A Quasi Aradhana Distribution with Properties and …A Quasi Aradhana Distribution with Properties...

International Journal of Statistics and Systems

ISSN 0973-2675 Volume 13, Number 1 (2018), pp. 61-80

© Research India Publications

http://www.ripublication.com

A Quasi Aradhana Distribution with Properties and

Application

Rama Shanker & Kamlesh Kumar Shukla

Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, India.

Ravi Shanker

Department of Mathematics, G.L.A. College, N.P University, Daltonganj, Jharkhand, India.

Abstract

A two-parameter quasi Aradhana distribution (QAD) which includes one

parameter Aradhana distribution introduced by Shanker (2016) as special case

has been proposed and investigated. Its moments based measures, reliability

properties, stochastic ordering, mean deviations, Bonferroni and Lorenz

curves, order statistics, Renyi entropy measure and stress-strength reliability

have been discussed. For estimating its parameters, the method of moments

and the method of maximum likelihood have also been discussed. Goodness of

fit of QAD has been discussed with a real lifetime data from engineering and

compared with other one parameter and two-parameter lifetime distributions.

Keywords: Aradhana distribution, Moments, Reliability measures, ,

Stochastic ordering, Mean Deviation, Bonferroni and Lorenz curves, Order

Statistics, Renyi entropy measure Stress-strength reliability, Estimation of

parameters, goodness of fit

1. INTRODUCTION

Shanker (2016) introduced a one parameter lifetime distribution named Aradhana

distribution for modeling lifetime data from biomedical sciences and engineering

defined by its probability density function (pdf) and cumulative distribution function

(cdf)

3

2

1 2; 1 ; 0, 0

2 2

xf x x e x

. (1.1)

mailto:[email protected];%[email protected]

mailto:[email protected]

62 Rama Shanker, Kamlesh Kumar Shukla, Ravi Shanker

1 2

2 2; 1 1 ; 0, 0

2 2

xx xF x e x

(1.2)

It has been shown by Shanker (2016) that in majority of real lifetime datasets it gives

better fit than exponential, Lindley (1958), Akash and Shanker distributions

introduced by Shanker (2015 a, 2015 b). The Aradhana distribution is a three-

component mixture of exponential , a gamma 2, and a gamma

3, distributions with their mixing proportions2

2 2 2

,

2

2

2 2

and

2

2

2 2 respectively.

The first four moments about origin of Ardadhana distribution (1.1) obtained by

Shanker (2016) are

2

1 2

4 6

2 2

,

2

2 2 2

2 6 12

2 2

,

2

3 3 2

6 8 20

2 2

,

2

4 4 2

24 10 30

2 2

The central moments of Aradhana distribution obtained by Shanker (2016) are given

by

4 3 2

2 22 2

8 24 24 12

2 2

6 5 4 3 2

3 33 2

2 12 54 100 108 72 24

2 2

8 7 6 5 4 3 2

4 44 2

3 3 48 304 944 1816 2304 1920 960 240

2 2

Important mathematical and statistical properties of Aradhana distribution including

shapes for varying values of parameter, hazard rate function, mean residual life

function, stochastic ordering, mean deviations, distribution of order Statistics,

Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability

have been discussed by Shanker (2016). Shanker (2016) has also discussed the

A Quasi Aradhana Distribution with Properties and Application 63

estimation of parameter and applications of Aradhana distribution for modeling

lifetime data. Shanker (2017) has obtained a Poisson mixture of Aradhana

distribution and named, ‘Poisson-Aradhana distribution’, discussed its various

statistical properties and estimation of parameter using both the method of moments

and the maximum likelihood estimation as well as applications to model count data

from different fields of knowledge.

