
ฯ
be the cdf of maxwell distribution and
193
๐ ๐ฅ = 2
ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3
be the pdf of the maxwell distribution.
By using equation 7.1, the density function of the beta-maxwell distribution is given by
๐ ๐ฅ; ๐, ๐, ฮฑ =
2ฮณ
32
,๐ฅ2
2ฮฑ
ฯ
aโ1
1 โ2ฮณ
32
,๐ฅ2
2ฮฑ
ฯ
bโ1
2ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3
๐ต ๐, ๐
=1
๐ต ๐, ๐
2
ฯฮณ
3
2,๐ฅ2
2ฮฑ
aโ1
1 โ2
ฯฮณ
3
2,๐ฅ2
2ฮฑ
bโ1
2
ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3______(7.11)
๐๐๐ ๐, ๐, ฮฑ, ๐ฅ > 0 ๐๐๐ ฮณ ๐, ๐ is an incomplete gamma function.
Equation 7.11 is the beta-Maxwell distribution introduced by Amusan (2010).
7.3.9 Beta-Pareto Distribution (Akinsete et.al (2008))
Akinsete et.al (2008) introduced the beta-Pareto distribution, based on a composition of
the classical beta distribution and the Pareto distribution. They defined and studied
properties of the four-parameter beta-pareto distribution.
The beta-pareto distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 โ ๐ฅ
๐
โk
be the cdf of pareto distribution and
๐ ๐ฅ =๐๐๐
๐ฅ๐+1
be the pdf of the pareto distribution.
By using equation 7.1, the density function of the beta-pareto distribution is given by
194
๐ ๐ฅ; ๐, ๐, ๐, ๐ = 1 โ
๐ฅ๐
โk
aโ1
1 โ 1 โ ๐ฅ๐
โk
bโ1
๐๐๐
๐ฅ๐+1
๐ต ๐, ๐
=๐
๐๐ต ๐, ๐ 1 โ
๐ฅ
๐ โk
aโ1
๐ฅ
๐ โ๐๐โ1
_______________________(7.12)
๐, ๐, ๐, ๐, ๐ฅ > 0
Equation 7.12 is the beta-pareto distribution defined by Akinsete et.al.
7.3.10 Beta-Rayleigh Distribution (Akinsete and Lowe (2009))
Akinsete and Lowe (2009) introduced the beta-Rayleigh distribution, based on a
composition of the classical beta distribution and the Rayleigh distribution.
The beta-Rayleigh distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 โ ๐โ
๐ฅ2
2ฮฑ2
be the cdf of Rayleigh distribution and
๐ ๐ฅ =๐ฅ
ฮฑ2 ๐
โ ๐ฅ
ฮฑ 2
2
be the pdf of the Rayleigh distribution.
By using equation 7.1, the density function of the beta-Rayleigh distribution is given by
๐ ๐ฅ; ๐, ๐, ๐ผ =
1 โ ๐โ
๐ฅ2
2ฮฑ2
aโ1
1 โ 1 โ ๐โ
๐ฅ2
2ฮฑ2
bโ1๐ฅฮฑ2 ๐
โ ๐ฅ
ฮฑ 2
2
๐ต ๐, ๐
=๐ฅ
ฮฑ2๐ต ๐, ๐ 1 โ ๐
โ๐ฅ2
2ฮฑ2
aโ1
๐โ๐
๐ฅ
ฮฑ 2
2
__________________________(7.13)
๐, ๐, ๐, ๐ผ > 0
195
Equation 7.13 is the beta-Rayleigh distribution. If ๐ = ๐ = 1, equation 7.13 reduces to
Rayleigh distribution with parameter ๐ผ. When ๐ = 1, the beta-rayleigh distribution
reduces to the Rayleigh distribution with parameter ๐ =๐ผ
๐.
7.3.11 Beta-generalized-logistic of type IV distribution (Morais (2009))
Morais (2009) introduced the beta-generalized logistic of type IV distribution, based on a
composition of the classical beta distribution and the generalized logistic of type IV
distribution. She discussed some of its special cases that belong to the beta-G such as the
beta โgeneralized logistic of types I, II, and III and some related distributions like the
beta-beta prime and the beta-F. The generalized logistic of type IV distribution was
proposed by Prentice (1976) as an alternative to modeling binary response data with the
usual symmetric logistic distribution.
The beta-generalized logistic of type IV distribution is obtained as follows:
Let
๐บ ๐ฅ =
B 11+eโx
(p, q)
B(p, q)
be the cdf of the generalized logistic of type IV distribution given by a normalized
incomplete beta function and
๐ ๐ฅ =eโqx
B p, q (1 + eโx )p+q
be the pdf of the generalized logistic of type IV distribution.
By using equation 7.1, the density function of the beta-generalized logistic of type IV
distribution is given by
๐ ๐ฅ; ๐, ๐, p, q
=B a, b 1โaโb
B a, b
eqx
1 โ eโx p+q[B 1
1+eโx(p, q)]aโ1[B eโx
1+eโx
(q, p)]bโ1 ______(7.14)
๐, ๐, p, q, > 0, ๐ฅ > ๐
196
Equation 7.14 is the beta-generalized logistic of type IV distribution introduced by
Morais (2009).
7.3.12 Beta-generalized-logistic of type I distribution (Morais (2009))
Morais (2009) introduced the beta-generalized logistic of type I distribution which is a
special case of the beta-generalized logistic of type IV distribution with q = 1.
The density function of the beta-generalized logistic of type I distribution is given by
๐ ๐ฅ; ๐, ๐, p =peโx[ 1 + eโax p โ 1]bโ1
B a, b 1 + eโx a+pb, _____________________________(7.15)
๐, ๐, p > 0, ๐ฅ > ๐
Equation 7.15 is the beta-generalized logistic of type I distribution introduced by Morais
(2009).
7.3.13 Beta-generalized-logistic of type II distribution (Morais (2009))
Morais (2009) introduced the beta-generalized logistic of type II distribution which is a
special case of the beta-generalized logistic of type IV distribution with p = 1. This
distribution can also be obtained through the transformation ๐ = โ๐ where Y follows the
beta-generalized logistic of type I distribution.
The density function of the beta-generalized logistic of type II distribution is given by
๐ ๐ฅ; ๐, ๐, q =qeโbqx
B a, b 1 + eโx qb +1 1 โ
eโqx
B a, b 1 + eโx q ________(7.16)
๐, ๐, q, > 0, ๐ฅ > ๐
Equation 7.16 is the beta-generalized logistic of type II distribution introduced by Morais
(2009).
197
7.3.14 Beta-generalized-logistic of type III distribution (Morais (2009))
Morais (2009) introduced the beta-generalized logistic of type III distribution which is a
special case of the beta-generalized logistic of type IV distribution with p = q. This
distribution is symmetric when a=b.
The density function of the beta-generalized logistic of type III distribution is given by
๐ ๐ฅ; ๐, ๐, q =B q, q 1โaโb
B a, b
eโqx
1 + eโx 2q[B 1
1+eโx(q, q)]aโ1[B eโx
1+eโx
(q, q)]bโ1 ___(7.17)
๐, ๐, q, > 0, ๐ฅ > ๐
Equation 7.17 is the beta-generalized logistic of type III distribution introduced by
Morais (2009).
7.3.15 Beta-beta prime distribution (Morais (2009))
The beta-beta prime distribution can be obtained from the beta-generalized logistic of
type IV distribution using the transformation
๐ฅ = eโy
where Y is a random variable following the beta-generalized logistic of type IV
distribution.
The density function of the beta-beta prime distribution is given by
๐ ๐ฅ; ๐, ๐, ๐, q =๐ต ๐, ๐ 1โ๐โ๐
๐ต ๐, ๐
๐ฅ๐โ1
1 + ๐ฅ ๐+๐[๐ต ๐ฅ
1+๐ฅ(๐, ๐)]๐โ1[๐ต 1
1+๐ฅ(๐, ๐)]๐โ1 ____(7.18)
๐, ๐, ๐, > 0, ๐ฅ > ๐
Equation 7.18 is the beta-beta prime distribution introduced by Morais (2009).
198
7.3.16 Beta-F distribution (Morais (2009))
The beta-F distribution can be obtained as follows:
Let
๐บ ๐ฅ =1
B ๐ข2 ,
๐ฃ2
B
vu x
1+ vu x
๐ฃ
2,๐ข
2
be the cdf of the F distribution and
๐ ๐ฅ =
vu
q2
B ๐ข2 ,
๐ฃ2
xv2โ1
1 + vu x
(u+v)/2
be the pdf of the F distribution.
By using equation 7.1, the density function of the beta-F distribution is given by
๐ ๐ฅ; ๐, ๐, u, v
=B a, b โ1
B u2 ,
v2
vu
v2
xv2โ1
1 + vu x
(u+v)/2
I
vu x
1+ vu x
v
2,u
2
aโ1
I 1
1+ vu x
v
2,u
2
bโ1
__(7.19)
๐, ๐, u, v, x > 0
Equation 7.19 is the beta-F distribution. It can easily be verified that if Y follows the beta
generalized logistic of type IV distribution with parameters ๐, ๐, u, v then F =v
ueโy
follows the beta F distribution with parameters ๐, ๐, 2u, 2v,.
7.3.17 Beta-Burr XII distribution (Paranaรญba et.al (2010))
Paranaรญba et.al (2010) introduced the beta-burr XII distribution, based on a composition
of the classical beta distribution and the burr XII distribution. They defined and studied
the five parameter beta-burr XII distribution. They also derived some of its properties like
moment generating function, mean and proposed methods of estimating its parameters.
199
The beta- burr XII distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 โ 1 + ๐ฅ
q
p
โ๐
be the cdf of burr XII distribution and
๐ ๐ฅ = ๐๐๐โ๐ 1 + ๐ฅ
q
p
โ๐โ1
๐ฅ๐โ1
be the pdf of the burr XII distribution.
By using equation 7.1, the density function of the beta- burr XII distribution is given by
๐ ๐ฅ; ๐, ๐, ๐, ๐, ๐
=
1 โ 1 + ๐ฅq
p
โ๐
aโ1
1 + ๐ฅq
p
โ๐
bโ1
๐๐๐โ๐ 1 + ๐ฅq
p
โ๐โ1
๐ฅ๐โ1
๐ต ๐, ๐
=๐๐๐โ๐๐ฅ๐โ1
๐ต ๐, ๐ 1 โ 1 +
๐ฅ
q
p
โ๐
aโ1
1 + ๐ฅ
q
p
โ๐๐โ1
_________(7.20)
for ๐, ๐, ๐, ๐, ๐, ๐ฅ > 0.
Equation 7.20 is the beta-burr XII distribution introduced by Paranaรญba et.al (2010). The
beta-burr XII distribution contains well-known distributions as special sub-models.
For ๐ = ๐ = 1, equation 7.20 gives the burr XII distribution
๐ ๐ฅ; ๐, ๐, ๐ = ๐๐๐โ๐๐ฅ๐โ1 1 + ๐ฅ
q
p
โ๐โ1
For ๐ = 1, equation 7.20 gives the exponentiated burr XII distribution
๐ ๐ฅ; ๐, ๐, ๐, ๐ = ๐๐๐๐โ๐๐ฅ๐โ1 1 โ 1 + ๐ฅ
q
p
โ๐
aโ1
1 + ๐ฅ
q
p
โ๐โ1
For ๐ = ๐ = 1, ๐ = ๐โ1 and ๐ = 1 equation 7.20 gives the log logistic distribution
๐ ๐ฅ; ๐, ๐, = ๐๐๐๐ฅ๐โ1 1 + ๐ฅ๐ p โ2
200
For ๐ = ๐ = 1 equation 7.20 gives the beta-pareto type II
๐ ๐ฅ; ๐, ๐, ๐, =๐๐
๐ 1 +
๐ฅ
q
โ๐๐โ1
For ๐ = ๐ = ๐ = 1 equation 7.20 gives the beta-pareto type II
๐ ๐ฅ; ๐, ๐, =๐
๐ 1 +
๐ฅ
q
โ๐โ1
7.3.18 Beta-Dagum distribution (Condino and Domma (2010))
Condino and Domma (2010) introduced the beta-dagum distribution, based on a
composition of the classical beta distribution and the dagum distribution. They defined
and studied its properties.
The beta- dagum distribution is obtained as follows:
Let
๐บ ๐ฅ = 1 + ๐๐ฅโ๐ โ๐
be the cdf of dagum distribution and
๐ ๐ฅ = ๐ ๐๐๐ฅโ๐โ1
(1 + ๐๐ฅโ๐)1+a
be the pdf of the dagum distribution.
By using equation 7.1, the density function of the beta- dagum distribution is given by
๐ ๐ฅ =1
๐ต ๐, ๐ 1 + ๐๐ฅโ๐ โ๐ aโ1 1 โ 1 + ๐๐ฅโ๐ โ๐ bโ1
๐ ๐๐๐ฅโ๐โ1
(1 + ๐๐ฅโ๐)1+a______(7.21)
for ๐, ๐, ๐, ๐, ๐ฅ > 0.
Equation 7.21 is the beta-dagum distribution introduced by Condino and Domma (2010).
The beta-dagum distribution contains the beta-fisk (log-logistic) distribution when
๐ = 1 and ๐ =1
๐ and the dagum distribution when ๐ = ๐ = 1.
201
7.3.19 Beta-(fisk) log logistic distribution
This is a new distribution referred to by Paranaรญba et.al (2010) as a special sub-model of
beta-burr XII distribution. It can be constructed based on a composition of the classical
beta distribution and the log logistic (fisk) distribution as follows.
Let
๐บ ๐ฅ = ๐๐ฅ p
1 + ๐๐ฅ p
be the cdf of log logistic distribution and
๐ ๐ฅ =๐๐ ๐๐ฅ pโ1
1 + ๐๐ฅ p 2
be the pdf of the log logistic distribution.
By using equation 7.1, the density function of the beta- log logistic distribution is given
by
๐ ๐ฅ; ๐, ๐, ๐, ๐ =
๐๐ฅ p
1 + ๐๐ฅ p aโ1
1 โ ๐๐ฅ p
1 + ๐๐ฅ p bโ1
๐๐ ๐๐ฅ pโ1
1 + ๐๐ฅ p 2
๐ต ๐, ๐
= ๐๐ ๐๐ฅ apโ1
๐ต ๐, ๐ 1 + ๐๐ฅ p a+b___________________________________________(7.22)
for ๐, ๐, ๐, ๐, ๐ฅ > 0.
Equation 7.22 is the beta-log logistic distribution referred to by Paranaรญba et.al (2010).
7.3.20 Beta-generalized half normal Distribution (Pescim, R.R, et.al (2009))
Pescim, R.R, et.al (2009) introduced the beta-generalized half normal distribution, based
on a composition of the classical beta distribution and the generalized half normal
distribution. They derived expansions for the cumulative distribution and density
functions. They also obtained formal expressions for the moments of the four-parameter
beta- generalized half normal distribution.
202
The beta- generalized half normal distribution is obtained as follows:
Let
๐บ ๐ฅ = 2๐ ๐ฅ
๐
ฮฑ
โ 1 = erf ๐ฅ๐
๐ผ
2
Where
๐ ๐ฅ =1
2 1 + erf
๐ฅ
2 and erf ๐ฅ =
2
๐ ๐๐ฅ๐ โ๐ก2
๐ฅ
0
๐๐ก
be the cdf of generalized half normal distribution and
๐ ๐ฅ = 2
ฯ ฮฑ
๐ฅ
๐ฅ
๐
ฮฑ
๐โ12 ๐ฅ๐
๐๐ถ
, 0 โค ๐ฅ
be the pdf of the generalized half normal distribution.
