BEYOND BETA, Other Continuous Distribution with Bounded …dorpjr/BEYONDBETA/1 - BEYOND BET… · A...

BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 1

BEYOND BETA, Other Continuous Distribution withBounded Support and Applications

"Presentation Short Course: Beyond Beta and Applications"November 20th, 2018, La Sapienza

Presentation by: J. René van Dorp‡

‡ Department of Engineering Management and Systems Engineering,School of Engineering and Applied Science,The George Washington University,800 22nd Street, N.W. Suite 2800,ßWashington D.C. 20052. E-mail: dorpjr@gwu.edu

OUTLINE

1. Review properties of Triangular Distribution.

2. Early extensions of the symmetric Triangular Distribution.

3. The Two-Sided Power Distribution (TSP)

4. A link between TSP and Asymmetric Laplace distributions

5. Moment Ratio Diagram Comparison

6. Mazimum Likelihood Estimation STSP distributions

7. A general framework of Two-Sided Distributions

8. Generalized Trapezoidal Distributions

9. Generalized Two-Sided Power Distributions

1. TRIANGULAR DISTRIBUTION... Review

• One of the first records that mentions the continuous uniformdistribution is the famous paper by the reverend ThomasBayes (1763) (only a few years after Simpson's written records in1757).

• One of the earliest mentions of the triangular distributionsseems to be by , a colorfulThomas Simpson(1755, 1757)personality in Georgian England.

• Hence, is certainlythe continuous triangular distributionamongst the first distributionscontinuous to have beennoticed by investigators during the 18-th century.

1. TRIANGULAR DISTRIBUTION... Review

0.00 0.50 1.00x

• R. Schmidt (1934) was possibly the first to notice that thesymmetric triangular distribution follows from twoindependent uniform random variables and on , viaY Y Ò!ß "Ó" #

\ œY Y

#" # .

• Asymmetric standard triangular distributions in Figure Babove with were studied by .support Ò!ß "Ó Ayyangar (1941)

1. TRIANGULAR DISTRIBUTION... pdf, cdf, qf

• probability density function (pdf):

0ÐDl+ß7ß ,Ñ œ

ß + Ÿ D Ÿ 7ß

ß 7 Ÿ D Ÿ ,ß

,+ 7+# ,D

for for elsewhere

• cumulative distribution function (cdf):

JÐDÑ œ T<Ð^ Ÿ D œ"

l+ß7ß ,Ñß + Ÿ D Ÿ 7ß

7 Ÿ D Ÿ ,Þ

7+ D+,+ D+

,7 ,D,+ ,7

• quantile function (qf):

J ÐCl+ß7ß ,ß 8Ñ œ+ CÐ7 +ÑÐ, +Ñß ! Ÿ C Ÿ

, Ð" CÑÐ, 7ÑÐ, +Ñß Ÿ C Ÿ "" for

1. TRIANGULAR DISTRIBUTION... Mean, MP, Variance

• The mean is of the parameters and the simple average +ß7 ,Þ

IÒ^Ó œ Þ+ 7 ,

• The mode probability (MP) to the left of the mode equals therelative distance of the mode to the lower bound over7 +the whole range Ò+ß ,ÓÞ

T <Ð^ Ÿ 7Ñ œ œ7 +

• The variance is a function of that mode probability ; À

Z +<Ò^Ó œ Ð, +Ñ † ; ‚ Ð" ;Ñ

OUTLINE

2. Early extensions of the Triangular Distribution.

2. EARLY TRIANGULAR EXTENSIONS... Trapezoidal

THE SYMMETRIC TRAPEZOIDAL DISTRIBUTION

0.00 0.33 0.67 1.00x

\ œ Y Y" #$ $" #

2. EARLY TRIANGULAR EXTENSIONS... Uniform mixture

A finite mixture of Uniform Distributions may thus beinterpreted as an extension of the symmetric Triangular

distribution with support .Ò!ß "Ó

] œ A Y ß A œ "ß A !ß 3œ" 3œ"

3 3 3 3

T <Ð] Ÿ CÑ œ á @ œ! @ œ!

Ð "Ñ3œ"

3@ (C A @ Ñ

7x A" ÐC A @ ÑÞ3œ"

Ò!ß∞Ñ

• First derived by . Later by Mitra (1971) Barrow and Smith (1979).Their proofs are geared towards mathematically oriented readers,are very concise and somewhat difficult to follow.