In this paper, a two - parameter quasi Aradhana distribution (QAD), of which one

parameter Aradhana distribution introduced by Shanker (2016) is a particular case,

has been proposed. Its raw moments and central moments have been obtained and

coefficients of variation, skewness, kurtosis and index of dispersion have been

discussed. Its hazard rate function, mean residual life function, stochastic ordering,

mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy

measure and stress-strength reliability have been discussed. Both the method of

moments and the method of maximum likelihood have been discussed for estimating

the parameters of QAD. The goodness of fit of the QAD has been presented with a

real lifetime dataset and the fit has been compared with other well known one

parameter and two-parameter lifetime distributions.

2. A QUASI ARADHANA DISTRIBUTION

A two - parameter quasi Aradhana distribution (QAD) having parameters and is

defined by its pdf

2 2

2 2; , ; 0, 0, 2 2 0.

2 2

xf x x e x

(2.1)

It can be easily verified that at , (2.1) reduces to the Aradhana distribution (1.1)

and at 0 , it reduces to gamma 3, distribution. It can be easily verified that

QAD (2.1), like Aradhana distribution, is also a convex combination of exponential

gamma 2, and gamma 3, distributions. We have

2 1 1 2 2 1 2 3; , ; ; ,2 1 ; ,3f x p g x p g x p p g x (2.2)

where 2

1 22 2

2,

2 2 2 2p p

,

1 ; ; 0, 0xg x e x

22 1

2 ; , 2 ; 0, 02

xg x e x x

33 1

3 ; ,3 ; 0, 03

xg x e x x

.


The corresponding cdf of QAD (2.1) can be obtained as

2

2 2

2 0,

2; , 1 1 ; 0, 02 2

2 2

xx xF x xe

(2.3)

Behaviors of the pdf and the cdf of QAD for varying values of parameters are

presented in figures1 and 2.It is clear from the graphs of the pdf of QAD that it takes

different shapes for varying values of parameters and ranging from positively

skewed to monotonically decreasing and hence QAD can be used for modeling real

lifetime data of various nature.

Figure.1. Behaviors of the pdf of QAD for varying values of parameters and


Figure. 2. Behaviors of the cdf of QAD for varying values of parameters and

3. STATISTICAL CONSTANTS

The r th moment about origin of QAD can be obtained as

2

2

! 2 1 1 2; 1,2,3,...

2 2r r

r r r rr

(3.1)

Thus the first four moments about origin of QAD are given by

2

1 2

4 6

2 2

,

2

2 2 2

2 6 12

2 2

,

2

3 3 2

6 8 20

2 2

,

2

4 4 2

24 10 30

2 2


Using the relationship between central moments and moments about mean, the central

moments of QAD are thus obtained as

4 3 2

2 22 2

8 24 24 12

2 2

6 5 4 3 2

3 33 2

2 12 54 100 108 72 24

2 2

8 7 6 5 4 3 2

4 44 2

3 3 48 304 944 1816 2304 1920 960 240

2 2

The coefficient of variation .C V , coefficient of skewness 1 , coefficient of

kurtosis 2 and index of dispersion of QAD are obtained as

4 3 2

2

1

8 24 24 12.

4 6C V

6 5 4 3 2

31 3 23/2 4 3 2

2

2 12 54 100 108 72 24

8 24 24 12

8 7 6 5 4 3 2

42 22 4 3 2

2

3 3 48 304 944 1816 2304 1920 960 240

8 24 24 12

2 4 3 2

2 2

1

8 24 24 12

2 2 4 6

.

One of the interesting features of QAD is that the expressions for C.V, 1 and

2 depends only on the parameter and whereas depends on the parameter and

both. The graphs of C.V, 1 , 2 , and of QAD have been drawn for varying

values of the parameters and presented in figure 3. It is obvious from the graph of

coefficient of variation that it is slowly increasing for increased values of the


parameter and slowly it reaches towards the constant value of 1. The coefficient of

skewness starts increasing for increased value of the parameter but once the value

of is greater than 2, it starts decreasing. The coefficient of kurtosis is always

increasing for increased value of the parameter . If the value of is less than 2 and

increases, index of dispersion is monotonically decreasing and its shape is concave

upward. If the value of is less than 1 and 3 , the index of dispersion is

increasing and as 3 , it starts decreasing and its shape is concave downward.