By using equation 7.1, the density function of the beta-generalized half normal
distribution is given by
๐ ๐ฅ; ๐, ๐, ฮฑ, ๐ = 2๐
๐ฅ๐
ฮฑ
โ 1 aโ1
2 โ 2๐ ๐ฅ๐
ฮฑ
bโ1
2ฯ
ฮฑ๐ฅ
๐ฅ๐
ฮฑ
๐โ12 ๐ฅ๐
๐๐ถ
๐ต ๐, ๐
๐, ๐, ฮฑ, ๐ > 0 ______________________________________________________(7.23)
Equation 7.23 is the beta- generalized half normal distribution defined by Pescim, R.R,
et.al. (2009) It contains as special sub-models well known distributions.
for ฮฑ = 1, it reduces to beta half normal distribution given as
๐ ๐ฅ; ๐, ๐, ๐ = 2๐
๐ฅ๐ โ 1
aโ1
2 โ 2๐ ๐ฅ๐
bโ1 2
ฯ๐โ
12 ๐ฅ๐ ๐
๐
๐ต ๐, ๐
๐, ๐, ๐ > 0
for a = b = 1, it reduces to generalized half normal distribution given by
203
๐ ๐ฅ; ฮฑ, ๐ = 2
ฯ ฮฑ
๐ฅ
๐ฅ
๐
ฮฑ
๐โ12 ๐ฅ๐ ๐๐ถ
ฮฑ, ๐ > 0
further if ฮฑ = 1, the generalized half normal distribution gives the half normal
distribution
๐ ๐ฅ; ฮฑ, ๐ = 2
ฯ
๐โ12 ๐ฅ๐ ๐
๐
ฮฑ, ๐ > 0
for ๐ = 1, it leads to the exponentiated generalized half normal distribution given as
๐(๐ฅ; ๐, ๐ผ, ๐) = ๐ 2
ฯ ฮฑ
๐ฅ
๐ฅ
๐
ฮฑ
๐โ12 ๐ฅ๐ ๐๐ถ
2๐ ๐ฅ
๐
ฮฑ
โ 1 aโ1
7.4 Beta Exponentiated Generated Distributions
Beta exponentiated generated distributions are based on the power of the cdfs.
7.4.1 Exponentiated generated distribution
Let
๐น ๐ฅ = ๐บ ๐ฅ ๐ผ
Where ๐บ ๐ฅ is the old or parent cdf and ๐น ๐ฅ is the new cdf
The new cdf in terms of the old is therefore given by
๐(๐ฅ) = ๐ผ ๐บ ๐ฅ ๐ผโ1๐(๐ฅ)
The following are some of the examples of the exponentiated exponential distributions
204
7.4.2 Exponentiated exponential distribution (Gupta and Kundu (1999))
Let
๐ ๐ฅ =1
ฮฒ ๐
โ ๐ฅ๐ฝ
, ฮฒ > 0, 0 < ๐ฅ < โ
๐บ ๐ฅ = 1 โ ๐โ(
๐ฅ๐ฝ
)
๐น ๐ฅ = 1 โ ๐โ
๐ฅ๐ฝ
๐ผ
and
๐ ๐ฅ =๐ผ
ฮฒ 1 โ ๐
โ ๐ฅ๐ฝ
๐ผโ1
๐โ
๐ฅ๐ฝ
________________________(7.24)
Equation 7.24 is an example of the exponentiated generated distribution known as the
exponentiated exponential distribution which has been extensively covered by Gupta and
Kundu (1999).
7.4.3 Exponentiated Weibull distribution (Mudholkar et.al. (1995))
Let
๐ ๐ฅ = ๐
ฮฒp ๐ฅ๐โ1๐
โ ๐ฅ๐ฝ
๐
๐บ ๐ฅ = 1 โ ๐โ
๐ฅ๐ฝ
๐
๐น ๐ฅ = 1 โ ๐โ
๐ฅ๐ฝ
๐
๐ผ
๐ ๐ฅ =๐ผ
๐ฝ๐ 1 โ ๐
โ ๐ฅ๐ฝ
๐
๐ผโ1
๐โ
๐ฅ๐ฝ
๐
_______________________(7.25)
Equation 7.25 is the exponentiated Weibull distribution proposed by Mudholkar et.al.
(1995).
205
7.4.4 Exponentiated Pareto distribution (Gupta et.al (1998))
Let
๐ ๐ฅ = ๐qp๐๐ฅโ๐โ1 1 โ ๐ฅ
๐
โ๐
๐โ1
๐บ ๐ฅ = 1 โ ๐ฅ
๐ โ๐
๐
๐น ๐ฅ = 1 โ ๐ฅ
๐ โ๐
๐
๐ผ
๐ ๐ฅ = ๐ผ 1 โ ๐ฅ
๐
โ๐
๐
๐ผโ1๐๐
๐๐ฅโ๐๐โ1 ___________________________(7.26)
Equation 7.26 is the exponentiated Pareto distribution.
7.4.5 Beta exponentiated generated distribution
๐ ๐ฅ =๐ ๐ฅ ๐ผ ๐บ ๐ฅ ๐ผโ1 ๐บ ๐ฅ ๐ผ ๐โ1 1 โ ๐บ ๐ฅ ๐ผ ๐โ1
๐ต ๐, ๐
=๐ ๐ฅ ๐ผ ๐บ ๐ฅ ๐๐ผโ1 1 โ ๐บ ๐ฅ ๐ผ ๐โ1
๐ต(๐, ๐), ๐ > 0, ๐ผ > 0, ๐ > 0,
โโ < ๐ฅ < โ, 0 < ๐บ ๐ฅ ๐ผ < 1
is the exponentiated generator distribution or exponentiated generated distribution. Below
are some of the examples on exponentiated generated distributions.
7.4.6 Generalized beta generated distributions
To obtain generalized beta distributions, we simply replace ๐บ ๐ฅ by [๐บ ๐ฅ ]๐ in the beta
generated distribution. Some of the many distributions which can arise within the
generalized beta generated class of distributions can be derived from an arbitrary parent
cumulative distribution function (cdf) ๐บ ๐ฅ , and the probability density function (pdf)
๐ ๐ฅ of the new class of generalized beta distributions is defined as
206
๐(๐บ ๐ฅ); ๐, ๐ ) = ๐ ๐(๐ฅ)[๐บ(๐ฅ)]๐๐โ1 1 โ ๐บ ๐ฅ
๐ ๐โ1
๐ต ๐, ๐ ______________(7.27)
where ๐ > 0, ๐ > 0 and ๐ > 0 are additional shape parameters which aim to introduce
skewness and to vary tail weight and ๐ ๐ฅ = ๐๐บ(๐ฅ)/๐๐ฅ. the new class of generalized
beta distributions generated by equation 7.27 includes as special sub-models the beta
generated distributions discussed in section 7.2 above. Eugene et.al (2002) introduced the
beta generated distributions for ๐ = 1 i.e
๐(๐ฅ) =๐(๐ฅ)[๐บ(๐ฅ)]๐โ1 1 โ ๐บ ๐ฅ
๐โ1
๐ต ๐, ๐
and Cordeiro and Castro (2010) introduced Kumaraswamy (Kw) generated distributions
for ๐ = 1 i. e
๐(๐ฅ) = ๐ ๐ ๐(๐ฅ)[๐บ(๐ฅ)]๐โ1 1 โ ๐บ ๐ฅ ๐ ๐โ1
Alternatively, the generalized beta generated distribution in equation 7.27 can be derived
from the generalized beta distribution of the first kind in equation 5.1 as follows:
Letting ๐ = 1
๐ ๐ฆ = ๐ ๐ฆ๐๐โ1 1 โ
๐ฆ1
๐
๐โ1
. ๐ ๐ฅ
1๐๐๐ต ๐, ๐ , 0 < ๐ฆ๐ < 1, ๐, ๐, ๐ > 0
Now consider
๐น(๐ฅ) = ๐ ๐ฆ๐๐โ1 1 โ ๐ฆ๐ ๐โ1
๐ต ๐, ๐
๐บ(๐ฅ)
0
๐๐ฆ
โด ๐(๐ฅ) = ๐ ๐(๐ฅ) ๐บ ๐ฅ
๐๐โ1 1 โ ๐บ ๐ฅ
๐ ๐โ1
๐ต ๐, ๐ , โโ < ๐ฅ < โ, ๐, ๐, ๐ > 0
7.4.7 Generalized beta exponential distribution (Barreto-Souza et al. (2010))
Let
๐บ ๐ฅ = 1 โ exp(โฮปx)
be the cdf of exponential distribution with parameter ฮป, and
๐ ๐ฅ = ฮป exp(โฮปx)
207
be the pdf of exponential distribution.
By using equation 7.27, the density function of the generalized beta-exponential
distribution is given by
๐ ๐ฅ; ๐, ๐, ๐, ฮป
= ๐ ฮป exp(โฮปx)[1 โ exp(โฮปx) ]๐๐โ1 1 โ 1 โ exp โฮปx ๐ ๐โ1
๐ต ๐, ๐ _________(7.28)
Equation 7.28 is the generalized beta-exponential distribution.
7.4.8 Generalized beta-Weibull distribution
Let
๐บ ๐ฅ = 1 โ eโ
xฮฒ
c
be the cdf of Weibull distribution and
๐ ๐ฅ = ๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
be the pdf of Weibull distribution.
By using equation 7.27, the density function of the generalized beta-weibull distribution
is given by
๐ ๐ฅ; ๐, ๐, ๐, ๐, ๐ฝ
=
๐ ๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
[1 โ eโ
xฮฒ
c
]๐๐โ1 1 โ 1 โ eโ
xฮฒ
c
๐
๐โ1
๐ต ๐, ๐ ___(7.29)
Equation 7.29 is the generalized beta-Weibull distribution.
208
7.4.9 Generalized beta-hyperbolic secant distribution
Let
๐บ ๐ฅ =2
ฯarctan ๐
๐ฅโ๐๐
be the cdf of the hyperbolic secant distribution and
๐ ๐ฅ =1
ฯฯCosh ๐ฅ โ ๐
๐
be the pdf of the hyperbolic secant distribution
By using equation 7.27, the density function of the generalized beta-hyperbolic secant is
given by
๐ ๐ฅ; ๐, ๐, ฮผ, ฯ, ฯ = ๐
2ฯ arctan ๐
๐ฅโ๐๐
๐๐โ1
1 โ 2ฯ arctan ๐
๐ฅโ๐๐
๐
๐โ1
๐ต ๐, ๐ ฯฯ cosh ๐ฅ โ ๐
๐ ___(7.30)
๐ > 0, ๐ > 0, ฮผ > 0, ๐ > 0 and ๐ฅ โ โ
Equation 7.30 is the generalized beta-hyperbolic secant distribution. If ฮผ = 0 and ฯ = 1,
the generalized beta-hyperbolic secant distribution reduces to the standard beta-
hyperbolic distribution in equation 7.7.
7.4.10 Generalized beta-normal distribution
The generalized beta-normal distribution is obtained as follows:
Let
๐บ ๐ฅ = ฮฆ ๐ฅ โ ๐
๐
be the cumulative density function of normal distribution with parameters ๐ and ๐, and
๐ ๐ฅ = ๐ ๐ฅ โ ๐
๐
be the probability density function of the normal distribution.
209
By using equation 7.27, the density function of the generalized beta-normal distribution is
given by
๐ ๐ฅ; ๐, ๐, ๐, ๐, ๐ =๐๐โ1[ฮฆ
๐ฅ โ ๐๐ ]๐๐โ1 1 โ ฮฆ
๐ฅ โ ๐๐
p
๐โ1
๐ต ๐, ๐ ๐
๐ฅ โ ๐
๐ ___(7.31)
Where ๐ > 0, ๐ > 0, ๐ > 0, ๐ > 0 ๐ โ โ ๐๐๐ ๐ฅ โ โ
Equation 7.31 is the generalized beta-normal distribution. If ฮผ = 0 , ฯ = 1 and ๐ = 1 ,
the generalized beta-normal distribution reduces to the standard beta-normal distribution
proposed by Eugene et.al (2002). If ฮผ = 0, ฯ = 1, ๐ = 1 and ๐๐ = 2 the generalized
beta-normal distribution coincides with the skew-normal distribution.
7.4.11 Generalized beta-log normal distribution
The generalized beta- log normal distribution is obtained as follows:
Let
๐บ ๐ฅ = ฮฆ log ๐ฅ
be the cumulative density function of standard log normal distribution, and
๐ ๐ฅ =๐ log ๐ฅ
๐ฅ
be the probability density function of the standard log normal distribution.
By using equation 7.27, the density function of the generalized beta-log normal
distribution is given by
๐ ๐ฅ; ๐, ๐, ๐ =๐๐ log ๐ฅ [ฮฆ log๐ฅ ]๐๐โ1 1 โ ฮฆ log ๐ฅ ๐ ๐โ1
๐ฅ๐ต ๐, ๐ _______(7.32)
Where ๐ > 0, ๐ > 0, ๐ > 0 and ๐ฅ โ โ
Equation 7.32 is the generalized beta-log normal distribution.
210
7.4.12 Generalized beta-Gamma distribution
The generalized beta-gamma distribution is obtained as follows:
Let
๐บ ๐ฅ =ฮxฮป ฯ
ฮ ฯ
be the cumulative density function of gamma distribution, and
๐ ๐ฅ = x
ฮป
ฯโ1
eโxฮป
be the probability density function of the gamma distribution.
By using equation 7.27, the density function of the generalized beta-gamma distribution
is given by
๐ ๐ฅ; ๐, ๐, ๐, ฯ, ฮป =
xฮป
ฯโ1
eโxฮป
ฮxฮป ฯ
ฮ ฯ
๐๐โ1
1 โ ฮxฮป ฯ
ฮ ฯ
๐
๐โ1
๐ต ๐, ๐ ___(7.33)
Where ๐ > 0, ๐ > 0, ๐ > 0, ฯ > 0, ๐ > 0 and ๐ฅ โ โ
Equation 7.33 is the generalized beta-gamma distribution.
7.4.13 Generalized beta-Frรจchet distribution
The generalized beta- Frรจchet distribution is obtained as follows:
Let
๐บ ๐ฅ = exp โ x
ฯ
โฮป
be the cumulative density function of Frรจchet distribution, and
๐ ๐ฅ =ฮป
ฯexp โ
x
ฯ
โฮป
๐ฅ๐โ1
211
be the probability density function of the Frรจchet distribution.
By using equation 7.27, the density function of the generalized beta-Frรจchet distribution
is given by
๐ ๐ฅ; ๐, ๐, ๐, ฯ, ฮป
=
๐ฮปฯ exp โ
xฯ
โฮป
๐ฅ๐โ1 exp โ xฯ
โฮป
๐๐โ1
1 โ exp โ xฯ
โฮป
๐
๐โ1
๐ต ๐, ๐ ___(7.34)
Where ๐ > 0, ๐ > 0, ๐ > 0, ฯ > 0, ๐ > 0 and ๐ฅ โ โ
Equation 7.34 is the generalized beta-Frรจchet distribution.
212
8 Chapter VIII: Generalized Beta Distributions
Based on Special Functions
8.1 Introduction
Beta, F, Gamma functions are among the many special functions. Most of these special
functions can be expressed as hypergeometric function. This chapter looks at the
generalized beta distributions based on special functions such as the hypergeometric
distribution function, the confluent hypergeometric function, gauss Hypergeometric
function among others.
8.2 Confluent Hypergeometric Function
The confluent hypergeometric (Kummerโs) function is denoted by the symbol
1๐น1 ๐; ๐; ๐ฅ and represent the series
1๐น1 ๐; ๐, ๐ฅ = 1 +๐
๐
๐ฅ
1!+
๐ ๐ + 1
๐ ๐ + 1
๐ฅ2
2!+
๐ ๐ + 1 (๐ + 2)
๐ ๐ + 1 (๐ + 2)
๐ฅ3
3!+ โฏ
= ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
n=0
for ๐ โ 0, โ1, โ2, โฆ It can simply be expressed in the following form
1๐น1 ๐; ๐, ๐ฅ = ๐ n ๐ฅ๐
๐ ๐ ๐!
โ
n=0
, c โ 0, โ1, โ2, โฆ
where ๐ n is the Pochhammerโs symbol defined by
๐ n = a a + 1 โฆ a + n โ 1
=ฮ(a + 1)
ฮ(a); n is a positive integer
๐ 0 = 1
If a is a non-positive integer, the series terminates.
The integral representation is given as
1๐น1 ๐; ๐, ๐ฅ =ฮ(c)
ฮ(a)ฮ(c โ a) ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข ๐๐ข
1
0
_______________________(8.1)
213
where ๐ > ๐ > 0
We verify that equation 8.1 is a distribution
๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข ๐๐ข1
0
= ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1 (๐ฅ๐ข)๐
๐!
โ
๐=0
๐๐ข1
0
= 1
๐! ๐ฅ๐๐ข๐+๐โ1
1
0
1 โ ๐ข ๐โ๐โ1
โ
๐=0
๐๐ข
= ๐ฅ๐
๐! ๐ข๐+๐โ1
1
0
1 โ ๐ข ๐โ๐โ1
โ
๐=0
๐๐ข
= ๐ฅ๐
๐!B(n + a, c โ a)
โ
๐=0
โน ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข ๐๐ข =1
0
๐ฅ๐
๐!