2. EARLY TRIANGULAR EXTENSIONS... Uniform mixture

0.00 0.25 0.50 0.75 1.00x

Bates Mitra Case 1Mitra Case 2 Uniform

Figure 2.8. Examples of the pdf's of (2.0) for . A: , (Bates);8 œ % A œ 3 œ "ßáß%3"%

B: , Mitra Case 1);A œ !Þ'ßA œ !Þ$ A œ !Þ!(&ß A œ !Þ!#& Ð" # $ %

C: , Mitra Case );A œ !Þ)ßA œ !Þ" A œ !Þ!(&ß A œ !Þ!#& Ð #" # $ %

D: (Uniform).A œ "ß A œ !ß 3 œ #ß $ß %" 3

2. EARLY TRIANGULAR EXTENSIONS... Geometric Proof

• Based on the time honored inclusion-exclusion principle

T< E œ T<ÐE Ñ T<ÖE ∩ E × 3 œ " 3 4

3 4 5T<ÖE ∩ E ∩ E × á Ð "Ñ T< E

3 œ "

3 3 3 4

3 4 5 37

for arbitrary events (not necessarily disjoint) (see, e.g.,E ßá ßE" 8

Feller (1990) ).

• The geometric nature of the proof allows for an efficientalgorithm for evaluation of the cdf of ] useful for itsapplication in Monte Carlo based uncertainty analyses.

2. EARLY TRIANGULAR EXTENSIONS... Geometric Proof

• Let . LetG œ Ö7 7? l ! Ÿ ? Ÿ "×3 be the unit hyper cube in ‘@ œ Ð@ ßá ß @ Ñß @ − Ö!ß "×" 7 3 be a vertex of the unit(corner-point) hyper cube asG7 and define the simplex at vertex W ÐCÑ@ @

W ÐCÑ œ Ö A ? Ÿ Cß ? @ ß 3 œ "ßá ß7×ß@ ? | 3œ"

3 3 3 3

where , A ! A œ "Þ3 33œ"

7• The hyper-volume of the simplex W ÐCÑ@ denoted by isZ ÖW ÐCÑ×@

given by:

Z ÖW ÐCÑ× œ@

C A @ Ñ

† " ÐC A @ Ñ

Ò!ß∞Ñ

2. EARLY TRIANGULAR EXTENSIONS 2D Example...

w1U1 + w2U2 = x1

0 = (0,0) e1 = (1,0)

v = (1,1)e2 = (0,1)

S0(x1)

2. EARLY TRIANGULAR EXTENSIONS... 2D Example

u2w1U1 + w2U2 = x2

x2 –w2w1

x2 –w1w2

0 = (0,0) e1 = (1,0)

v = (1,1)e2 = (0,1)

Se1(x2)

Se2(x2)

2. EARLY TRIANGULAR EXTENSIONS... 3D Example

u3w1u1 + w2u2 + w3u3 = y2

e3=(0,0,1)

e1=(1,0,0)

e2=(0,1,0)

v =(1,1,0)

v =(1,0,1)

v =(0,1,1)

Evaluating for KÐC Ñ 7 œ $Þ#

• Trapezoidal distributions have been advocated in risk analysis problemsby Pouliquen 1970Ð Ñ. Its pdf is given by À

0 ÐBl+ß ,ß -ß .Ñ œ Ð+ß ,ß -ß .Ñ ‚

ß + Ÿ B ,

ß , Ÿ B -

ß - Ÿ B .\

for 1 for

where and .+ , - . Ð+ß ,ß -ß .Ñ œ #Ö- . Ð+ ,Ñ×V "

• Cumulative distribution function (cdf):

J ÐBl+ß ,ß -ß .Ñ œ

Ð+ß ,ß -ß .Ñ ß + Ÿ B ,

Ð+ß ,ß -ß .ÑÐB Ð, +ÑÎ#Ñ , Ÿ B -

" Ð+ß ,ß -ß .Ñ - Ÿ B .