Figure.3. Behaviors of C.V, 1 , 2 , and of QAD for varying values of

parameters and


4. STOCHASTIC ORDERING

Stochastic ordering of positive continuous random variables is an important tool for

judging their comparative behavior. A random variable X is said to be smaller than a

random variable Y in the

(i) stochastic order stX Y if X YF x F x for all x

(ii) hazard rate order hrX Y if X Yh x h x for all x

(iii) mean residual life order mrlX Y if X Ym x m x for all x

(iv) likelihood ratio order lrX Y if

X

Y

f xf x

decreases in x .

The following results due to Shaked and Shanthikumar (1994) are well known for

establishing stochastic ordering of distributions

lr hr mrlX Y X Y X Y

stX Y

The QAD is ordered with respect to the strongest ‘likelihood ratio ordering’ as

established in the following theorem:

Theorem: Let X QAD 1 1, and Y QAD 2 2, . If 1 2 1 2and or

1 2 1 2and , then lrX Y and hence hrX Y , mrlX Y and stX Y .

Proof: We have

1 1 1 2

2 2

22

1 2 2; , 1 1

; , 22 22 1 1

2 2

2 2

X

Y

f x xf x

x ex

; 0x

Now

1 1

2 2

2

1 2 2; , 1 11 2; , 2

2 22 1 1

2 2ln ln 2ln

2 2

X

Y

f xf x

x xx

.

This gives

1 1

2 2

; , 1 2 2 1

1 2; ,

1 1 2 2

2ln X

Y

f xf x

ddx x x

.

Thus if 1 2 1 2and or 1 2 1 2and ,

1 1

2 2

; ,

; ,ln 0X

Y

f xf x

ddx

. This means that

lrX Y and hence hrX Y , mrlX Y and stX Y . This shows flexibility of QAD

over Aradhana distribution introduced by Shanker (2016).


5. MEAN DEVIATIONS

The amount of variation in a population is generally measured to some extent by the

totality of deviations usually either from the mean or the median. These are known as

the mean deviation about the mean and the mean deviation about the median and are

defined as

1

0

X x f x dx

and 2

0

X x M f x dx

, respectively, where

E X and Median M X . The measures 1 X and 2 X can be

calculated using the following simplified relationships

1

0

X x f x dx x f x dx

0

1F x f x dx F x f x dx

2 2 2F x f x dx

0

2 2F x f x dx

(5.1)

and

2

0

M

M

X M x f x dx x M f x dx

0

1

M

M

M F M x f x dx M F M x f x dx

2M

x f x dx

0

2

M

x f x dx (5.2)

Using pdf (2.1) and the mean of QAD, we get

3 3 2 2 2 2

2 2

0

2 3 4 6 4 6; ,

2 2

ex f x dx

(5.3)


3 3 2 2 2 2

2 2

0

2 3 4 6 4 6; ,

2 2

MM M M M ex f x dx

(5.4)

Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about

mean, 1 X and the mean deviation about median, 2 X of QAD are obtained as

2 2 2

1 2

2 2 2 4 6

2 2

eX

(5.5)

3 3 2 2 2 2

2 2

2 2 3 4 6 4 6

2 2

MM M M eX

(5.6)

6. BONFERRONI AND LORENZ CURVES

The Bonferroni and Lorenz curves ( Bonferroni, 1930) and Bonferroni and Gini

indices have applications not only in economics to study the variation in income and

poverty, but also in other fields like reliability, demography, insurance and medicine.

The Bonferroni and Lorenz curves are defined as

0 0

1 1 1q

q q

B p x f x dx x f x dx x f x dx x f x dxp p p

(6.1)

and 0 0

1 1 1q

q q

L p x f x dx x f x dx x f x dx x f x dx

(6.2)

respectively or equivalently

1

0

1p

B p F x dxp

(6.3)

and 1

0

1p

L p F x dx

(6.4)

respectively, where E X and 1q F p .