ฮ(n + a)ฮ(c โ a)
ฮ(n + c)
โ
๐=0
but ฮ n + a = (n + a โ 1)ฮ n + a โ 1
= a a + 1 a + 2 โฆ n + a โ 1 ฮ(a)
โน ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข ๐๐ข =1
0
a a + 1 a + 2 โฆ n + a โ 1 ฮ(a)ฮ(c โ a)
c c + 1 c + 2 โฆ n + c โ 1 ฮ(c)
๐ฅ๐
๐!
= 1๐น1 ๐; ๐, ๐ฅ B(a, c โ a)
Thus
1๐น1 ๐; ๐, ๐ฅ =1
๐ต(๐, ๐ โ ๐) ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข ๐๐ข
1
0
8.3 The Differential Equations
The first and second order differential equations can be expressed as follows;
๐
๐๐ฅ 1๐น1 ๐; ๐; ๐ฅ = 0 +
๐
๐
1
1!+
๐ ๐ + 1
๐ ๐ + 1
2๐ฅ
2!+
๐ ๐ + 1 (๐ + 2)
๐ ๐ + 1 (๐ + 2)
3๐ฅ
3!+ โฏ
=๐
๐+
๐
๐
๐ + 1
๐ + 1
๐ฅ
1!+
๐
๐
๐ + 1 (๐ + 2)
๐ + 1 (๐ + 2)
๐ฅ2
2!+ โฏ
=๐
๐ 1 +
๐ + 1
๐ + 1
๐ฅ
1!+
๐ + 1 ๐ + 2
๐ + 1 ๐ + 2
๐ฅ2
2!+ โฏ
214
=๐
๐ 1๐น1 ๐ + 1; ๐ + 1; ๐ฅ
๐2
๐๐ฅ2 1๐น1 ๐; ๐; ๐ฅ =
๐
๐ 0 +
๐ + 1
๐ + 1 +
๐ + 1 ๐ + 2
๐ + 1 ๐ + 2
๐ฅ
1!+ โฏ
=๐
๐
๐ + 1
๐ + 1 1 +
๐ + 2
๐ + 2
๐ฅ
1!+ โฏ
=๐ ๐ + 1
๐ ๐ + 1 1๐น1 ๐ + 2; ๐ + 2; ๐ฅ
In applied mathematics, special functions are used in solving differential equation. Thus
1๐น1 ๐; ๐; ๐ฅ is a solution to a certain differential equation which is derived as follows:-
Using the operator
๐ฟ = ๐ฅ๐
๐๐ฅ
Therefore
๐ฟ๐ฅ๐ = ๐ฅ๐
๐๐ฅ๐ฅ๐ = ๐ฅ๐๐ฅ๐โ1 = ๐๐ฅ๐
๐ฟ + ๐ โ 1 ๐ฅ๐ = ๐ฟ๐ฅ๐ + (๐ โ 1)๐ฅ๐
โด ๐ฟ + ๐ โ 1 ๐ฅ๐ = (๐ + ๐ โ 1)๐ฅ๐
๐ฟ ๐ฟ + ๐ โ 1 ๐ฅ๐ = ๐ฟ(๐ + ๐ โ 1)๐ฅ๐
= ๐ฟ๐ฅ๐ (๐ + ๐ โ 1)
= ๐๐ฅ๐ (๐ + ๐ โ 1)
โด ๐ฟ ๐ฟ + ๐ โ 1 ๐ฅ๐ = ๐(๐ + ๐ โ 1)๐ฅ๐
๐ฟ + ๐ ๐ฟ + ๐ ๐ฅ๐ = ๐ฟ + ๐ ๐ฟ๐ฅ๐ + ๐๐ฅ๐
= ๐ฟ + ๐ ๐๐ฅ๐ + ๐๐ฅ๐
= ๐ฟ + ๐ ๐๐ฅ๐ + ๐ฟ + ๐ ๐๐ฅ๐
= ๐ฟ๐๐ฅ๐ + ๐๐๐ฅ๐ + ๐ฟ๐๐ฅ๐ + ๐๐๐ฅ๐
= ๐2๐ฅ๐ + ๐๐๐ฅ๐ + ๐๐๐ฅ๐ + ๐๐๐ฅ๐
= (๐2 + ๐๐ + ๐๐ + ๐๐)๐ฅ๐
= (๐(๐ + ๐) + ๐(๐ + ๐))๐ฅ๐
โด ๐ฟ + ๐ ๐ฟ + ๐ ๐ฅ๐ = ๐ + ๐ (๐ + ๐) ๐ฅ๐
215
๐ฟ ๐ฟ + ๐ โ 1 1๐น1 ๐; ๐; ๐ฅ = ๐ฟ ๐ฟ + ๐ โ 1 ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
n=0
= ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 1)๐ฟ ๐ฟ + ๐ โ 1
๐ฅ๐
๐!
โ
n=0
= ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 1)๐ ๐ + ๐ โ 1
๐ฅ๐
๐!
โ
n=0
= ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 2)
๐ฅ๐
(๐ โ 1)!
โ
n=0
= ๐ฅ ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐ โ 1)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 2)
๐ฅ๐โ1
(๐ โ 1)!
โ
n=1
= ๐ฅ ๐ ๐ + 1 (๐ + 2) โฆ (๐ + ๐)
๐ ๐ + 1 ๐ + 2 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
n=1
by replacing n with n+1
= ๐ฅ ๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1 (๐ + ๐)
๐ฅ๐
๐!
โ
n=0
= ๐ฅ ๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1 (๐ + ๐ฟ)
๐ฅ๐
๐!
โ
n=0
= ๐ฅ(๐ + ๐ฟ) ๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 ๐ + 2 โฆ ๐ + ๐ โ 1
๐ฅ๐
๐!
โ
n=0
โด ๐ฟ ๐ฟ + ๐ โ 1 1๐น1 ๐; ๐; ๐ฅ = ๐ฅ(๐ + ๐ฟ) 1๐น1 ๐; ๐; ๐ฅ
i.e
๐ฟ ๐ฟ + ๐ โ 1 ๐ฆ = ๐ฅ(๐ + ๐ฟ)๐ฆ
where
๐ฆ = 1๐น1 ๐; ๐; ๐ฅ
โด [๐ฟ2 + ๐ฟ(๐ โ 1)]๐ฆ = ๐ฅ(๐ + ๐ฟ)๐ฆ
โน ๐ฟ2๐ฆ + [๐ฟ ๐ โ 1 โ ๐ฅ(๐ + ๐ฟ)]๐ฆ = 0
โด ๐ฟ2๐ฆ + ๐ โ 1 ๐ฟ๐ฆ โ ๐ฅ๐๐ฆ โ ๐ฅ๐ฟ๐ฆ = 0
216
โด ๐ฟ2๐ฆ + ๐ โ 1 โ ๐ฅ ๐ฟ๐ฆ โ ๐ฅ๐๐ฆ = 0
๐ฟ(๐ฟ๐ฆ) + ๐ โ 1 โ ๐ฅ ๐ฟ๐ฆ โ ๐ฅ๐๐ฆ = 0
๐ฟ ๐ฅ๐
๐๐ฅ๐ฆ + ๐ โ 1 โ ๐ฅ ๐ฅ
๐
๐๐ฅ๐ฆ โ ๐ฅ๐๐ฆ = 0
๐ฅ๐
๐๐ฅ ๐ฅ
๐
๐๐ฅ๐ฆ + (๐ โ 1 โ ๐ฅ)๐ฅ
๐๐ฆ
๐๐ฅโ ๐ฅ๐๐ฆ = 0
โด ๐
๐๐ฅ ๐ฅ
๐
๐๐ฅ๐ฆ + (๐ โ 1 โ ๐ฅ)
๐๐ฆ
๐๐ฅโ ๐๐ฆ = 0
๐ฅ๐2๐ฆ
๐๐ฅ2+
๐๐ฆ
๐๐ฅ+ (๐ โ 1 โ ๐ฅ)
๐๐ฆ
๐๐ฅโ ๐๐ฆ = 0
๐ฅ๐2๐ฆ
๐๐ฅ2+ (๐ โ ๐ฅ)
๐๐ฆ
๐๐ฅโ ๐๐ฆ = 0
Which is the Kummerโs differential equation. The confluent hypergeometric function
satisfies this equation.
The confluent hypergeometric distribution due to Gordy (1988a) is a special case of the
confluent hypergeometric distribution given in equation 8.1 where
๐ = ๐ + ๐, ๐ข = ๐ฅ and ๐ฅ = โ๐พ such that
1๐น1 ๐; ๐ + ๐, โ๐พ =ฮ(a + b)
ฮ(a)ฮ(a + b โ a) ๐ฅ๐โ1(1 โ ๐ฅ)๐+๐โ๐โ1๐โ๐พ๐ฅ ๐๐ฅ
1
0
1๐น1 ๐; ๐ + ๐, ๐ฅ =ฮ(a + b)
ฮ(a)ฮ(b) ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐โ๐พ๐ฅ ๐๐ฅ
1
0
1 = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1๐โ๐พ๐ฅ ๐๐ฅ
1๐น1 ๐; ๐ + ๐, โ๐พ B(a, b)
1
0
๐ ๐ฅ =๐ฅ๐โ1[1 โ ๐ฅ]๐โ1๐ โ๐พ๐ฅ
๐ต ๐, ๐ 1๐น1(๐; ๐ + ๐; โ๐พ), _________________________________________(8.2)
for 0 < x < 1, a > 0, b > 0, โโ < ๐พ < โ and 1๐น1 is the confluent hypergeometric function.
If ๐พ = 0, then the moment generating function denoted by
๐ ๐ก =๐ต ๐ โ ๐ก, ๐ 1๐น1 ๐ โ ๐ก; ๐ + ๐ โ ๐ก; โ๐พ
๐ต ๐, ๐ 1๐น1 ๐; ๐ + ๐; โ๐พ _______________________(8.3)
reduces to the standard beta pdf given by equation 2.1
217
8.4 Gauss Hypergeometric Function.
The Gaussian hypergeometric function denoted by the symbol 2๐น1 ๐, ๐; ๐; ๐ฅ represents
the series
2๐น1 ๐, ๐; ๐; ๐ฅ = 1 +๐๐
๐
๐ฅ
1!+
๐ ๐ + 1 ๐ ๐ + 1
๐ ๐ + 1
๐ฅ2
2!+ โฏ
where ๐ โ 0, โ1, โ2, โฆ
It can simply be expressed in the following form;
2๐น1 ๐, ๐; ๐; ๐ฅ = ๐ n ๐ n๐ฅ๐
๐ ๐ ๐!
โ
n=0
, c โ 0, โ1, โ2, โฆ ______________(8.4)
where ๐ n is the Pochhammerโs symbol defined by
๐ n = a a + 1 โฆ a + r โ 1
=ฮ(a + 1)
ฮ(a); (n is a positive integer)
๐ 0 = 1
If a is a non-positive integer, then ๐ n is zero for n > โ๐, and the series terminates.
The Eulerโs integral representation is given as
2๐น1 ๐, ๐; ๐; ๐ฅ =ฮ(c)
ฮ(a)ฮ(c โ a) ๐ข๐โ1(1 โ ๐ข)๐โ๐โ1(1 โ ๐ฅ๐ข)โ๐๐๐ข
1
0
______________(8.5)
where ๐ > ๐ > 0
8.5 The Differential Equations
The first and second order differentials can be expressed as follows;
๐
๐๐ฅ 2๐น1 ๐, ๐; ๐; ๐ฅ =
๐๐
๐+
๐ ๐ + 1 ๐ ๐ + 1
๐ ๐ + 1 ๐ฅ + โฏ
=๐๐
๐ 1 +
๐ + 1 ๐ + 1
๐ + 1 ๐ฅ +
๐ + 1 ๐ + 2 ๐ + 1 (๐ + 2)
๐ + 1 (๐ + 2)
๐ฅ2
2!โฆ
=๐๐
๐ 2๐น1 ๐ + 1, ๐ + 1; ๐ + 1; ๐ฅ
218
๐2
๐๐ฅ2 2๐น1 ๐, ๐; ๐; ๐ฅ =
๐๐
๐ ๐ + 1 ๐ + 1
๐ + 1 +
๐ + 1 ๐ + 2 ๐ + 1 (๐ + 2)
๐ + 1 (๐ + 2)๐ฅ + โฏ
=๐๐
๐
๐ + 1 ๐ + 1
๐ + 1 1 +
๐ + 2 (๐ + 2)
(๐ + 2)๐ฅ + โฏ
=๐ ๐ + 1 ๐ ๐ + 1
๐ ๐ + 1 2๐น1 ๐ + 2, ๐ + 2; ๐ + 2; ๐ฅ
We now wish to derive a differential equation whose solution is 2๐น1 ๐, ๐; ๐; ๐ฅ
Let
๐ฟ = ๐ฅ๐
๐๐ฅ
then
๐ฟ ๐ฟ + ๐ โ 1 2๐น1 ๐, ๐; ๐; ๐ฅ
= ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 โฆ (๐ + ๐ โ 1)๐ฟ ๐ฟ + ๐ โ 1
๐ฅ๐
๐!
โ
๐=0
= ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 โฆ (๐ + ๐ โ 2)
๐ฅ๐
(๐ โ 1)!
โ
๐=0
= ๐ฅ ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 โฆ (๐ + ๐ โ 2)
๐ฅ๐โ1
(๐ โ 1)!
โ
๐=1
replace n by n+1. Then
๐ฟ ๐ฟ + ๐ โ 1 2๐น1 ๐, ๐; ๐; ๐ฅ
= ๐ฅ ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ + ๐ (๐ + ๐)
๐ ๐ + 1 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
๐=1
= ๐ฅ ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ + ๐ฟ (๐ + ๐ฟ)
๐ ๐ + 1 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
๐=0
= ๐ฅ ๐ + ๐ฟ (๐ + ๐ฟ) ๐ ๐ + 1 โฆ ๐ + ๐ โ 1 ๐ ๐ + 1 โฆ ๐ + ๐ โ 1
๐ ๐ + 1 โฆ (๐ + ๐ โ 1)
๐ฅ๐
๐!
โ
๐=0
โด ๐ฟ ๐ฟ + ๐ โ 1 2๐น1 ๐, ๐; ๐; ๐ฅ = ๐ฅ ๐ + ๐ฟ (๐ + ๐ฟ) 2๐น1 ๐, ๐; ๐; ๐ฅ
๐ฟ ๐ฟ + ๐ โ 1 ๐น = ๐ฅ ๐ + ๐ฟ (๐ + ๐ฟ) ๐น
where
219
๐น = 2๐น1 ๐, ๐; ๐; ๐ฅ and ๐ฟ = ๐ฅ๐
๐๐ฅ
โด ๐ฟ ๐ฟ๐น + ๐ โ 1 ๐น = ๐ฅ ๐ + ๐ฟ (๐๐น + ๐ฟ๐น)
โน ๐ฟ ๐ฅ๐๐น
๐๐ฅ+ ๐ โ 1 ๐น = ๐ฅ ๐ + ๐ฟ ๐๐น + ๐ฅ
๐
๐๐ฅ๐น
โน ๐ฟ ๐ฅ๐๐น
๐๐ฅ + ๐ โ 1 ๐ฟ๐น = ๐ฅ ๐ + ๐ฟ ๐๐น + ๐ฅ
๐๐น
๐๐ฅ
โน ๐ฅ๐
๐๐ฅ ๐ฅ
๐๐น
๐๐ฅ + ๐ โ 1 ๐ฅ
๐๐น
๐๐ฅ= ๐ฅ ๐ + ๐ฟ ๐๐น + ๐ฅ
๐๐น
๐๐ฅ
โน๐
๐๐ฅ ๐ฅ
๐๐น
๐๐ฅ + ๐ โ 1
๐๐น
๐๐ฅ= ๐ + ๐ฟ ๐๐น + ๐ฅ
๐๐น
๐๐ฅ
๐ฅ๐2๐น
๐๐ฅ2 +
๐๐น
๐๐ฅ+ ๐
๐๐น
๐๐ฅโ
๐๐น
๐๐ฅ= ๐๐๐น + ๐๐ฅ
๐๐น
๐๐ฅ+ ๐๐ฟ๐น + ๐ฟ๐ฅ
๐๐น
๐๐ฅ
โด ๐ฅ๐2๐น
๐๐ฅ2 + ๐
๐๐น
๐๐ฅ= ๐๐๐น + ๐๐ฅ
๐๐น
๐๐ฅ+ ๐๐ฅ
๐๐น
๐๐ฅ+ ๐ฅ
๐๐น
๐๐ฅ ๐ฅ
๐๐น
๐๐ฅ
= ๐๐๐น + (๐ + ๐)๐ฅ๐๐น
๐๐ฅ+ ๐ฅ ๐ฅ
๐2๐น
๐๐ฅ2+
๐๐น
๐๐ฅ
โด ๐ฅ 1 โ ๐ฅ ๐2๐น
๐๐ฅ2 + ๐ โ ๐๐ฅ โ ๐๐ฅ โ ๐ฅ
๐๐น
๐๐ฅโ ๐๐๐น = 0
โด ๐ฅ 1 โ ๐ฅ ๐2๐น
๐๐ฅ2 + ๐ โ (๐ + ๐ + 1)๐ฅ
๐๐น
๐๐ฅโ ๐๐๐น = 0
This is the differential equation whose solution is 2๐น1 ๐, ๐; ๐; ๐ฅ i.e the Gaussian
hypergeometric function satisfies this second order linear differential equation.