Ð,+Ñ# ,+

.- .B# .-

for , for

, for .

a b c d xa b c d x

( )Xf x

2( )c d a b+ − +

Probability density function of a .Trapezoidal Distribution

OUTLINE

3. TSP DISTRIBUTION... Symmetric Examples

0 0.2 0.4 0.6 0.8 1

Symmetric Two-Sided Power Distribution Examples

3. TSP DISTRIBUTION... Asymmetric Examples

0 0.2 0.4 0.6 0.8 1

Asymmetric Two-Sided Power Distribution Examples

3. TSP DISTRIBUTION... pdf, cdf, qf

• for probability density function (pdf) + Ÿ 7 Ÿ ,ß 8 !:

0 ÐBl+ß7ß ,ß 8Ñ œß + D Ÿ 7

ß 7 Ÿ D ,^

8 D+,+ 7+

8 ,D,+ ,7

• cumulative distribution function (pdf):

J ÐBl+ß7ß ,ß 8Ñ œß + Ÿ D Ÿ 7

" ß 7 Ÿ D Ÿ ,^

7+ D+,+ 7+

,7 ,D,+ ,7

• :quantile function (pdf)ß ; œ Ð7 +ÑÎ, +Ñ

J ÐCl+ß7ß ,ß 8Ñ œ+ CÐ7 +Ñ Ð, +Ñß ! Ÿ C Ÿ ;

, Ð" CÑÐ, 7Ñ Ð, +Ñß ; Ÿ C Ÿ ""^

for for .

For the above expressions reduce to the triangular equivalents.8 œ #

3. TSP DISTRIBUTION... Mean, MP, Variance

• The mean is now of the parameters and a weighted average +ß7 ,Þ

IÒ^Ó œ+ Ð8 "Ñ7 ,

• The mode probability (MP) to the left of the mode equals therelative distance of the mode to the lower bound over the7 +range Ò+ß ,Óß regardless of the value of the power parameter 8Þ

T <Ð^ Ÿ 7Ñ œ œ ;7 +

• The variance is a function of the mode probability above; À

Z +<Ò^Ó œ Ð, +Ñ †8 #Ð8 "Ñ ‚ ;Ð" ;Ñ

Ð8 #ÑÐ8 "Ñ#

# • By substituting above we obtain the equivalent expressions for8 œ "

the uniform distribution.

OUTLINE

3. The Two-Sided Power Distribution (TSP).

4. TSP DISTRIBUTION... Link with Asymmetic Laplace

• Suppose and anda lower percentile upper percentile + ,: <

the mode 7 are specified and we wish to solve for the lowerbound + and where upper bound ,ß + + 7 , ,Þ: <

• Recalling for TSP distributions:the mode probability

T<Ð^ 7l+ß7ß ,ß 8Ñ ´ ; œ Ð7 +ÑÎÐ, +Ñ

one obtains with after some algebraic manipulations: œ " ; À

+ ´ +Ð;l8Ñ œ œ + Þ+ + :Î;

" :Î;

, ´ ,Ð;l8Ñ œ œ , Þ, ,

+ 7 :Î;

" :Î;

< <"<";

• Substituting and back in +Ð;l8Ñ ,Ð;l8Ñ the mode probabilityequation:

; œ Ð7 +ÑÎÐ, +Ñ

one obtains the equation

; œ ;l8Ñ7 +Ð;l8Ñ

,Ð;l8Ñ +Ð;l8Ñ´ 1Ð

1Ð ´Ð7 + Ñ "

Ð, 7Ñ " Ð7 + Ñ " ;l8Ñ

< ::; ";

• The equation above defines a continuous implicit function ;Ð8Ñwith domain ,8 ! where ;Ð8Ñ is the probability mass to the leftof the mode 7 for a given value of the shape/power parameter 8Þ

0 1 2 3 4 5 6 7 8 9 10

( ) 0.244q ∞ ≈(7) 0.237q ≈(2) 0.220q ≈

Graph of for the case the implicit function ;Ð8Ñ : œ !Þ"!ß< œ " : œ !Þ*!ß + œ 'Þ& 7 œ ( , œ "!Þ&: <, and .