The Bonferroni and Gini indices are thus defined as

1

0

1B B p dp (6.5)

and 1

0

1 2G L p dp (6.6)

respectively.

Using pdf of QAD (2.1), we get

3 3 2 2 2 2

2 2

2 3 4 6 4 6; ,

2 2

q

q

q q q ex f x dx

(6.7)

Now using equation (6.7) in (6.1) and (6.2), we get

3 3 2 2 2 2

2

2 3 4 6 4 611

4 6

qq q q eB p

p

(6.8)

and 3 3 2 2 2 2

2

2 3 4 6 4 61

4 6

qq q q eL p

(6.9)

Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini

indices of QAD are thus obtained as

3 3 2 2 2 2

2

2 3 4 6 4 61

4 6

qq q q eB

(6.10)

3 3 2 2 2 2

2

2 2 3 4 6 4 61

4 6

qq q q eG

(6.11)

7. RELIABILITY MEASURES

There are three important reliability measures namely hazard rate function, mean

residual life function and stress-strength reliability and each will be discussed in this

section.

7.1. Hazard Rate Function and Mean Residual Life Function

Let X be a continuous random variable with pdf f x and cdf F x . The hazard rate


function (also known as the failure rate function) and the mean residual life function

of X are respectively defined as

0lim

1x

P X x x X x f xh x

x F x

(7.1.1)

and

1

11 x

m x E X x X x F t dtF x

(7.1.2)

The corresponding hazard rate function, h x and the mean residual life

function, m x of QAD (2.1) are obtained as

2

22 2 2 2

xh x

x x

(7.1.3)

and

2

2

12 2 2 2

2 2 2 2

tx

x

m x t t e dtx x e

2 2 2

2

2 2 4 6

2 2 2 2

x x

x x

(7.1.4)

It can be easily verified that 2

20 0

2 2h f

and

2

12

4 60

2 2m

. The graphs of h x and m x of QAD for varying

values of parameters are shown in figure 4. The graphs of h x is monotonically

increasing whereas the graphs of m x is monotonically decreasing.


Figure. 4. Graphs of h x and m x of QAD for varying values of

parameters and

7.2. STRESS-STRENGTH RELIABILITY

The stress- strength reliability describes the life of a component which has random

strength X that is subjected to a random stress Y . When the stress applied to it

exceeds the strength, the component fails instantly and the component will function

satisfactorily till X Y . Therefore, R P Y X is a measure of component

reliability and in statistics literature it is known as stress-strength parameter. It has

wide applications in biomedical sciences and engineering.

Let X and Y be independent strength and stress random variables having QAD (2.1)


with parameter 1 1, and 2 2, respectively. Then the stress-strength reliability

R can be obtained as

0

| XR P Y X P Y X X x f x dx

2 1 1 2 2 2

0

; , ; ,f x F x dx

2 2 2 2

1 2 2 1 2 1 1 2 1 2

22 2 2 2

2 2 1 1 2 1 2 1 2 1 2

1 32 2

1 1 2 2 2 1 2 1 2

42 2

1 2 2 1 2

52 2

1 1 2 2 1 2

24 6 2 1 2

2 2 2 4 1

2 2 2 2 1

2 21

2 2 2 2

.

It can be easily verified that above expression reduces to the corresponding expression

for Aradhana distribution (1.1) at 1 1 and 2 2 .

8. ORDER STATISTICS AND RENYI ENTROPY MEASURE

8.1. Order Statistics

Let 1 2, ,..., nX X X be a random sample of size n from QAD (2.1). Let

1 2... nX X X denote the corresponding order statistics. The pdf and the cdf of

the k th order statistic, say kY X are given by

1!1

1 ! !

n kkY

nf y F y F y f yk n k

1

0

!1

1 ! !

n kl k l

l

n kn F y f ylk n k

and

1n n jj

Yj k

nF y F y F y

j

0

1n jn

l j l

j k l

n n jF y

j l

,

respectively, for 1,2,3,...,k n .