The Gauss hypergeometric distribution due to Armero and Bayarri (1994) is a special
case of the Gaussian hypergeometric distribution given in equation 8.5 where
๐ = ๐ + ๐, ๐ฅ = โ๐ง , ๐ข = ๐ฅ, and โ๐ = ๐พ such that
2๐น1 ๐พ, ๐; ๐ + ๐; โ๐ง =ฮ(a + b)
ฮ(a)ฮ(a + b โ a) ๐ฅ๐โ1(1 โ ๐ฅ)๐+๐โ๐โ1(1 + ๐ง๐ฅ)โ๐๐๐ฅ
1
0
1 = ๐ฅ๐โ1(1 โ ๐ฅ)๐โ1(1 + ๐ง๐ฅ)๐พ๐๐ฅ
2๐น1 ๐พ, ๐; ๐ + ๐; โ๐ง B(a, b)
1
0
220
๐ ๐ฅ =๐ฅ๐โ1(1 โ ๐ฅ)๐โ1/(1 + ๐ง๐ฅ)๐พ
๐ต ๐, ๐ 2๐น1(๐พ, ๐; ๐ + ๐; โ๐ง)_________________(8.6)
for 0 < x < 1, a > 0, b > 0, and โโ < ๐พ < โ. If ๐พ = 0, then equation 8.6 reduces to beta of
the first kind given by equation 2.1
Nadarajah and Kotz (2004) also studied the beta distribution based on the Gauss
hypergeometric function. They defined the distribution as follows:
๐ ๐ฅ =๐๐ต ๐, ๐
๐ต ๐, ๐ + ๐พ ๐ฅ๐+๐โ1 2๐น1(1 โ ๐พ, ๐; ๐ + ๐; ๐ฅ), _________________(8.7)
When ๐ = 0 in equation 8.7, then it reduces to the Classical beta distribution with
parameters a and ๐พ.
8.6 Appell Function
Nadarajah and Kotz (2006) introduced a beta function based on the Appell function. The
distribution is given by
๐ ๐ฅ =๐ถ๐ฅ๐ผโ1(1 โ ๐ฅ)๐ฝโ1
(1 โ ๐ข๐ฅ)๐ (1 โ ๐ฃ๐ฅ)๐, ________________________________________(8.8)
for 0 < ๐ฅ < 1, ๐ผ > 0, ๐ฝ > 0, ๐ > 0, ๐ > 0, โ1 < ๐ข < 1, โ1 < ๐ฃ < 1
where C denotes the normalizing constant given by
1
๐ถ= ๐ต ๐ผ, ๐ฝ ๐น1(๐ผ, ๐, ๐, ๐ผ + ๐ฝ; ๐ข, ๐ฃ)
and ๐น1 denotes the Appell function of the first kind. This distribution is flexible and
contains several of the known generalizations of the Classical beta distribution as
particular cases. The Classical beta distribution is the particular case for either ๐ =
0 and ๐ = 0 or ๐ = 0 and ๐ฃ = 0 or ๐ข = 0 and ๐ = 0 or ๐ข = 0 and ๐ฃ = 0.Libby and
Novickโs beta distribution in equation 4.1 is the particular case for either ๐ = ๐ผ +
๐ฝ and ๐ = 0 or ๐ข = ๐ฃ and ๐ + ๐ = ๐ผ + ๐ฝ. Armero and Bayarriโs Gauss hypergeometric
distribution in equation 8.6 is the particular case for either ๐ = 0 or ๐ = 0
221
8.7 Nadarajah and Kotzโs Power Beta Distribution
The generalization due to Nadarajah and Kotz (2004) is based on the basic
characterization that if X and Y are independent gamma distributed random variables
with common scale parameter then the ratio X/(X+Y) is a beta distribution of the first
kind. If one consider the distribution of
๐ = ๐๐ / (๐๐ + ๐๐ ) ๐ถ > 0, referred to as power beta, then the pdf and cdf of W can be
expressed as
๐(๐ค) =๐คโ 1+๐/๐ 1 โ ๐ค ๐/๐โ1
๐๐๐ต(๐, ๐) ๐ 2๐น1 ๐, ๐ + ๐; ๐ + 1; โ
1 โ ๐ค
๐ค
1๐
โ 1 โ ๐ค
๐ค
1๐
2๐น1 ๐ + 1, ๐ + ๐ + 1; ๐ + 2; โ 1 โ ๐ค
๐ค
1๐
and
๐น ๐ค = 1 โ1
๐๐ต(๐, ๐)
1 โ ๐ค
๐ค
๐/๐
2๐น1 ๐, ๐ + ๐; ๐ + 1; โ 1 โ ๐ค
๐ค
1/๐
.
Simpler expressions can be obtained for integer values of a, b and c.
8.8 Compound Confluent Hypergeometric Function
The compound confluent hypergeometric distribution due to Gordy (1988b) is given by
๐ ๐ฅ =๐ฅ๐โ1[1 โ ๐ฃ๐ฅ]๐โ1{๐ + (1 โ ๐)๐ฃ๐ฅ}โ๐๐ โ๐ ๐ฅ
๐ต ๐, ๐ ๐ฃโ๐ exp โ๐ ๐ฃ ฮฆ1(๐, ๐, ๐ + ๐,
๐ ๐ฃ , 1 โ ๐)
, ______________________(8.9)
for 0 < a < 1, b > 0, โโ < ๐ < โ, โโ < ๐ < โ, 0 โค ๐ฃ โค 1, ๐ > 0 and ฮฆ1 is the
confluent hypergeometric function of two variables defined by
ฮฆ1 ๐ผ, ๐ฝ, ๐พ, ๐ฅ, ๐ฆ = ๐ผ ๐+๐ (๐ฝ)๐
๐พ ๐+๐๐! ๐!๐ฅ๐๐ฆ๐
โ
๐=0
โ
๐=0
222
When ๐ = 0, ๐ = ๐ + ๐, ๐ฃ = 1โ๐
๐ and ๐ = 1 โ ๐ in Equation 8.9, then it reduces to
McDonald and Xuโs five parameter generalized beta distribution in equation 6.1.
When ๐ = 0, and ๐ฃ = 1 in Equation 8.9, then it reduces to gauss hypergeometric
distribution due to Armero and Bayarri (1994) in equation 8.6. When ๐ฃ = 1, and ๐ = 1
in Equation 8.9, then it reduces to confluent hypergeometric distribution due to Gordy
(1988a) in equation 8.2. When ๐ฃ = 0, and > 0 , Equation 8.9 becomes the gamma
distribution.
8.9 Beta-Bessel Distribution (A.K Gupta and S.Nadarajah (2006))
A.K Gupta and S. Nadarajah (2006) introduced three types of beta-bessel distributions,
based on a composition of the classical beta distribution and the Bessel function. They
derived expressions for their shapes, particular cases and the nth
moments. They also
discussed estimation by the method of maximum likelihood and Bayes estimation.
The beta- bessel distribution is obtained as follows:
The first generalization of equation 2.1 is given by the pdf
๐ ๐ฅ = C๐ฅ๐ถโ1 1 โ ๐ฅ ๐ทโ1๐ผ๐ฃ ๐๐ฅ ________________________________(8.10)
For 0 < ๐ฅ < 1, ๐ฃ > 0, ๐ผ > 0, ๐ฝ > 0 ๐๐๐ ๐ โฅ 0 , where C denotes the normalizing
constant and can be determined as
1
๐ถ=
๐๐ฃ๐ค ๐ผ + ๐ฃ ๐ค ๐ฝ
2๐ฃ๐ค ๐ผ + ๐ฝ + ๐ฃ ๐ค ๐ฃ + 1
2๐น3 ฮฑ + ๐ฃ
2,ฮฑ + ๐ฃ + 1
2; ๐ฃ
+ 1,ฮฑ + ๐ฃ + ฮฒ
2,ฮฑ + ๐ฃ + ฮฒ + 1
2;๐2
4
๐ผ๐ฃ ๐ฅ is the modified Bessel function defined by
๐ผ๐ฃ ๐ฅ =๐ฅ๐ฃ
๐ 2๐ฃ ๐ฃ + 1/2 1 โ ๐ก2 ๐ฃ+1/2 exp ๐ฅ๐ก
1
โ1
๐๐ก
The classical beta pdf (2.1) arises as particular case of 8.10 for ๐ฃ = 0 and ๐ = 0.
The second generalization of equation 2.1 is given by the pdf
223
๐ ๐ฅ = C๐ฅ๐ถโ1 1 โ ๐ฅ ๐ทโ1exp(๐๐ฅ)๐ผ๐ฃ ๐๐ฅ _________________________(8.11)
For 0 < ๐ฅ < 1, ๐ฃ > 0, ๐ผ > 0, ๐ฝ > 0 ๐๐๐ ๐ โฅ 0 , where C denotes the normalizing
constant and can be determined as
1
๐ถ=
๐๐ฃ๐ค ๐ผ + ๐ฃ ๐ค ๐ฝ
2๐ฃ๐ค ๐ผ + ๐ฝ + ๐ฃ ๐ค ๐ฃ + 1
2๐น2 ๐ฃ +1
2, ๐ผ + ๐ฃ; 2๐ฃ + 1, ๐ผ + ๐ฝ + ๐ฃ; 2๐
The classical beta pdf (2.1) arises as the particular case of 8.11 for ๐ฃ = 0 and ๐ = 0.
The third and final generalization of equation 2.1 is given by the pdf
๐ ๐ฅ = C๐ฅ๐ถโ1 1 โ ๐ฅ ๐ทโ1exp(โ๐๐ฅ)๐ผ๐ฃ ๐๐ฅ _____________________(8.12)
For 0 < ๐ฅ < 1, ๐ฃ > 0, ๐ผ > 0, ๐ฝ > 0 ๐๐๐ ๐ โฅ 0 , where C denotes the normalizing
constant and can be determined as
1
๐ถ=
๐๐ฃ๐ค ๐ผ + ๐ฃ ๐ค ๐ฝ
2๐ฃ๐ค ๐ผ + ๐ฝ + ๐ฃ ๐ค ๐ฃ + 1
2๐น2 ๐ฃ +1
2, ๐ผ + ๐ฃ; 2๐ฃ + 1, ๐ผ + ๐ฝ + ๐ฃ; 2๐
The classical beta pdf (2.1) arises as the particular case of 8.12 for ๐ฃ = 0 and ๐ = 0
224
9 Chapter IX: Summary and Conclusion
9.1 Summary
This project has looked at three methods of generalizing the classical beta distribution. The work
could be summarized as follows:
In Chapter I, a general introduction covering the work on statistical distributions in general is
given followed by a brief background on classical beta distribution. The limitation of the
classical beta distribution is noted leading to the main objective of adding more parameters to
make the distribution more flexible for analyzing empirical data. The recent works in literature is
also highlighted.
In Chapter II, the classical beta distribution is constructed from the beta function and the
relationship between the beta function and the Gamma function is shown. Its construction from
ratio transformation of two independent gamma variables is also done. Other constructions from
the rth order statistic of the Uniform distribution and from the stochastic processes are also
looked at. Some properties, shapes, and special cases of the classical beta are also covered under
this chapter.
In Chapter III, looks at the transformation of the classical beta distribution on the domain (0,1) to
the extended beta distribution of the second kind on the domain (0, โ). It also looked at some
properties, shapes, and special cases of the beta distribution of the second kind.
In Chapter IV, highlights some of the three parameter beta distributions used by various authors
in literature. In particular, the well known Libby and Novick (1982) three parameter beta
distribution and the McDonald (1995) three parameter beta distribution.
In Chapter V, the four parameter beta distributions are constructed. Their properties and special
cases are also given.
In Chapter VI, covers the construction of the five parameter beta distributions due to McDonald
(1995) giving some of its properties and special cases. It further extends the five parameter beta
distribution due to McDonald (1995) by introducing a sixth parameter to give a five parameter
weighted beta distribution. Some of the properties and special cases of this five parameter
225
weighted beta distribution have also been looked at. The chapter also looked at the construction
of the five parameter exponential generalized beta distribution and its special cases.
In Chapter VII, gives a new family of generalized beta distributions based on the work of Eugene
et al. (2002) using the generator approach. It covers the beta generated distributions,
exponentiated generated distributions and the generalized beta generated distributions.
In Chapter VIII, the generalized beta distributions based on some special functions are covered.
In particular, the Confluent Hypergeometric function, the Gauss Hypergeometric function, the
Bessel and the Appell function.
In Chapter IX, a summary and concluding remark of the work is given followed by references
and an appendix.
9.2 Conclusion
This project has looked at both the two kinds of two parameter beta distributions. It has reviewed
various methods of constructing the beta distributions, their properties which include expressions
for the rth
moments, first four moments, mode, coefficient of variation, coefficient of skewness,
coefficient of kurtosis and their shapes. It has also presented the special or limiting cases.
However, there are other methods which can be explored in extending this work e.g. using
mixtures of distributions. Other areas of research which need to be looked at are: further
properties of the beta distributions, estimation and applications.