0 1 2 3 4 5 6 7 8 9 10n

= ap = 6.5

≈ 5.464

≈ 2.687

0 1 2 3 4 5 6 7 8 9 10n

)} ≈ 20.891

≈ 12.452

br =10.5= 11

The and lower bound function +Ö;Ð8Ñl8× the upper bound function,Ö;Ð8Ñl8× where is the implicit function depicted in the previous;Ð8Ñfigure for the case , and .: œ !Þ"!ß < œ !Þ*!ß + œ 'Þ& 7 œ ( , œ "!Þ&: <

• Note that, as where is a fixed value.;Ð8Ñ Ä ;Ð∞Ñ 8 Ä ∞ß ;Ð∞Ñ

• Note that, as +Ö;Ð8Ñl8× Ä ∞ 8 Ä ∞Þ

• Note that, as ,Ö;Ð8Ñl8× Ä ∞ 8 Ä ∞Þ

• Introducing the relative distance of the mode $ 7 to the lowerquantile +: over ,the range from to the upper quantile + ,: <

$ œ7 +

(and thus ,} it can be shown that $ − Ð!ß "Ñ ;Ð∞Ñ is the uniquesolution of the equation

;Ð∞Ñ ;Ð∞Ñ " ;Ð∞Ñ " ;Ð∞ÑP91 œ P91

: " " <$ $ .

• In addition, the TSP distribution with parameters +Ö;Ð8Ñl8× 7, ,,Ö;Ð8Ñl8× 8 and converges as (in distribution) to the pdf8 Ä ∞

0 ÐBl+ ß7ß , Ñ œ;Ð∞Ñ IB: Ð7 BÑ B Ÿ 7

Ö" ;Ð∞Ñ× IB: ÐB 7Ñ B 7\ : <

where the coefficients and areT U

T Uœ œP91 P91

7 + , 7

;Ð∞Ñ ";Ð∞Ñ: "<

: < and .

• Pdf is a 0 ÐBl+ ß7ß , Ñ\ : < reparameterized asymmetric Laplacedistribution with lower and upper quantile and and mode+ ,: <

7.Kotz, S. and Van Dorp, J.R. (2005). A Link between Two-Sided Power and Asymmetric Laplace Distributions: with

Applications to Mean and Variance Approximations, Statistics and Probability Letters

OUTLINE

5. MOMENT RATIO DIAGRAM... Comparison

• Moment ratio plots provide a of useful visual assessment theskewness indicative of the and the kurtosis. Their coverage is flexibility a particular family of asymmetric distributions.of

• The classical form of the diagram shows the values of the ratio

" ". .

. ." #

# $ # %

$ # # #$$ ##

œ œ ß œ œI ÒÐ\ IÒ\ÓÑ Ó IÒÐ\ IÒ\ÓÑ Ó

I ÒÐ\ IÒ\ÓÑ Ó I ÒÐ\ IÒ\ÓÑ Ó,

where and ."" is plotted on the abscissa "# on the ordinate

• This diagram possesses e that indicatinga disadvantag the sign of.$ Ðleft skewness or right skewness) disappears.

• A moment ratio diagram that retains this information is a plot with"" on the abscissa and , with the convention"# on the ordinatethat (See, e.g., Kotz and Johnson (1985))." ." $ retains the sign of

5. MOMENT RATIO DIAGRAM... Comparison

• Values for and for (with support" "" # Standard TSP distributionsÒ!ß "Óß 8Ñmode and shape/power parameter can be calculated using)the general expression for their moments around the origin.w 55 œ IÒ\ Óß 5 œ "ßá ß %ß given by

.w5 œ IÒ\ Ó œ

8 5 8Ð "Ñ

8 5 5 3 8 35

5) ) .

and their relationship with central moments .5 œ IÒÐ\ Ñ Ó."5 ,

5 œ #ß $ %, given by. . .

. . . . .

. . . . . . .

#w w# "

$w w w w$ # " "

%w w w w w w% $ " # " "

œ $ #

œ % ' $

(see, e.g., Stuart and Ord (1994)).

5. MOMENT RATIO DIAGRAM... Beta Distribution

0123456789

-3 -1 1 31β

infeasible infeasible

LEGEND

Asymmetric Laplace

(0,1.8) Uniform

x(0,3) Gaussian

(2,9) Right Exponential

(-2,9) Left Exponential

Unimodal Beta

U-Shaped Beta

LEGEND

Reflected Power

Bernouilli

J-Shaped Beta

5. MOMENT RATIO DIAGRAM... TSP Distribution

0123456789

-3 -1 1 31β

infeasible infeasible

Unimodal TSP

U-Shaped TSP

LEGEND LEGEND

Asymmetric Laplace

(0,1.8) Uniform

x(0,3) Gaussian

(2,9) Right Exponential

(-2,9) Left Exponential

Reflected Power

Bernouilli

OUTLINE

6. MAXIMUM LIKELIHOOD... STSP Distribution

• Standard TSP (STSP) distribution pdf with support :Ò!ß "Ó

0ÐBl ß 8Ñ œ

8 ! Ÿ B Ÿ

8 Ÿ B Ÿ "

, for .