Thus, the pdf and the cdf of k th order statistic of QAD (2.1) are obtained as

2

20

12

2

!1

2 2 1 ! !

2 2 2 21

2 2

x n kl

Yl

k lx

n kn x ef y

lk n k

x x e

and

2

20

2 2 2 21 1

2 2

j lxn jn

lY

j k l

x x en n jF y

j l

8.2. Renyi Entropy Measure

An entropy of a random variable X is a measure of the variation of uncertainty. A

popular entropy measure is Renyi entropy (1961). If X is a continuous random

variable having probability density function .f , then Renyi entropy is defined as

1log

1RT f x dx

where 0 and 1 .

Thus, the Renyi entropy for QAD (2.1) can be obtained as

2

20

1log

1 2 2

xRT x e dx

2

2

20

1log 1

1 2 2

xx e dx

2

200

21log

1 2 2

jx

j

x e dxj

21 1

20 0

21log

1 2 2

j jx j

je x dx

j


2

12

0

2 11log

1 2 2

j j

jj

jj

12

12

0

2 11log

1 2 2

jj

j

jj

9. ESTIMATION OF PARAMETERS

9.1. Method of Moments Estimates (MOME) of parameters

Since the QAD (2.1) has two parameters to be estimated, using the first two moments

about origin, we get

2 2

2

2 22

1

2 6 12 2 2Say)

4 6k

This gives a fourth degree polynomial equation in as

4 3 22 8 2 4 7 13 24 2 3 12 3 4 0k k k k k (9.1.1)

Replacing the first and second population moments about origin 1 and 2 by their

respective sample moments, an estimate of k can be obtained and substituting the

value of k in equation (9.1.1), the MOME of parameter can be obtained.

Substituting this estimate of in the expression for the mean of QAD (2.1), MOME

of parameter can be obtained as

2

2

4 6

2 2 x

(9.1.2)

9.2. Maximum Likelihood Estimates (MLE) of parameters

Let 1 2, , ... , nx x x be a random sample of size n from QAD (2.1)). The likelihood

function, L of QAD is given by

2

212 2

n nn x

ii

L x e

The natural log likelihood function is thus obtained as

21

ln ln 2 ln2 2

n

ii

L n x n x


The maximum likelihood estimates (MLEs) ˆ ˆ, of , are then the solutions

of the following non-linear equations

1

ln2 0

ni

i i

xL n n xx

21

2 1ln 12 0

2 2

n

i i

nLx

,

where x is the sample mean.

These two natural log likelihood equations do not seem to be solved directly because

they are not in closed form. However, the Fisher’s scoring method can be applied to

solve these equations. We have

22

22 21

ln2

ni

i i

xL nx

2

2 22 21

2 2ln2

2 2

ni

i i

n xLx

2 2

21

ln ln2

ni

i i

xL Lx

The iterative solution of the following equations will be the MLEs ˆ ˆ, of

parameters , of QAD

00

00

2 2

2

0

2 2

0

ˆ2ˆ

ˆˆ

ln ln lnˆ

lnˆln ln

L L L

LL L

where 0 and 0 are the initial values of and , respectively given by the MOME

of the parameters of QAD. These equations are solved iteratively till sufficiently

close values of ̂ and ̂ are obtained.

10. AN ILLUSTRATIVE EXAMPLE

The goodness of fit of QAD has been discussed with a real lifetime dataset from

engineering and the fit has been compared with one parameter and two-parameter

lifetime distributions. The following dataset represents the failure times (in minutes)


for a sample of 15 electronic components in an accelerated life test, Lawless (2003,

pp. 204)

1.4 5.1 6.3 10.8 12.1 18.5 19.7 22.2 23.0 30.6 37.3 46.3

53.9 59.8 66.2

The pdf and the cdf of the fitted distributions have been given in table 1. Recall that

the quasi Shanker distribution (QSD) has been introduced by Shanker and Shukla

(2017) and the exponentiated exponential distribution (EED) has been introduced by

Gupta and Kundu (1999). The Lindley distribution has been introduced by Lindley

(1958) and its detailed study has been done by Ghitany et al (2008).