226
10 Appendix
A.1 Classical Beta Distribution
Pdf ๐ ๐ฅ =
๐ฅ๐โ1(1 โ ๐ฅ)๐โ1
๐ต(๐, ๐), ๐ > 0, ๐ > 0,
0 < ๐ฅ < 1
Cumulative distribution function
๐น ๐ฅ =1
๐ต(๐, ๐) ๐ก๐โ1(1 โ ๐ก)๐โ1
๐ฅ
0
๐๐ก
rth
order moment ฮ(a + r) ฮ(a + b)
ฮ(a + b + r) ฮ(a)
Mean a
a + b
Mode (๐ โ 1)
(๐ + ๐ โ 2)
Variance ab
a + b + 1 a + b 2
Skewness 2 b โ a
a + b + 2
a + b + 1
ab
Kurtosis a + b + 1
๐๐
3[a2(b + 2) โ 2ab + b2(a + 2)]
a + b + 2 a + b + 3
A.2 Power Distribution (b=1 in table A.1)
Pdf ๐ ๐ฅ = ๐๐ฅ๐โ1 , ๐ > 0, 0 < ๐ฅ < 1
Cumulative distribution function ๐น(๐ฅ) = ๐ฅ๐
rth
order moment ๐
a + r
Mean ๐
a + 1
Mode 1
Variance ๐
(a + 2)(a + 1)2
Skewness 2๐(1 โ ๐)
(a + 3)
(a + 2)
a
Kurtosis ๐ + 2
๐
3 3๐2 โ ๐ + 2
a + 3 a + 4
227
A.3 Uniform Distribution (a=b=1 in table A.1)
Pdf ๐ ๐ฅ = 1, 0 < ๐ฅ < 1
rth
order moment 1
1 + r
Mean 1
2
Mode
Any value in the interval [0,1]
Variance 1
12
Skewness 0
Kurtosis 1.8
A.4 Arcsine Distribution (a=b= ๐
๐ in table A.1)
Pdf ๐ ๐ฅ =
1
๐ ๐ฅ(1 โ ๐ฅ), 0 < ๐ฅ < 1
Cumulative distribution function F x =
2
๐arcsin ๐ฅ
rth
order moment ฮ 12 + ๐
rฮ r ฮ 12
Mean 1
2
Mode 1
2
Variance 0.125
Skewness 0
Kurtosis 1.5
228
A.5 Triangular Shaped (a=1, b=2 in table A.1)
๐ ๐ฅ = 2 โ 2๐ฅ, 0 < ๐ฅ < 1
Cumulative distribution function
๐น ๐ฅ = 2๐ฅ โ ๐ฅ2
rth
order moment 2rฮ ๐
ฮ r + 3
Mean
0.3333
Mode
0
Variance
0.0556
Skewness
0.5657
Kurtosis
2.4
A.6 Triangular Shaped (a=2, b=1 in table A.1)
Pdf ๐ ๐ฅ = 2๐ฅ, 0 < ๐ฅ < 1
Cumulative distribution function ๐น ๐ฅ = ๐ฅ2
rth
order moment 2ฮ 2 + ๐
ฮ r + 3
Mean 0.6667
Mode 1
Variance 0.0556
Skewness -0.5657
Kurtosis 2.4
A.7 Parabolic Shaped (a=b=2 in table A.1)
Pdf ๐ ๐ฅ = 6๐ฅ 1 โ ๐ฅ , 0 < ๐ฅ < 1
Cumulative distribution function ๐น ๐ฅ = 3๐ฅ2 โ 2๐ฅ3
rth
order moment 6
r + 3 (r + 2)
Mean 0.5
Mode 0.5
Variance 0.05
Skewness 0
Kurtosis 2.1429
229
A.8 Wigner Semicircle (a=b=๐
๐, ๐๐ง๐ ๐ฑ =
๐ฒ+๐
๐๐ in table A.1)
Pdf ๐ ๐ฅ =
2 ๐ 2 โ ๐ฅ2
๐ 2๐, โ๐ < ๐ฅ < ๐
rth
order moment ๐
2
2๐ 1
๐ + 1
2๐
๐
Mean 0
Mode 0
Variance ๐ 2
4
Skewness
0
Kurtosis 2
A.9 Beta Distribution of the Second Kind (using ๐ฑ =๐ฒ
๐+๐ฒ or
๐
๐+๐ฒ in table A.1)
Pdf ๐ ๐ฅ =
๐ฅ๐โ1
๐ต ๐, ๐ 1 + x a+b, ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
rth
order
moment
ฮ ๐ + ๐ ฮ ๐ โ ๐
ฮ ๐ ฮ ๐
Mean ๐
๐ โ 1
Mode ๐ โ 1
๐ + 1
Variance ๐(๐ + ๐ โ 1)
๐ โ 1 2(๐ โ 2)
Skewness 2๐[2๐2 + 3๐ ๐ โ 1 + (๐ โ 1)2]
๐ โ 1 3 ๐ โ 2 (๐ โ 3)
Kurtosis 3a(ab3 + 2b3+2a2b2 + 5ab2 โ 6b2+a3b + 8a2b โ 13ab + 6b + 5a3 โ 10a2
+7a โ 2)
b โ 1 4 b โ 2 b โ 3 b โ 4
230
A.10 Lomax (Pareto II) (a=1 in table A.9)
pdf ๐ ๐ฅ =
๐
1 + x 1+b, ๐ > 0, 0 < ๐ฅ < โ
rth
order moment ฮ 1 + ๐ ฮ ๐ โ ๐
ฮ ๐
Mean 1
๐ โ 1
Mode ๐ โ 1
2
Variance ๐
(๐ โ 2)(๐ โ 1)2
Skewness 2๐ ๐ + 1
๐ โ 1 3 ๐ โ 2 ๐ โ 3
Kurtosis 3(3b3 + b2 + 2b)
b โ 1 4 b โ 2 b โ 3 b โ 4
A.11 Log Logistic (Fisk) (a=b=1 in table A.9)
pdf ๐ ๐ฅ =
1
1 + ๐ฅ 2, 0 < ๐ฅ < โ
rth
order moment
rฯ
sin rฯ
Mean ฯ
sin ฯ
Mode 0
Variance 2ฯ
sin 2ฯโ
ฯ
sin ฯ
2
Skewness 3ฯ
sin 3ฯโ 3
2ฯ
sin 2ฯ
ฯ
sin ฯ + 2
ฯ
sin ฯ
3
Kurtosis 4ฯ
sin 4ฯโ 4
ฯ
sin ฯ
3ฯ
sin 3ฯ+ 6
ฯ
sin ฯ
2 2ฯ
sin 2ฯ
โ 3 ฯ
sin ฯ
4
231
A.12 Libby and Novick (1982) Beta Distribution (๐ฑ =๐ฒ๐
๐โ๐ฒ+๐ฒ๐ in table A.1)
pdf ๐ ๐ฅ =
1
B a, b
๐๐๐ฆ๐โ1 1 โ ๐ฆ ๐โ1
1 โ 1 โ c y a+b,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < 1
A.13 McDonaldโs (1995) beta distribution of the first kind (๐ฑ =๐ฒ
๐ in table A.1)
๐ ๐ฅ =๐ฅ๐โ1 1 โ
๐ฅ๐
๐โ1
๐๐B a, b ,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < ๐
A.14 Three Parameter Beta Distribution (๐ฑ =๐๐ฒ
๐+๐ฒ in table A.1)
Pdf ๐ ๐ฅ =
๐๐๐ฅ๐โ1 1 + (1 โ ๐)๐ฅ ๐โ1
B a, b (1 + ๐ฅ)a+b,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < 1
A.15 Chotikapanich et.al (2010) Beta of the Second Kind Distribution (๐ฑ =๐ฒ
๐ in
table A.9)
Pdf ๐ ๐ฅ =
๐ฅ๐โ1
๐๐B a, b 1 + ๐ฅ๐
๐+๐ ,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
A.16 General Three Parameter Beta Distribution (๐ฑ = ๐๐ฒ in table A.9)
Pdf ๐ ๐ฅ =
๐๐๐ฅ๐โ1
B a, b 1 + ๐๐ฅ ๐+๐,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
232
A.17 Three Parameter Beta Distribution (๐ฑ = ๐ฒ๐ in table A.1)
Pdf ๐ ๐ฅ =
๐๐ฅ๐๐โ1 1 โ ๐ฅ๐ ๐โ1
B a, b ,
๐ > 0, ๐ > 0, ๐ > 0, 0 < ๐ฅ < 1
A.18 Generalized Four Parameter Beta Distribution of the First Kind
(๐ฑ = ๐ฒ
๐ช ๐
๐ข๐ง ๐ญ๐๐๐ฅ๐ ๐. ๐)
๐ ๐ฅ = ๐ ๐ฅ๐๐โ1 1 โ
๐ฅ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ , ๐ > 0, ๐ > 0, ๐ > 0,
๐ > 0, 0 โค ๐ฅ๐ โค ๐๐
rth
order moment ๐๐ฮ a + b ฮ ๐ +๐๐
ฮ ๐ + ๐ +๐๐ ฮ a
Mean ๐ฮ a + b ฮ a +1๐
ฮ a + b +1๐ ฮ a
Mode
๐๐ ๐๐ โ 1
๐๐ + ๐๐ โ 1 โ ๐
1๐
Variance
๐2 ฮ a +
2p ฮ b ฮ a + b
ฮ a + b +2p
โฮ2 a +
1p ฮ2 b ฮ2 a + b
ฮ2 a + b +1p
Skewness
๐3
ฮ a + b ฮ a +
3๐
ฮ a + b +3๐ ฮ a
โ 3ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
ฮ a + b ฮ a +2๐
ฮ a + b +2๐ ฮ a
+ 2 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
3
233
Kurtosis
q4
ฮ a + b ฮ a +
4๐
ฮ a + b +4๐ ฮ a
โ 4ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
ฮ a + b ฮ a +3๐
ฮ a + b +3๐ ฮ a
+ 6 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
2
ฮ a + b ฮ a +2๐
ฮ a + b +2๐ ฮ a
โ 3 ฮ a + b ฮ a +
1๐
ฮ a + b +1๐ ฮ a
4
A.19 Pareto Type I (b=1, p=-1 in table A.1)
Pdf ๐ ๐ฅ =
๐๐๐
๐ฅ๐+1 , ๐ > 0, ๐ > 0,
0 < ๐ โค ๐ฅ
Cumulative distribution function 1 โ
๐
๐ฅ
๐
rth
order moment ๐๐a
๐ โ ๐
Mean ๐๐
๐ โ 1
Mode
๐
Variance ๐2a
a โ 2 a โ 1 2
Skewness 2(1 + a)
a โ 3
a โ 2
a
Kurtosis 3(3a3 โ 5a2 โ 4)
a a โ 3 a โ 4
234
A.20 Stoppa (Generalized Pareto Type I) (a=1, p=-p in table A.1)
Pdf ๐ ๐ฅ = p๐๐๐๐ฅโ๐โ1 1 โ
๐ฅ
๐
โ๐
bโ1
, ๐ > 0, ๐ > 0, ๐ > 0,
0 < ๐ โค ๐ฅ
Cumulative
distribution function 1 โ ๐ฅ
๐
โ๐
b
rth
order moment qr๐ฮ ๐ ฮ 1 โ๐๐
ฮ 1 + ๐ โ๐๐
Mean ๐bฮ ๐ ฮ 1 โ1๐
ฮ 1 + ๐ โ1๐
Mode
๐ 1 + ๐๐
1 + ๐
1๐
Variance
q2 ฮ 1 โ
2p ฮ b ฮ 1 + b
ฮ 1 + b โ2p
โฮ2 1 โ
1p ฮ2 b ฮ2 1 + b
ฮ2 1 + b โ1p
Skewness
๐3
bฮ b ฮ 1 โ
3๐
b โ3๐ ฮ b โ
3๐
โ 3bฮ b ฮ 1 โ
1๐
b โ1๐ ฮ b โ
1๐
bฮ b ฮ 1 โ2๐
b โ2๐ ฮ b โ
2๐
+ 2 bฮ b ฮ 1 โ
1๐
b โ1๐ ฮ b โ
1๐
3
Kurtosis
q4
bฮ b ฮ 1 โ
4๐
ฮ 1 + b โ4๐
โ 4bฮ b ฮ 1 โ
1๐
ฮ 1 + b โ1๐ ฮ 1
bฮ b ฮ 1 โ3๐
ฮ 1 + b โ3๐
+ 6 ฮ 1 + b ฮ 1 โ
1๐
ฮ 1 + b โ1๐
2
ฮ 1 + b ฮ 1 โ2๐
ฮ 1 + b โ2๐
โ 3 ฮ 1 + b ฮ 1 +
1๐
ฮ 1 + b +1๐
4
235
A.21 Kumaraswamy (a=1, q=1 in table A.1)
๐ ๐ฅ = pb๐ฅ๐โ1 1 โ ๐ฅ๐ bโ1, ๐ > 0, ๐ > 0
rth
order moment bฮ ๐๐ + 1 ฮ(๐)
ฮ(1 +๐๐ + ๐)
Mean bฮ 1๐ + 1 ฮ(๐)
ฮ(1 +1๐ + ๐)
Mode
1 โ ๐
1 โ ๐๐
1๐
Variance
b! b โ 1 !
2p !
2p + b !
โ b! b โ 1 !
1p !
1p + b !
2
Skewness bฮ b ฮ a +3๐
ฮ 1 + b +3๐
โ 3bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
bฮ b ฮ 1 +2๐
ฮ 1 + b +2๐
+ 2 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
3
Kurtosis bฮ b ฮ 1 +4๐
ฮ 1 + b +4๐
โ 4bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
bฮ b ฮ 1 +3๐
ฮ 1 + b +3๐
+ 6 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
2
bฮ b ฮ 1 +2๐
ฮ 1 + b +2๐
โ 3 bฮ b ฮ 1 +
1๐
ฮ 1 + b +1๐
4
236
A.22 Generalized Power (b=1, p=1 in table A.1)
pdf ๐ ๐ฅ =
๐๐ฅ๐โ1
๐๐, ๐ > 0, ๐ > 0, 0 < ๐ฅ < ๐
rth
order moment ๐๐a
a + r
Mean ๐๐
a + 1
Variance ๐2
(a + 1)2 a + 2 aฮ a โ a2ฮ2 a
(a + 1)2 a + 2 2
Skewness 2๐ โa3 โ 3a2 + 4
a + 3
1
a + 1 2 a + 2 aฮ a โ a2ฮ2 a
32
Kurtosis
(a + 1)2 a + 2 2
๐2(a + 1)2 a + 2 aฮ a โ a2ฮ2 a
2
. q4 a
a + 4โ
4a2
(a + 1)(a + 3)
+6a3
a + 1 2(a + 2)โ
3a4
a + 1 4
237
A.23 Generalized Four Parameter Beta Distribution of the Second Kind
(๐ฑ = ๐ฑ
๐ช ๐ฉ
in table A.9)
Pdf ๐ ๐ฅ =
๐ ๐ฅ๐๐โ1
๐๐๐๐ต ๐, ๐ (1 + (๐ฅ๐)๐)a+b
, ๐ > 0, ๐ > 0, ๐ > 0,
๐ > 0, 0 < ๐ฅ < โ
rth
order moment ๐๐ฮ ๐ +๐๐ ฮ ๐ โ
๐๐
ฮ(๐)ฮ(๐)
Mean ๐ฮ ๐ +1๐ ฮ ๐ โ
1๐
ฮ ๐ ฮ ๐
Mode
๐ ๐๐ โ 1
๐๐ + 1
1๐
Variance
๐2
๐ต ๐ +
2๐ , ๐ โ
2๐
๐ต ๐, ๐ โ
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
Skewness
๐3
๐ต ๐ +
3๐ , ๐ โ
3๐
๐ต ๐, ๐ โ 3
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
๐ต ๐ +2๐ , ๐ โ
2๐
๐ต ๐, ๐
+ 2 ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
3
Kurtosis
๐4
๐ต ๐ +
4๐ , ๐ โ
4๐
๐ต ๐, ๐ โ 4
๐ต ๐ +1๐ , ๐ โ
1๐
๐ต ๐, ๐
๐ต ๐ +3๐ , ๐ โ
3๐
๐ต ๐, ๐
+ 6 ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
2
๐ต ๐ +2๐ , ๐ โ
2๐
๐ต ๐, ๐
โ 3 ๐ต ๐ +
1๐ , ๐ โ
1๐
๐ต ๐, ๐
4
238
A.24 Singh Maddala (Burr 12) (๐ = ๐ in table A.23)
Pdf ๐ ๐ฅ =
๐ ๐๐ฅ๐โ1
๐๐ 1 + (๐ฅ๐)๐ 1+b
, ๐ > 0, ๐ > 0, ๐ > 0
0 < ๐ฅ < โ
Cumulative
distribution function ๐น ๐ฅ = 1 โ 1 + (
๐ฅ
๐)๐ โb
rth
order moment ๐๐ฮ 1 +๐๐ ฮ ๐ โ
๐๐
ฮ b
Mean ๐ฮ 1 +1๐ ฮ ๐ โ
1๐
ฮ b
Mode
๐ ๐ โ 1
๐๐ + 1
1๐
Variance
๐2 ฮ(b)ฮ 1 +
2๐ ฮ ๐ โ
2๐ โ ฮ2(1 +
1๐)ฮ2 ๐ โ
1๐
ฮ2(๐)
Skewness
๐3 ๐๐ต 1 +3
๐, ๐ โ
3
๐ โ 3๐๐ต 1 +
1
๐, ๐ โ
1
๐ ๐๐ต 1 +
2
๐, ๐ โ
2
๐
+ 2 ๐๐ต 1 +1
๐, ๐ โ
1
๐
3
Kurtosis
๐4 ๐๐ต 1 +4
๐, ๐ โ
4
๐ โ 4๐2๐ต 1 +
1
๐, ๐ โ
1
๐ ๐ต 1 +
3
๐, ๐ โ
3
๐
+ 6 ๐๐ต 1 +1
๐, ๐ โ
1
๐
2
๐๐ต 1 +2
๐, ๐ โ
2
๐
โ 3 ๐๐ต ๐ +1
๐, ๐ โ
1
๐
4
239
A.25 Generalized Lomax (Pareto II) (a=1, p=1, q = ๐
๐ in table A.23)
Pdf ๐ ๐ฅ =
๐๐
1 + ๐๐ฅ 1+b, ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
Cumulative