• Likelihood function for order statistics is byÐ\ ßá ß\ ÑÐ"Ñ Ð=Ñ

definition:

PÐ à 8 œ 8 ÖLÐ à Ñ× ß

LÐ à Ñ œ

\ Ð" \ Ñ

Ð" Ñ

, ) where) )

3œ" 3œ<"

Ð3Ñ Ð3Ñ

and with \ Ÿ Ÿ \ \ ´ !ß \ ´ "ÞÐ<Ñ Ð<"Ñ Ð!Ñ Ð="Ñ)

1.01.7

2.33.0

3.74.3

Graph of the likelihood , )PÐ à 8\ ) for the dataÐ\ ßá ß\ Ñ œ Ð!Þ"!ß !Þ#&ß !Þ$!ß !Þ%!ß !Þ%&ß !Þ'!ß !Þ(&ß !Þ)!ÑÐ"Ñ Ð)Ñ

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(8)

1.10E-02

H( X ; θ)

Graph of ; LÐ\ Ñ) for the same data (Maxima over the setsÒ\ ß\ ÓÐ<Ñ Ð<"Ñ are attained here solely at the order statistics).

Theorem: Let ( ) be an i.i.d. sample from a \ œ \ ßá ß\ WXWTÐ ß 8Ñ" = )distribution. The ML estimators of and maximizing the likelihood ) 8 PÐ à 8\ , ))over the parameter domain and are:! Ÿ Ÿ " 8 ")

)s œ \

8 œ Q+B ß "s

Ð<Ñs

=P91QÐ<Ñs

< œ +<17+B QÐ<Ñs< − Ö"ßá ß =×

QÐ<Ñ œ\ " \

\ " \ 3œ" 3œ<"

<" =Ð3Ñ Ð3Ñ

Ð<Ñ Ð<Ñ.

• when maximizing the likelihoodModifications have to be madePÐ à ß 8Ñ ! Ÿ Ÿ " Þ\ ) ) over the parameter domain and ! 8 Ÿ "

6. MAXIMUM LIKELIHOOD... Interest Rate Example

30-YEAR US MORTGAGE INTEREST RATES

• Let the be denoted by ; interest rate after year 5 35 one of thesimplest financial engineering models for the randombehavior of the interest rates is the multiplicative model:

3 œ 35" 5 5% ,

where are random%5 identical independent distributed (i.i.d.)variables (see, e.g., Leunberger, D.G. (1998). Investment Science.New York, NY: Oxford University Press).

• Taking logarithms on LHS and RHS of yields:

P8Ð3 Ñ œ P8Ð3 Ñ P8Ð Ñ Í P8Ð3 Ñ P8Ð3 Ñ œ P8Ð Ñ5" 5 5 5" 5 5% % ,

where as before the are i.i.d.P8Ð Ñ%5

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Empirical PDF Normal PDF AS Laplace

Empirical pdf of two-month lag log differences of 30-yearconventional mortgage interest rates together with the ML fitted

Gaussian asymmetric Laplace distributions. and

-0.3 -0.1 0.1 0.3

Empirical PDF Beta PDF TSP PDF

Empirical pdf of two-month lag log differences of 30-year

conventional mortgage interest rates together with ML fittedbeta TSP distributions and .

SUMMARY OF MORE FORMAL FIT ANALYSIS

Normal AS Laplace Beta TSPChi-Squared Statistic 18.99 13.27 20.56 12.01Degrees of Freedom 12 11 12 12P-value 0.089 0.276 0.057 0.445K-S Statistic 8.90% 6.44% 9.21% 4.66%SS 0.276 0.091 0.320 0.062Log-Likelihood 307.42 319.16 304.71 318.36

Except for the theLog-Likelihood StatisticTSP distribution outperforms the other four distributions.

OUTLINE

7. GENERAL FRAMEWORK... Two-Sided Distributions

• can be viewed as a particular case of theSTSP distributionsgeneral Standard Two-Sided (STS) continuous family with supportÒ!ß "Ó given by the density

1ÖBl ß :Ð † l Ñ× œ:Ð l Ñ ! B Ÿ

:Ð l Ñ B "ß) G

, for , for

where is an appropriately selected :Ð † l ÑG continuous pdfdefined on with parameter(s) Ò!ß "Ó G, which may in principlebe vector-valued.

• The density will be referred to as the :Ð † l ÑG generatingdensity of the resulting STS family of distributions and theparameter reflection parameter) is termed the (an alternativedesignation could be the hinge thresholdor parameter).

• Triangular distribution has generating density:

:ÐCÑ œ #C ! Ÿ C Ÿ "ß,

• Standard Two-Sider Power distribution has generating density:

:ÐCl8Ñ œ 8C ß ! Ÿ C Ÿ "ß 8 !ß8"

• Two-Sided Slope distribution has generating density:

:ÐCl Ñ œ #Ð" ÑCß ! Ÿ C Ÿ "ß ! Ÿ Ÿ #ßα α α α

• Two-Sided Ogive distribution has generating density:

:ÐCl8Ñ œ 8C Ö C ×ß ! Ÿ C Ÿ "ß 8 "Î#Þ%8 # #8 #

$8 " $8 "8" 8

Triangular Distribution

Two-Sided Power Distribution

0.00 0.50 1.00

Two-Sided Ogive Distribution

0.00 0.50 1.00

Two-Sided Slope Distribution

0.00 0.50 1.00TWo-

• with\ µ 1Ö † l ß :Ð † l Ñ× two-sided distributed) G one obtains for generating density :Ð † l ÑG the cdf of \

KÖBl ß T Ð † Ñ× œTÐ l Ñ ! B

" Ð" ÑT Ð l Ñ Ÿ B "ß) G

) G ) B

, for , for

where is the cdf of the generating density .TÐ † l Ñ :Ð † l ÑG G

• is thatAn important and revealing property of the STS family

KÐ l ß Ñ œ TÐ"l Ñ œ Þ) ) G ) G )

• If with] µ :Ð † l Ñ ß 5 œ "ß #ß $ßáG moments IÒ] Ó5

IÒ\ l ß Ó œ Ð "Ñ Ð" Ñ5

35 5" 3 3"

) G ) )IÒ] l Ó IÒ] l Ó5 3G G 0

IÒ\l ß Ó œ Ð# "Ñ Ð" Ñ) G ) )IÒ] l ÓG

Z +<Ð\l ß Ñ œ Ö" $ $ × Ð" ÑÖ "×) G ) ) ) )# #Z +<Ð] l Ñ IÒ] l ÓG G .

OUTLINE

8. GT DISTRIBUTIONS... Density Function

Having generalized the Triangular distribution can wegeneralize the Trapezoidal distribution in a similar manner?

a b c d xa b c d x

( )Xf x

2( )c d a b+ − +

Probability density function of a Trapezoidal Distribution.

8. GT DISTRIBUTIONS... Density Function

• Pdf of Generalized Trapezoidal Distributions:

0 ÐBl Ñ œ

+ Ÿ B ,

Ð "Ñ " ,

# 78 B+# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-Ñ7 ,+

#78 -B# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-Ñ7 -,

+ß ,ß -ß .ß7ß 8ßα

αα α

α α α Ÿ B -

- Ÿ B .#78 .B# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-Ñ7 .-

α α where and .7 !ß 8 !ß ! + , - .α

• The pdf above is constructed using three probabilities 3œ"

31 œ "ß

13 ! and a mixture of three densities

0 ÐBÑ œ 0 ÐBÑ\ 3 \

8. GT DISTRIBUTIONS... Mixture Elements

• Setting equal tothe mixture density elements

0 ÐBl+ß ,ß7Ñ œ + Ÿ B , 7 !ß7 B +

, + , +

0 ÐBl,ß -ß Ñ œ Ð" ÑB - , ß , Ÿ B Ÿ -ß !ß#

Ð "Ñ - ,

0 ÐBl-ß .ß 8Ñ œ ß - Ÿ B .8 . B

. - . -

( )α α α α

ß 8 !

the problem of solving for the mixture probabilitiesß1 1 11 2 3ß ß !, such that the density is continuous over the0 ÐBÑ\

support Ò+ß .Ó is not trivial. See:

J.R. van Dorp and S. Kotz (2003). “Generalized Trapezoidal Distributions”. , Vol. 58,MetrikaIssue 1, pp. 85-97. R-package for Generalized Trapezoidal distribution is available onmy faculty page, Courtesy of Jeremy Thoms Hetzel (April 25, 2011).