distribution function ๐น ๐ฅ = 1 โ 1 + ๐๐ฅ โb
rth
order moment
1๐
๐
ฮ(1 + ๐)ฮ(๐ โ ๐)
ฮ(b)
Mean 1
๐ b โ 1
Mode 0
Variance b
๐2 b โ 1 2 b โ 2
Skewness 2(b + 1)
(b โ 3)
b โ 2
b
Kurtosis
1
๐
4
24
b โ 1 b โ 2 b โ 3 (b โ 4)โ
24
(b โ 1)2 b โ 2 (b โ 3)
+12
(b โ 1)3(b โ 2)โ
3
(b โ 1)4
240
A.26 Inverse Lomax (b=1, p=1 in table A.23)
Pdf ๐ ๐ฅ =
๐๐ฅ๐โ1
๐๐ 1 +๐ฅ๐
1+a , ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
rth
order moment ๐ ๐ฮ(๐ + ๐)ฮ(1 โ ๐)
ฮ(๐)ฮ(1)
Mode ๐
๐ โ 1
2
Variance ๐2 ๐๐ต ๐ + 2, โ1 โ ๐๐ต ๐ + 1,0 2 Skewness
๐3 ๐ต ๐ + 3,1 โ 3
๐ต ๐, 1 โ 3
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
๐ต ๐ + 2,1 โ 2
๐ต ๐, 1
+ 2 ๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
3
Kurtosis ๐4
๐ต ๐ + 4,1 โ 4
๐ต ๐, 1 โ 4
๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
๐ต ๐ + 3,1 โ 3
๐ต ๐, 1
+ 6 ๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
2๐ต ๐ + 2,1 โ 2
๐ต ๐, 1
โ 3 ๐ต ๐ + 1,1 โ 1
๐ต ๐, 1
4
A.27 Dagum (Burr 3) (b=1 in table A.23)
Pdf ๐ ๐ฅ =
๐ ๐๐ฅ๐๐โ1
๐๐๐ (1 + (๐ฅ๐)๐)1+a
, ๐ > 0, ๐ > 0, ๐ > 0
0 < ๐ฅ < โ
rth
order moment ๐๐ฮ(๐ +๐๐)ฮ(1 โ
๐๐)
ฮ(a)
Mean ๐ฮ(๐ +1๐)ฮ(1 โ
1๐)
ฮ(a)
Mode
๐ ๐๐ โ 1
๐ + 1
1๐
Variance
๐2 ฮ(a)ฮ ๐ +
2๐ ฮ 1 โ
2๐ โ ฮ2(๐ +
1๐)ฮ2 1 โ
1๐
ฮ2(๐)
241
A.28 Para-Logistic (a=1, b = p in table A.23)
Pdf ๐ ๐ฅ =
๐2๐ฅ๐โ1
๐๐(1 + (๐ฅ๐)๐)1+p
, ๐ > 0, ๐ > 0
0 < ๐ฅ < โ
A.29 General Fisk (Log โLogistic) (a=b=1, q = ๐
๐ in table A.23)
pdf ๐ ๐ฅ =
๐ ๐ ๐๐ฅ ๐โ1
(1 + (๐๐ฅ)๐)2, ๐ > 0, ๐ > 0, 0 < ๐ฅ < โ
rth
order moment
๐๐ฮ 1 +๐
๐ ฮ 1 โ
๐
๐
Mean ๐
1
๐ฮ
1
๐ ฮ 1 โ
1
๐
Mode
๐ ๐ โ 1
๐ + 1
1๐
Variance ๐2 ฮ 1 +
2
๐ ฮ 1 โ
2
๐ โ ฮ2(1 +
1
๐)ฮ2 1 โ
1
๐
A.30 Fisher (F) (a=๐
๐, b=
๐
๐, q =
๐
๐ in table A.23)
๐ ๐ฅ =ฮ
๐ข + ๐ฃ2
๐ข๐ฃ
๐ข2๐ฅ
๐ข2โ1
ฮ ๐ข2 ฮ
๐ฃ2 1 +
๐ฅ๐ข๐ฃ
๐ข+๐ฃ2
, ๐ข > 0, ๐ฃ > 0
0 < ๐ฅ < โ
rth
order moment ๐ฃ๐ข
๐
ฮ ๐ข2 + ๐ ฮ
๐ฃ2 โ ๐
ฮ ๐ข2 ฮ
๐ฃ2
Mean ๐ฃ
๐ฃ โ 2
Mode ๐ข โ 2
๐ข
๐ฃ
๐ฃ + 2
Variance 2๐ฃ2
๐ข + ๐ฃ โ 2
๐ข ๐ฃ โ 2 2 ๐ฃ โ 4
242
A.31 Half Studentโs t (a=๐
๐, b=
๐
๐, q = ๐ in table A.23)
๐ ๐ฅ =2ฮ
1 + ๐2
rฯฮ ๐2 1 +
๐ฅ2
๐
1+๐2
, โโ < ๐ฅ < โ
A.32 Logistic (x = ๐๐in table A.23)
pdf ๐ ๐ฅ =
๐ ๐๐๐(๐ฅโ๐๐๐ (๐)
๐ต ๐, ๐ (1 + ๐๐(๐ฅโlog ๐ ))๐+๐, โโ < ๐ฅ < โ
A.33 Generalized Gamma (๐ช = ๐ท๐๐
๐, ๐ฅ๐ข๐ฆ๐โโ
in table A.23)
pdf ๐ ๐ฅ =
๐ ๐ฅ๐๐โ1
ฮฒ๐๐
ฮ ๐ ๐
โ ๐ฅ๐ฝ
๐
, ๐ > 0, ๐ > 0, ฮฒ > 0,
0 < ๐ฅ < โ
rth
order moment ฮฒ๐ฮ ๐ +
๐๐
ฮ ๐
Mean ฮฒฮ ๐ +
1๐
ฮ ๐
Mode
๐ฝ ๐ โ1
๐
1๐
Variance ๐ฝ2
ฮ2 a ฮ a ฮ ๐ +
2
๐ โ ฮ2 ๐ +
1
๐
Skewness ฮฒ
3ฮ2 ๐ ฮ ๐ +
3๐ โ 3ฮฒ
3ฮ a ฮ ๐ +2๐ ฮ ๐ +
1๐ + 2ฮฒ
3ฮ3 ๐ +
1๐
ฮ3 ๐
Kurtosis
ฮฒ4
ฮ ๐ +4๐
ฮ ๐ โ
4ฮ ๐ +3๐ ฮ ๐ +
1๐
ฮ2 ๐ +
6ฮ ๐ +2๐ ฮ2 ๐ +
1๐
ฮ3 ๐
โ 3ฮ4 ๐ +
1๐
ฮ4 ๐
243
A.34 Gamma (๐ฉ = ๐, in table A.33)
pdf ๐ ๐ฅ =
๐ฅ๐โ1
ฮฒ๐
ฮ ๐ ๐
โ ๐ฅ
๐ฝ , ๐ > 0, ฮฒ > 0, 0 < ๐ฅ < โ
rth
order moment ฮฒ๐ฮ ๐ + ๐
ฮ ๐
Mean ฮ๐
Mode ฮฒ a โ 1
Variance a๐ฝ2
Skewness 2
๐
Kurtosis 3 +
6
๐
A.35 Inverse Gamma (๐ฉ = โ๐, in table A.33)
pdf ๐ ๐ฅ =
ฮฒ๐
๐ฅ1+๐ฮ ๐ ๐
โ ๐ฝ๐ฅ , ๐ > 0, ฮฒ > 0, 0 < ๐ฅ < โ
rth
order moment ฮฒ๐ฮ ๐ โ ๐
ฮ ๐
Mean ฮฒ
a โ 1
Mode ฮฒ
a + 1
Variance ๐ฝ2
๐ โ 1 2 a โ 2
Skewness 4 (a โ 2)
a โ 3
Kurtosis 3(๐ โ 2)(๐ + 5)
๐ โ 3 ๐ โ 4
244
A.36 Weibull (๐ = ๐, in table A.33)
pdf ๐ ๐ฅ =
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
, ๐ > 0, ฮฒ > 0, 0 < ๐ฅ < โ
Cumulative
distribution function 1 โ ๐โ
๐ฆ๐ฝ
๐
rth
order moment ฮฒ
๐ฮ 1 +
๐
๐
Mean ฮฒ
pฮ
1
p
Mode
๐ฝ ๐ โ 1
๐
1๐
Variance ฮฒ
2ฮ 1 +
2
๐ โ ฮผ2
A.37 Exponential (๐ = ๐ฉ = ๐, in table A.33)
pdf ๐ ๐ฅ =
1
ฮฒ ๐
โ ๐ฅ๐ฝ
, ฮฒ > 0, 0 < ๐ฅ < โ
Cumulative
distribution function ๐น ๐ฅ = 1 โ ๐
โ(๐ฅ๐ฝ
)
rth
order moment ฮฒ๐ rฮ ๐
Mean ฮฒ
Mode 0
Variance ฮฒ2
A.38 Chi-Squared (๐ =๐ง
๐, ๐ฉ = ๐, ๐ = ๐, in table A.33)
pdf ๐ ๐ฅ =
1
2n2ฮ
n2
๐ฅn2โ1๐โ
๐ฅ2 , 0 < ๐ฅ < โ
rth
order moment 2๐ฮ n2 + ๐
ฮ n2
Mean n
Mode n โ 2
Variance 2n
245
A.39 Rayleigh (๐ = ๐, ๐ฉ = ๐, ๐ = ๐ ๐, in table A.33)
pdf ๐ ๐ฅ =
๐ฅ
ฮฑ2 ๐
โ ๐ฅ
ฮฑ 2
2
, ฮฑ > 0, 0 < ๐ฅ < โ
Cumulative
distribution function ๐น(๐ฅ) = 1 โ ๐โ
๐ฅ2
2ฮฑ2
rth
order moment ฮฑ 2 ๐ฮ 1 +
๐
2
Mean ฮฑ
ฯ
2
Mode ฮฑ
Variance ฮฑ2
4 โ ๐
2
A.40 Maxwell (๐ =๐
๐, ๐ฉ = ๐, ๐ = ๐ ๐, in table A.33)
๐ ๐ฅ = 2
ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3, ฮฑ > 0, 0 < ๐ฅ < โ
rth
order moment ฮฑ 2 ๐ฮ
32
+๐2
12 ฯ
Mean 2ฮฑ 2
ฯ
Mode 2ฮฑ
Variance ฮฑ2
3ฯ โ 8
ฯ
A.41 Half Standard Normal (๐ =๐
๐, ๐ฉ = ๐, ๐ = ๐, in table A.33)
pdf ๐ ๐ฅ =
2
2ฯ ๐โ
๐ฅ2
2 , 0 < ๐ฅ < โ
rth
order moment 2 ๐ฮ
12 +
๐2
ฮ 12
Mean
2
ฯ
Mode 0
Variance 1
246
A.42 Half Normal (๐ =๐
๐, ๐ฉ = ๐, ๐ = ๐ ๐, in table A.33)
๐ ๐ฅ = 2
ฯ
๐โ
๐ฅ2
2๐2
๐, 0 < ๐ฅ < โ
rth
order moment ฯ 2 ๐ฮ
12 +
๐2
ฮ 12
Mean ฯ 2
ฯ
Mode 0
Variance ฯ2 1 โ
2
ฯ
A.43 Half Log Normal (๐ = (๐๐ ๐ โ ๐, ๐ =๐
๐, ๐ = ๐ ๐ in table A.33)
pdf ๐ ๐ฅ =
2
ฯ๐ฅ 2ฯ ๐
โ ln ๐ฅ โ๐ 2
2ฯ2 , 0 < ๐ฅ < โ
rth
order moment e๐๐+
12๐2๐2
Mean e๐+
12๐2
Mode e๐ +๐2
Variance e2๐ +๐2 e๐2
โ 1
A.44 Four Parameter Generalized Beta Distribution (๐ =๐โ๐
๐โ๐ in table A.1)
pdf ๐ ๐ฅ =
๐ฅ โ ๐ ๐โ1 ๐ โ ๐ฅ ๐โ1
๐ต ๐, ๐ ๐ โ ๐ ๐+๐โ1, ๐, ๐, ๐, ๐ > 0, 0 < ๐ฅ < โ
Mean ๐๐ + ๐๐
๐ + ๐
Mode ๐ โ 1 ๐ + (๐ โ 1)๐
๐ + ๐ โ 2
Variance ๐๐(๐ โ ๐)2
๐ + ๐ 2(๐ + ๐ + 1)
247
A.45 Five Parameter Generalized Beta Distribution (๐ =๐๐
๐๐+๐๐๐ in table A.1)
๐ ๐ฅ =๐๐ฅ๐๐โ1 1 โ 1 โ ๐
๐ฅ๐
๐
๐โ1
๐๐๐๐ต ๐, ๐ 1 + ๐ ๐ฅ๐
๐
๐+๐ , ๐, ๐, ๐, ๐ > 0, 0 < ๐ < 1
rth
order moment ๐๐๐ต(๐ +
๐๐ , ๐)
๐ต(๐, ๐) 2๐น1
๐ +๐
๐,
๐
๐; ๐
๐ + ๐ +๐
๐
Mean ๐๐ต(๐ +
1๐ , ๐)
๐ต(๐, ๐) 2๐น1
๐ +
1
๐,
1
๐; ๐
๐ + ๐ +1
๐
Mode ๐ โ 1 ๐ + (๐ โ 1)๐
๐ + ๐ โ 2
Variance q2B(a +
2p , b)
1 โ c a+
2pB(a, b)
2F1
a +
2
p,
2
p; c
a + b +2
p
โ
qB(a +
1p
, b)
1 โ c a+
1p B(a, b)
2F1
a +
1
p,
1
p; c
a + b +1
p
2
248
A.46 Five Parameter Weighted Generalized Beta Distribution (๐ = ๐ +๐
๐, ๐ =
๐ โ๐
๐ in table A.18)
pdf ๐ ๐ฅ =
๐๐ฅ๐๐+๐โ1
๐๐๐ +๐๐ต ๐ +๐๐ , ๐ โ
๐๐ 1 +
๐ฅ๐
๐
๐+๐ , ๐, ๐, ๐, ๐ > 0,
0 < ๐ฅ < โ, โ๐๐ < ๐ < ๐๐
rth
order moment ๐๐๐ต ๐ +๐๐ +
๐๐ , ๐ โ
๐๐ โ
๐๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
Mean ๐๐ต ๐ +๐๐ +
1๐ , ๐ โ
๐๐ โ
1๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
Variance
๐๐
๐ต ๐ +
๐ + 2๐ , ๐ โ
๐ + 2๐
๐ต ๐ +๐๐ , ๐ โ
๐๐
โ ๐ +
๐ + 1๐ , ๐ โ
๐ โ 1๐
๐ +๐๐ , ๐ โ
๐๐
2
A.47 Weighted Beta of the Second Kind (๐ = ๐ = ๐ in table A.46)
pdf ๐ ๐ฅ =
๐ฅ๐+๐โ1
๐๐+๐๐ต ๐ + ๐, ๐ โ ๐ 1 + ๐ฅ ๐+๐, ๐, ๐, > 0,
0 < ๐ฅ < โ, โ๐ < ๐ < ๐
A.48 Weighted Singh-Maddala (Burr12) (๐ = ๐ in table A.46)
Pdf ๐ ๐ฅ =
๐๐ฅ๐+๐โ1
๐๐+๐๐ต 1 +๐๐ , ๐ โ
๐๐ 1 +
๐ฅ๐
๐
1+๐ , ๐, ๐, ๐ > 0,
0 < ๐ฅ < โ, โ๐ < ๐ < ๐๐
A.49 Weighted Dagum (Burr3) (๐ = ๐ in table A.46)
Pdf ๐ ๐ฅ =
๐๐ฅ๐๐+๐โ1
๐๐๐+๐๐ต ๐ +๐๐ , 1 โ
๐๐ 1 +
๐ฅ๐
๐
๐+1 , ๐, ๐, ๐ > 0,
0 < ๐ฅ < โ, โ๐๐ < ๐ < ๐
249
A.50 Weighted Generalized Fisk (Log-Logistic) (๐ = ๐ = ๐, ๐ =๐
๐ in table
A.46)
Pdf ๐ ๐ฅ =
๐๐ฅ๐+๐โ1
1๐ ๐+๐
๐ต 1 +๐๐ , 1 โ
๐๐ 1 + (๐๐ฅ)๐ 2
, ๐, ๐, ๐ > 0,
0 < ๐ฅ < โ, โ๐ < ๐ < ๐
A.51 Five Parameter Exponential Generalized Beta Distribution (๐ = ๐๐, ๐ =
๐
๐ and ๐ = ๐๐น in table A.18)
๐ ๐ฅ =๐
๐ ๐ฅโ๐ฟ ๐ 1 โ 1 โ ๐ ๐
๐ฅโ๐ฟ ๐
๐โ1
๐๐ต ๐, ๐ 1 + ๐ ๐๐ฅ
๐๐ฟ
1๐
๐+๐ , ๐, ๐, ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< ln
1
1 โ ๐
A.52 Exponential Generalized Beta Distribution of the First Kind (๐ = ๐ in
table A.51)
๐ ๐ฅ =๐
๐ ๐ฅโ๐ฟ ๐ 1 โ ๐
๐ฅโ๐ฟ ๐
๐โ1
๐๐ต ๐, ๐ , ๐, ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< 0
A.53 Exponential Generalized Beta Distribution of the Second Kind (๐ = ๐ in
table A.51)
๐ ๐ฅ =๐
๐ ๐ฅโ๐ฟ ๐
๐๐ต ๐, ๐ 1 + ๐๐ฅโ๐ฟ๐
๐+๐ , ๐, ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< โ
250
A.54 Exponential Generalized Gamma (Generalized Gompertz) (๐ โ
โ ๐๐ง๐ ๐นโ = ๐๐๐๐ + ๐น in table A.51)
๐ ๐ฅ =๐
๐ ๐ฅโ๐ฟ ๐ ๐โ๐ ๐ฅโ๐ฟ /๐
๐ฮฮ๐, ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< โ
A.55 Generalized Gumbell (๐ = ๐ ๐๐ง๐ ๐ = ๐ in table A.51)
Pdf ๐ ๐ฅ =
๐๐ ๐ฅโ๐ฟ /๐
๐๐ต ๐, ๐ 1 + ๐๐ฅโ๐ฟ๐
2๐ , ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< โ
A.56 Exponential Power (๐ = ๐ ๐๐ง๐ ๐ = ๐ in table A.51)
Pdf ๐ ๐ฅ =
๐๐๐ ๐ฅโ๐ฟ /๐
๐, ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< 0
A.57 Exponential (Two Parameter) (๐ = ๐, ๐ = ๐ ๐๐ง๐ ๐ = ๐ in table A.51)
Pdf ๐ ๐ฅ =
๐ ๐ฅโ๐ฟ /๐
๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< 0
251
A.58 Exponential (One Parameter) (๐ = ๐, ๐ = ๐, ๐ = ๐ ๐๐ง๐ ๐น = ๐ in table
A.51)
Pdf ๐ ๐ฅ =
๐๐ฅ/๐
๐, ๐ > 0,
โโ <๐ฅ
๐< 0
A.59 Exponential Fisk (Logistic) (๐ = ๐, ๐ = ๐, ๐ = ๐ in table A.51)
Pdf ๐ ๐ฅ =
๐(๐ฅโ๐ฟ)/๐
๐(1 + ๐(๐ฅโ๐ฟ)/๐ )2, ๐ > 0,
โโ <๐ฅ โ ๐ฟ
๐< โ
A.60 Standard Logistic (๐ = ๐, ๐ = ๐, ๐ = ๐, ๐น = ๐ in table A.51)
Pdf ๐ ๐ฅ =
๐๐ฅ/๐
๐(1 + ๐๐ฅ/๐)2, ๐ > 0,
โโ <๐ฅ
๐< โ
A.61 Exponential Weibull (๐ โ โ, ๐นโ = ๐๐๐๐ + ๐น ๐๐ง๐ ๐ = ๐ in table A.51)
๐ ๐ฅ =๐
๐ฅโ๐ฟ ๐ ๐โ๐ ๐ฅโ๐ฟ /๐
๐ฮฮ๐, ๐, ๐ฟ > 0,
โโ <๐ฅ โ ๐ฟ
๐< โ
Other Generalized Beta Distributions
A.62 Beta Generator (๐ = ๐ฎ ๐ in table A.1)
Pdf ๐ ๐ฅ =
๐(๐ฅ) ๐บ ๐ฅ ๐โ1
1 โ ๐บ ๐ฅ ๐โ1
๐ต(๐, ๐), ๐ > 0, ๐ > 0,
โโ < ๐ฅ < โ, 0 < ๐บ(๐ฅ) < 1
252
A.63 ith Order Statistics (a=i, b=n-i+1 in table A.62)
Pdf ๐ ๐ฅ =
n!