8. GT DISTRIBUTIONS... Mixture Probabilities

Proposition: The generelized trapezoidal probability density function follows fromexpressions for utilizing mixture0 ÐBl+ß ,ß7Ñ 0 ÐBl,ß -ß Ñ 0 ÐBl-ß .ß 8Ñ\ \ \" # $

ß ßα probabilities

"# Ð,+Ñ8

# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-Ñ7

#Ð "ÑÐ-,Ñ78

# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-Ñ7

$#Ð.-Ñ

# Ð,+Ñ8Ð "ÑÐ-,Ñ78#Ð.-

αα α

α αm

where , and the probability density+ , - . 8 !ß 8 !ß !" # αfunction given by is continuous.Ð$Ñ

• Note that the numerator of the first stage probability 1" has as amultiplier .the power parameter of the third stage

• Note that the numerator of the third stage probability 1$ has as amultiplier the power parameter of .the first stage

• Note that the numerator of the second stage probability 1# has bothpower parameters of the first and third stage as multipliers.

8. GT DISTRIBUTIONS... Unimodal Shapes

8. GT DISTRIBUTIONS... J-Shaped and U-Shaped

While fun mathematical exercises, these new distribution models areonly usefull if they can find application better than other models.

Data From: Wagner JC, DeHart MD (2000). Review of Axial Burnup Distributions for Burnup CreditCalculations. , ORNL/TM-1999/246, Oak Ridge, Tennessee.Oak Ridge National Laboratory

OUTLINE

9. GTSP DISTRIBUTIONS... Probability Density Function

• Setting in the Generalized Trapezoidal+ œ !ß , œ - œ ß œ ") αdistribution, we arrrive at the pdf of a Generalized Two-Sided Power(GTSP) distribution that allows for seperate power parameterswithin each of its branches 8ß7 ! ! Ÿ Ÿ ", :)

0 ÐBl Ñ œ ‚78

Ð Ñ7 8

ß ! B

ß Ÿ B "Þ\

8"@) )

• is now also a function of the power parametersThe mode probability

T<Ð\ Ÿ Ñ œ))

Ð" Ñ7 8

• are more involved. Mean, Variance and other moment expressions

IÒ\Ó œ 78Ð8 "Ñ Ð7 "ÑÐ" ÑÐ8 Ñ

Ð7 "ÑÐ8 "ÑÖÐ" Ñ7 8×

9. GTSP DISTRIBUTIONS... Unimodal Shapes

8. GTSP DISTRIBUTIONS... Unimodal Shapes

9. GTSP DISTRIBUTIONS... Moment Ratio

GTSP Distribution now shares the J-Shaped coverage with the Betadistribution that the TSP distribution did not cover.

BEYOND BETA

Publication Date: December 15, 2004

Co-authored by Samuel Kotz and Johan Rene van Dorp, both at The George Washington University, Department of Engineering Management and Systems Engineering, Washington, D.C., USA

Samuel Kotz is the co-author with Dr. N.L. Johnson of the recently published Leading Personalities in Statistical Sciences, the classic five-volume Distributions in Statistics (now in its second edition), amongst other books in Statistics and Probability.

J. Rene van Dorp is active in applications of statistical methodology such as probabilistic risk analysis, reliability and simulation, and is an author of over 20 scientific publications.

http://www.worldscientific.com/

Statistical distributions are fundamental to statistical science and are a prime indispensable tool for its applications. The BEYOND BETA monograph is the first to examine an important but somewhat neglected area in this field - univariate continuous distributions on a bounded domain, excluding the beta distribution. It provides an elementary but thorough discussion of "novel" contributions developed in recent years and some of their applications. It contains a comprehensive chapter on the triangular distribution as well as a chapter on earlier extensions of this distribution not emphasized in existing statistical literature. Special attention is given to estimation, in particular, non-standard maximum likelihood procedures. In the presentation, we shall concentrate on the properties and parameter estimation of the Two-Sided Power (TSP) distributions and their link to Generalized Trapezoidal and the Asymmetric Laplace distributions. The different shapes of TSP pdf’s are presented on the book cover and mimic those of the beta distributions. The general framework for Two-Sided distribution will be presented as well.

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