๐ โ 1 ! ๐ โ ๐ !๐(๐ฅ)[๐บ(๐ฅ)]๐โ1 1 โ ๐บ ๐ฅ ๐โ๐ , ๐ > 0,
๐ โ ๐ > 0, โโ < ๐ฅ < โ, 0 < ๐บ(๐ฅ) < 1
A.64 Beta-Normal (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ =๐โ1 ฮฆ
๐ฅ โ ๐๐
๐โ1
1 โ ฮฆ ๐ฅ โ ๐
๐ ๐โ1
๐ต ๐, ๐ ๐
๐ฅ โ ๐
๐
๐ > 0, ๐ > 0, ๐ > 0, ๐ โ โ ๐๐๐ ๐ฅ โ โ
Parent cdf
๐บ ๐ฅ = ฮฆ x โ ฮผ
ฯ
Parent pdf ๐ ๐ฅ = ๐ x โ ฮผ
ฯ
A.65 Beta-Exponential (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
ฮป exp(โ๐ฮปx)[1 โ exp(โฮปx)]๐โ1
๐ต ๐, ๐
๐ > 0, ๐ > 0, ฮป > 0, ๐ฅ >0
Parent cdf
๐บ ๐ฅ = 1 โ exp(โฮปx)
Parent pdf ๐ ๐ฅ = ฮปexp(โฮปx)
A.66 Beta-Weibull (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ =
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ๐ ๐ฅ๐ฝ
๐
1 โ eโ
xฮฒ
c
๐โ1
๐ต ๐, ๐ , ๐, ๐, ๐, ฮฒ > 0
Parent cdf
๐บ ๐ฅ = 1 โ eโ
xฮฒ
c
Parent pdf ๐ ๐ฅ =
๐
ฮฒ๐ ๐ฅ๐โ1e
โ xฮฒ
c
253
A.67 Beta-Hyperbolic Secant (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ = 2ฯ arctan ๐๐ฅ
๐โ1
1 โ2ฯ arctan ๐๐ฅ
๐โ1
๐ต ๐, ๐ ฯ cosh x
๐ > 0, ๐ > 0 and ๐ฅ โ โ
Parent cdf
๐บ ๐ฅ =2
ฯarctan ๐๐ฅ
Parent pdf ๐ ๐ฅ =
1
ฯ cosh x
A.68 Beta-Gamma (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ =
๐ฅ๐โ1๐โ๐ฅ๐ฮx
ฮป ฯ ๐โ1 1 โ
ฮxฮป ฯ
ฮ ฯ
๐โ1
๐ต ๐, ๐ ฮ ฯ aฮปฯ
๐, ๐, ฯ, ฮป, x > 0
Parent cdf
๐บ ๐ฅ =ฮxฮป ฯ
ฮ ฯ
Parent pdf ๐ ๐ฅ =
๐ฅ
๐ ๐โ1
๐โ
๐ฅ๐
A.69 Beta-Gumbel (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ = exp โ
x โ ฮผฯ
๐๐ต ๐, ๐ eโ๐ (exp โ
xโฮผฯ
) 1 โ eโ exp โxโฮผฯ
bโ1
๐, ๐, ๐ข, ฯ, ๐ฅ > 0
Parent cdf
๐บ ๐ฅ = exp โ exp โx โ ฮผ
ฯ
Parent pdf
๐ ๐ฅ =exp โ
x โ ฮผฯ
ฯexp โ(exp โ
x โ ฮผ
ฯ )
254
A.70 Beta-Frรจchet (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ = ฮป๐ฮป exp โa
xฯ
โฮป
1 โ exp โ xฯ
โฮป
bโ1
๐ฅ1+ฮป๐ต ๐, ๐
๐, ๐, ฮป, ฯ, x > 0
Parent cdf
๐บ ๐ฅ = exp โ x
ฯ
โฮป
Parent pdf ๐ ๐ฅ =
ฮป๐ฮป
๐ฅ1+ฮปexp โ
x
ฯ โฮป
A.71 Beta-Maxwell (Substituting G(X) and g(X) in table A.62)
๐ ๐ฅ =1
๐ต ๐, ๐
2
ฯฮณ
3
2,๐ฅ2
2ฮฑ
aโ1
1 โ2
ฯฮณ
3
2,๐ฅ2
2ฮฑ
bโ1
2
ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3
๐, ๐, ฮฑ, ๐ฅ > 0 ๐๐๐ ฮณ ๐, ๐ is an incomplete gamma function
Parent cdf
๐บ ๐ฅ =2
ฯฮณ
3
2,๐ฅ2
2ฮฑ
Parent pdf
๐ ๐ฅ = 2
ฯ
๐ฅ2๐โ
๐ฅ2
2โ2
ฮฑ3
A.72 Beta-Pareto (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
๐
๐๐ต ๐, ๐ 1 โ
๐ฅ
๐ โk
aโ1
๐ฅ
๐
โ๐๐โ1
๐, ๐, ๐, ๐, ๐ฅ > 0
Parent cdf
๐บ ๐ฅ = 1 โ ๐ฅ
๐
โk
Parent pdf ๐ ๐ฅ =
๐๐๐
๐ฅ๐+1
255
A.73 Beta-Rayleigh (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
๐ฅ
ฮฑ2๐ต ๐, ๐ 1 โ ๐
โ๐ฅ2
2ฮฑ2
aโ1
๐โ๐
๐ฅ
ฮฑ 2
2
๐, ๐, ๐, ๐ผ > 0
Parent cdf
๐บ ๐ฅ = 1 โ ๐โ
๐ฅ2
2ฮฑ2
Parent pdf ๐ ๐ฅ =
๐ฅ
ฮฑ2 ๐
โ ๐ฅ
ฮฑ 2
2
A.74 Beta-Generalized-Logistic of Type IV (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
B a, b 1โaโb
B a, b
eqx
1 โ eโx p+q[B 1
1+eโx(p, q)]aโ1[B eโx
1+eโx
(q, p)]bโ1
๐, ๐, p, q > 0, ๐ฅ > ๐
Parent cdf
๐บ ๐ฅ =
B 11+eโx
(p, q)
B p, q
Parent pdf ๐ ๐ฅ =
eqx
1 + eโx p+q
A.75 Beta-Generalized-Logistic of Type I (Substituting q=1 in table A.74)
Pdf ๐ ๐ฅ =
peโx[ 1 + eโax p โ 1]bโ1
B a, b 1 + eโx a+pb
๐, ๐, p > 0, ๐ฅ > ๐
A.76 Beta-Generalized-Logistic of Type II (Substituting q=1 in table A.74)
Pdf ๐ ๐ฅ =
qeโbqx
B a, b 1 + eโx qb +1 1 โ
eโqx
B a, b 1 + eโx q
p
๐, ๐, q > 0, ๐ฅ > ๐
A.77 Beta-Generalized-Logistic of Type III (Substituting q=1 in table A.74)
Pdf ๐ ๐ฅ =
B q, q 1โaโb
B a, b
eโqx
1 + eโx 2q[B 1
1+eโx(q, q)]aโ1[B eโx
1+eโx
(q, q)]bโ1
๐, ๐, q > 0, ๐ฅ > ๐
256
A.78 Beta-Beta Prime (๐ฑ = ๐โ๐ฒ ๐ข๐ง ๐ญ๐๐๐ฅ๐ ๐. ๐๐)
Pdf ๐ ๐ฅ =
๐ต ๐, ๐ 1โ๐โ๐
๐ต ๐, ๐
๐ฅ๐โ1
1 + ๐ฅ ๐+๐[๐ต ๐ฅ
1+๐ฅ(๐, ๐)]๐โ1[๐ต 1
1+๐ฅ(๐, ๐)]๐โ1
๐, ๐, ๐, > 0, ๐ฅ > ๐
A.79 Beta-F (Substituting G(X) and g(X) in table A.62) Pdf ๐ ๐ฅ
=B a, b โ1
B u2 ,
v2
vu
v2
xv2โ1
1 + vu x
(u+v)/2
I
vu x
1+ vu x
v
2,u
2
aโ1
I 1
1+ vu x
v
2,u
2
bโ1
๐, ๐, u, v, x > 0
Parent cdf
๐บ ๐ฅ =1
B ๐ข2 ,
๐ฃ2
B
vu x
1+ vu x
๐ฃ
2,๐ข
2
Parent pdf
๐ ๐ฅ =
vu
q2
B ๐ข2 ,
๐ฃ2
xv2โ1
1 + vu x
(u+v)/2
A.80 Beta-Burr XII (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
๐๐๐โ๐๐ฅ๐โ1
๐ต ๐, ๐ 1 โ 1 +
๐ฅ
q
p
โ๐
aโ1
1 + ๐ฅ
q
p
โ๐๐โ1
๐, ๐, ๐, ๐, ๐, ๐ฅ > 0.
Parent cdf
๐บ ๐ฅ = 1 โ 1 + ๐ฅ
q
p
โ๐
Parent pdf ๐ ๐ฅ = ๐๐๐โ๐ 1 +
๐ฅ
q
p
โ๐โ1
๐ฅ๐โ1
A.81 Beta-Dagum (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
1
๐ต ๐, ๐ 1 + ๐๐ฅโ๐ โ๐ aโ1 1 โ 1 + ๐๐ฅโ๐ โ๐ bโ1
๐ ๐๐๐ฅโ๐โ1
(1 + ๐๐ฅโ๐)1+a
๐, ๐, ๐, ๐, ๐ฅ > 0
Parent cdf
๐บ ๐ฅ = 1 + ๐๐ฅโ๐ โ๐
Parent pdf ๐ ๐ฅ =
๐ ๐๐๐ฅโ๐โ1
(1 + ๐๐ฅโ๐)1+a
257
A.82 Beta-Fisk (Log-Logistic) (Substituting G(X) and g(X) in table A.62)
Pdf ๐ ๐ฅ =
๐๐ ๐๐ฅ apโ1
๐ต ๐, ๐ 1 + ๐๐ฅ p a+b
๐, ๐, ๐, ๐, ๐ฅ > 0
Parent cdf
๐บ ๐ฅ = ๐๐ฅ p
1 + ๐๐ฅ p
Parent pdf ๐ ๐ฅ =
๐๐ ๐๐ฅ pโ1
1 + ๐๐ฅ p 2
A.83 Beta-Generalized Half Normal (Substituting G(X) and g(X) in table A.62) Pdf
๐ ๐ฅ = 2๐
๐ฅ๐
ฮฑ
โ 1 aโ1
2 โ 2๐ ๐ฅ๐
ฮฑ
bโ1
2ฯ
ฮฑ๐ฅ
๐ฅ๐
ฮฑ
๐โ12 ๐ฅ๐ ๐๐ถ
๐ต ๐, ๐
๐, ๐, ฮฑ, ๐ > 0
Parent cdf
๐บ ๐ฅ = 2๐ ๐ฅ
๐
ฮฑ
โ 1
Parent pdf
๐ ๐ฅ = 2
ฯ ฮฑ
๐ฅ
๐ฅ
๐
ฮฑ
๐โ12 ๐ฅ๐ ๐๐ถ
A.84 Generalized Beta Generator (๐ฑ = ๐ ๐ฒ ๐ in table A.1)
Pdf ๐ ๐ฅ =
๐ ๐ฅ ๐ผ ๐บ ๐ฅ ๐ผโ1 ๐บ ๐ฅ ๐ผ ๐โ1 1 โ ๐บ ๐ฅ ๐ผ ๐โ1
๐ต(๐, ๐), ๐ > 0,
๐ผ > 0, ๐ > 0, โโ < ๐ฅ < โ, 0 < ๐บ ๐ฅ ๐ผ < 1
A.85 Exponentiated Exponential (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ =
1 โ ๐โ
๐ฅ๐ฝ
๐ผ
1 โ ๐โ
๐ฅ๐ฝ
๐ผ
๐โ1
1 โ 1 โ ๐โ
๐ฅ๐ฝ
๐ผ
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = 1 โ ๐โ
๐ฅ๐ฝ
๐ผ
Parent pdf ๐ ๐ฅ =
๐ผ
๐ฝ 1 โ ๐
โ ๐ฅ๐ฝ
๐ผโ1
๐โ
๐ฅ๐ฝ
258
A.86 Exponentiated Weibull (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ =
๐ฮฒp ๐ฆ๐โ1๐
โ ๐ฆ๐ฝ
๐
[1 โ ๐โ
๐ฆ๐ฝ
๐
]๐โ1 1 โ 1 โ ๐โ
๐ฆ๐ฝ
๐
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = 1 โ ๐โ
๐ฆ๐ฝ
๐
Parent pdf ๐ ๐ฅ =
๐
ฮฒp ๐ฆ๐โ1๐
โ ๐ฆ๐ฝ
๐
A.87 Exponentiated Pareto (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ =
๐qp๐๐ฆโ๐โ1 1 โ ๐ฆ๐ โ๐
๐โ1
[ 1 โ ๐ฆ๐ โ๐
๐
]๐โ1 1 โ 1 โ ๐ฆ๐
โ๐
๐
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = 1 โ ๐ฆ
๐
โ๐
๐
Parent pdf ๐ ๐ฅ = ๐qp๐๐ฆโ๐โ1 1 โ
๐ฆ
๐
โ๐
๐โ1
A.88 Generalized Beta Exponential (Substituting G(X) and g(X) in table A.84)
Pdf ๐ ๐ฅ =
๐ ฮป exp(โฮปx)[1 โ exp(โฮปx) ]๐๐โ1 1 โ 1 โ exp โฮปx ๐ ๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = 1 โ exp(โฮปx) Parent pdf ๐ ๐ฅ = ฮป exp(โฮปx)
A.89 Generalized Beta Weibull (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ =
๐ ๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
[1 โ eโ
xฮฒ
c
]๐๐โ1 1 โ 1 โ eโ
xฮฒ
c
๐
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = 1 โ eโ
xฮฒ
c
Parent pdf ๐ ๐ฅ =
๐
ฮฒ๐ ๐ฅ๐โ1 ๐
โ ๐ฅ๐ฝ
๐
259
A.90 Generalized Beta Hyperbolic Secant (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ = ๐
2ฯ
arctan ๐๐ฅโ๐๐
๐๐โ1
1 โ 2ฯ
arctan ๐๐ฅโ๐๐
๐
๐โ1
๐ต ๐, ๐ ฯฯ cosh ๐ฅ โ ๐
๐
Parent cdf
๐บ ๐ฅ =2
ฯarctan ๐
๐ฅโ๐๐
Parent pdf ๐ ๐ฅ =
1
ฯฯ cosh ๐ฅ โ ๐
๐
A.91 Generalized Beta Normal (Substituting G(X) and g(X) in table A.84) Pdf
๐ ๐ฅ =๐๐โ1[ฮฆ
๐ฅ โ ๐๐ ]๐๐โ1 1 โ ฮฆ
๐ฅ โ ๐๐
p
๐โ1
๐ต ๐, ๐ ๐
๐ฅ โ ๐
๐
Parent cdf
๐บ ๐ฅ = ฮฆ ๐ฅ โ ๐
๐
Parent pdf ๐ ๐ฅ = ๐
๐ฅ โ ๐
๐
A.92 Generalized Beta Log Normal (Substituting G(X) and g(X) in table A.84) Pdf
๐ ๐ฅ =๐๐ log ๐ฅ [ฮฆ log ๐ฅ ]๐๐โ1 1 โ ฮฆ log ๐ฅ ๐ ๐โ1
๐ฅ๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = ฮฆ log ๐ฅ
Parent pdf ๐ ๐ฅ =
๐ log๐ฅ
๐ฅ
A.93 Generalized Beta Gamma (Substituting G(X) and g(X) in table A.84) Pdf
๐ ๐ฅ =
xฮป
ฯโ1
eโxฮป
ฮxฮป ฯ
ฮ ฯ
๐๐โ1
1 โ ฮxฮป ฯ
ฮ ฯ
๐
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = ฮxฮป ฯ
ฮ ฯ
Parent pdf ๐ ๐ฅ =
x
ฮป
ฯโ1
eโ
xฮป
260
A.94 Generalized Beta Frรจchet (Substituting G(X) and g(X) in table A.84)
๐ ๐ฅ =
๐ฮปฯ
exp โ xฯ
โฮป ๐ฅ๐โ1 exp โ
xฯ
โฮป
๐๐ โ1
1 โ exp โ xฯ
โฮป
๐
๐โ1
๐ต ๐, ๐
Parent cdf
๐บ ๐ฅ = exp โ x
ฯ โฮป
Parent pdf ๐ ๐ฅ =
ฮป
ฯexp โ
x
ฯ
โฮป
๐ฅ๐โ1
A.95 Confluent Hypergeometric
Pdf ๐ ๐ฅ =
1
๐ต(๐,๐ โ ๐) ๐ข๐โ1 1 โ ๐ข ๐โ๐โ1๐๐ฅ๐ข๐๐ข
1
0
A.96 Gauss Hypergeometric
Pdf ๐ ๐ฅ =
ฮ(c)
ฮ(a)ฮ(c โ a) ๐ข๐โ1(1 โ ๐ข)
๐โ๐โ1(1 โ ๐ฅ๐ข)
โ๐๐๐ข
1
0
A.97 Appell
Pdf ๐ ๐ฅ =
๐ถ๐ฅ๐ผโ1 1 โ ๐ฅ ๐ฝโ1
1 โ ๐ข๐ฅ ๐ 1 โ ๐ฃ๐ฅ ๐ ,
0 < ๐ฅ < 1, ๐ผ > 0, ๐ฝ > 0, ๐ > 0, ๐ > 0, โ1 < ๐ข < 1, โ1 < ๐ฃ < 1 and C
is the normalizing constant given by 1
๐ถ= ๐ต ๐ผ, ๐ฝ ๐น1(๐ผ, ๐, ๐, ๐ผ + ๐ฝ; ๐ข, ๐ฃ)
261
SUMMARY: FRAMEWORK FOR CLASSICAL TO GENERALIZED BETA DISTRIBUTIONS
Power
Beta of the first Kind Beta of the Second Kind Arcsine
Triangular Shaped
Parabolic
Wigner SemiCircle
3 Parameter generalized beta
Chotikapanich, McDonaldโs
Kum
araswam
y
3 Parameter generalized beta
Libby&Novickโs, McDonaldโs,
Beta type 3
4 Parameter generalized beta
of the first kind
4 Parameter generalized beta
of the second kind
Gen
eralized P
ow
er
Unit G
amm
a
Pareto
(Type I)
Generalized Gamma
Gam
ma
Weib
ull
Exponen
tial
Chi-S
quared
Ray
leigh
Half-N
orm
al
Max
well
Half L
og
-Norm
al
Uniform Lomax
(Pareto Type II)
Log-logistic
(Fisk)
Sto
pp
a
Sin
gh
-Mad
dala
Gen
eralized L
om
ax
(Pareto
Type II)
Inverse L
om
ax
Dag
um
Para-L
og
istic
Gen
eral Fisk
(Log-lo
gistic)
Fish
er (F)
Inverse G
amm
a
Logistic
5 Parameter generalized beta
262
11 References
1. Akinsete, Alfred, Felix Famoye, and Carl Lee. โThe beta Pareto distribution.โ
Statistics 42.6 (2008):547-563.
2. Akinsete,A., and C. Lowe. โBeta Rayleigh distribution in reliability
measure.Section on Physical and Engineering Sciences.โ Proceedings of the
American Statistical Association. (2009).
3. Alexander C., Cordeiro, G.M., Ortega, E.M.M., and Josรจ M.S (2010)
โGeneralized Beta Generated Distributionsโ
4. Amusan, Grace Ebunoluwa, โThe Beta Maxwell Distribution (2010)โ.
5. Barreto-Souza, W., Cordeiro, G. M., and Simas, A. B. Some results for beta
Frรจchet distribution, (2008).
6. C. Armero and M.J. Bayarri, Prior assessments for prediction in queues, The
Statistician 43, 139-153, (1994).
7. Catherine, F., Merran, E., Nicholas, H., and Brian, P., Statistical distributions, 4th
ed., Wiley, (2011)
8. Chen, J.C. and Novick, M.R., (1984), Bayesian analysis for binomial models
with generalized beta prior distributions. Journal of Educational Statistics, 9,
163-175.
9. Chotikapanich, D., W.E. Griffiths, D.S.P Rao, and V. Valecia (2010). Global
Income Distribution and Inequality, 1993 and 2000: Incorporating Country-
Level Inequality Modeled with Beta Distributions. Forthcoming in The Review of
economics and Statistics
10. Christian, C., and Samuel, K. (2003). Statistical Size Distributions in Economics
and Actuarial Sciences. New Jersey: John Wiley.
11. Condino, Francesca, and Filippo Domma. โThe beta-Dagum distribution.โ 45th
Scientific Meeting of the Italian Statistical Society, University of Padua. (2010).
12. Cummins, J. D., Dionne, G., McDonald, J. B., and Pritchett, B. M. (1990).
Applications of the GB2 family of distributions in modeling insurance loss
processes. Insurance: Mathematics & Economics, 9, 257โ272.
263
13. DE MORAIS, ALICE LEMOS. A class of generalized beta distributions, Pareto
power series and Weibull power series. Diss. Universidade Federal de
Pernambuco, (2009).
14. Dutka, J. (1981). The incomplete beta functionโa historical profile. Archive for
the History of Exact Sciences, 24, 11โ29.
15. Eugene, N., C. Lee, and F. Famoye, (2002). Beta-normal Distribution and its
applications. Communication in Statistics โ Theory and Methods, 31:497-512.
16. Famoye, F. , Lee, C. & Olugbenga Olumolade, (2005). The Beta-weibull
distribution. Journal of Statistical Theory and Applications, 121-138.
17. Famoye, F., Lee, C., and Eugene, N. Beta-Normal Distribution: Bimodality
Properties and Application. Journal of Modern Applied Statistical Methods 2
(Nov.2003), 314โ326.
18. Fischer, Matthias J.; Vaughan, David (2004) : The Beta-Hyperbolic Secant
(BHS) Distribution, Diskussionspapiere // Friedrich-Alexander-Universitรคt
Erlangen-Nรผrnberg, Lehrstuhl fรผr Statistik und รkonometrie, No. 64/2004,
http://hdl.handle.net/10419/29617
19. Gordy, M.B., (1988a), Computationally convenient distributional assumptions
for common-value auctions, Computational Economics 12, 61-78.
20. Gordy, M.B., (1988b), A generalization of generalized beta distributions. Paper
No. 1998-18, Finance and Economics Discussion Series. The Federal Reserve
Boeard.
21. Grassia, A., (1977), On a family of distributions with argument between 0 and 1
obtained by transformation of the gamma and derived compound distributions,
Australian Journal of Statistics 19, 108-114.
22. Gupta, A. K. and Nadarajah, S. (2004). On the moments of the beta normal
distribution. Communication in Statistics โ Theory and Methods, 31: 1-13.
23. Gupta, A. and Nadarajah, S. (2004). Handbook of Beta Distribution and Its
Applications. New York: Marcel Dekker.
24. Gupta, A. K. and Nadarajah, S.. Beta Bessel Distribution. International Journal
of Mathematics and Mathematical Sciences, (2006): 1-14.
25. Gupta, R.D., Kundu, D. (1999). Generalized exponential distributions.
264
Australian and New Zealand Journal of Statistic 41, 173-188
26. J.H. Sepanski and L.Kong, A Family of Generalized Beta Distribution For
Income, International Business, 1(10), (2007), 129-145.
27. Jones, M.C. โFamilies of distributions arising from distributions of order
statistics.โ Test 13.1 (2004):1-43.
28. Kalbfleisch, J. D., and Prentice, R. L. (1980). The Statistical Analysis of Failure
Time Data. New York: John Wiley.
29. Klugman, S. A., Panjer, H. H., and Willmot, G. E. (1998). Loss Models. New
York: John Wiley.
30. Kong, L., Lee, C., and Sepanski, J. H. On the Properties of Beta-Gamma
Distribution. Journal of Modern Applied Statistical Methods 6, 1 (2007), 187โ
211.
31. L.Cardeลo, D.K Nagar and L.E Sรกnchez, Beta Type 3 Distribution and Its
Multivariate Generalization, Tamsui Oxford Journal of Mathematical Sciences
21(2)(2005).225-241 Aletheia University
32. Lee, C., Famoye, F., and Olumolade, O. Beta-Weibull Distribution: Some
Properties and Applications to Censored Data. Journal Of Modern Applied
Statistical Methods 6 (May 2007), 173โ186.
33. Leemis, L. and Mcqueston, J., Univariate Distribution Relationships The
American Statistician, February (2008), vol.62, No.1 45
34. Libby, D.L and Novick, M., (1982), Multivariate generalized beta-distributions
with applications to utility assessment. Journal of Educational Statistics, 7, 271-
294.
35. McDonald, J. B., and Ransom, M. (1979a). Functional forms, estimation
techniques, and the distribution of income. Econometrica, 47, 1513โ1526.
36. McDonald, J. B., and Richards, D. O., (1987a), Model selection: some
generalized distributons. Communications in Statistics (volume A), 16, 1049-
1074.
37. McDonald, J. B., and Richards, D. O., (1987b), Some generalized models with
applications to reliability. Journal of Statistical Planning and Inference, 16, 365-
376.
265
38. McDonald, J. B., and Y.J.Xu, A Generalization of the Beta Distribution with
Application, Journal of Econometrics, 69(2), (1995), 133-152.
39. McDonald, J.B., (1984), Some generalized functions for the size distribution of
income. Econometrica, 52, 647-664.
40. Mudholkar, G.S., Srivastava, D.K., Freimer, M. (1995). The exponentiated
Weibull family. Technometrics 37 436-445.
41. Nadarajah, S. (2005). Exponentiated beta distributions. Computers
&Mathematics with Application. 49, 1029-1035.
42. Nadarajah, S. and Kotz, S., (2004): A generalized beta distribution. The
Mathematical Scientist, 29, 36-41.
43. Nadarajah, S., and Gupta, A. K. The Beta Frรจchet Distribution. Far East Journal
of Theoretical Statistics 14 (2004), 15โ24.
44. Nadarajah, S., and Kotz, S. The Beta Exponential Distribution. Reliability
Engineering and System Safety 91 (2006), 689-697.
45. Nadarajah, S., and Kotz, S. (2006), An F1 beta distribution with bathtub failure
rate function. American Journal of Mathematical and Management Sciences, 26,
113-131.
46. Nadarajah, S., and Kotz, S. (2007): Multitude of beta distributions with
applications, Statistics: A Journal of Theoretical and Applied Statistics, 41:2,
153-179
47. Nadarajah, S., and Kotz, S. The Beta Gumbel Distribution. Mathematical
Problems in Engineering 4 (2004), 323โ332.
48. Paranaiba P.F, Ortega E.M.M, Cordeiro G.M, Pescim R.R.,The beta Burr XII
distribution with application to lifetime data, Computational Statistics and Data
Analysis (2010)
49. Patil, G.P., M.T.Boswell, and M.V. Ratnaparkhi, (1984), Dictionary and
classified bibliography of statistical distributions in scientific work: Continuous
univariate models (International cooperative Publishing House, Burtonsville,
MD).
50. Pescim,R.R.,Demรจtrio C.G., Cordeiro, G.M., Ortega, E.M., and Urbano, M.R.
(2009). The beta generalized half-normal distribution. Computational Statistics
266
and data analysis, 54(4), 945-957.
51. Thurow, L. C. (1970). Analyzing the American Income Distribution. Papers and
proceedings, American Economics Association, 60, 261-269.
52. Vartia, P. L. I., and Vartia, Y. O. (1980). Description of the income distribution
by the scaled F distribution model. In: N. A. Klevmarken and J. A. Lybeck
(eds.): The Statics and Dynamics of Income. Clevedon, UK: Tieto, pp. 23โ36.
53. Venter, G. (1983). Transformed beta and gamma distributions and aggregate
losses. Proceedings of the Casualty Actuarial Society, 70, 156โ193.
54. Walck, C. (1996). Handbook on Statistical Distributions for Experimetalists.
Internal Report SUFโPFY/96โ01, Particle Physics Group, Fysikum, Stockholm
University, 11 December 1996.
55. Yuan; Ye, Broderick, O. Oluyede; Mavis, Pararai (2012) : Weighted generalized
beta distribution of the second kind and related distributions, Journal of
Statistical and Econometric Methods, ISSN 2241-0376, International Scientific
Press, Vol 1, Iss. 1, 13-31.