In order to compare the considered distributions, values of 2ln L , AIC(Akaike

Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) and p-

value for the dataset have been computed and presented in table 2. The AIC and K-S

Statistic has been computed using the following formulae:

2ln 2AIC L k and 0- Sup nx

K S F x F x , where k number of parameters,

n sample size, nF x is the empirical distribution function and 0F x is the

theoretical cumulative distribution function. The best distribution corresponds to the

lower values of 2ln L , AIC, and K-S statistic.

Table 1: The pdf and the cdf of the fitted distributions

Models p.d.f. c.d.f.

QSD

3

3

2

; ,2

x

f x

x x e

2 2

3

2; , 1 1

2

x

xx

F x e

Weibull 1; , xf x x e ; , 1 xF x e

Gamma

1; , xf x x e

,; , 1

xF x

Lognormal

21 log

21; ,

2

x

f x ex

log

; ,xF x

EED 1

; , 1 x xf x e e

; , 1 xF x e

Lindley

2

; 11

xf x x e

1

; 11

xxF x e

Exponential ; xf x e ; 1 xF x e


Table 2. MLE’s, 2ln L , Standard Error (S.E), AIC, K-S Statistics and p-value of the

fitted distributions of dataset.

Distributions ML Estimates S.E 2ln L AIC K-S p-value

QAD ˆ 0.07548 0.02308 128.07 132.07 0.095 0.997

ˆ 1.23504 1.17209

QSD ˆ 0.07389 0.04131 129.37 133.37 0.121 0.961

ˆ 0.00147 0.04401

Gamma ˆ 0.05236 0.02067 128.37 132.37 0.102 0.992

ˆ 1.44219 0.47771

Weibull ˆ 0.01190 0.01124 128.04 132.04 0.098 0.995

ˆ 1.30586 0.24925

Lognormal ˆ 2.93059 0.26472 131.23 135.23 0.161 0.951

ˆ 1.02527 0.18718

EED ˆ 0.04529 0.01372 128.47 132.47 0.108 0.986

ˆ 1.44347 0.51301

Aradhana ˆ 0.10511 0.01567 132.30 134.30 0.172 0.701

Lindley ˆ 0.07022 0.01283 128.81 130.81 0.110 0.983

Exponential ˆ 0.03631 0.00937 129.47 131.47 0.156 0.807

It can be easily observed from table 2 that the QAD gives better fit than one parameter

exponential, Lindley and Aradhana distributions and two-parameter QSD, Gamma,

Weibull lognormal and EED and thus it can be considered as an important two-

parameter lifetime distribution for modeling lifetime dataset from engineering over

these distributions.

11. CONCLUDING REMARKS

In the present paper a two - parameter quasi Aradhana distribution (QAD) which

includes one parameter Aradhana distribution introduced by Shanker (2016) as a

particular case has been proposed and studied. Its statistical properties including

moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate

function, mean residual life function, stochastic ordering, mean deviations, order

statistics, Bonferroni and Lorenz curves, and stress-strength reliability have been


discussed. The method of moments and the method of maximum likelihood have also

been discussed for estimating its parameters. Finally, a numerical example of real

lifetime dataset from engineering has been presented to test the goodness of fit of the

QAD over one parameter exponential, Lindley and Aradhana , and two-parameter

EED, Lognormal, Gamma, Weibull and QSD.

REFERENCES

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[2] Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008): Lindley distribution and its

Application, Mathematics Computing and Simulation, 78, 493 - 506

[3] . Gupta, R.D. and Kundu, D. (1999): Generalized Exponential Distributions,

Australian and New Zealand Journal of Statistics, 41(2), 173 – 188.

[4] Lawless, J.F. (2003): Statistical models and methods for lifetime data, John

Wiley and Sons, New York.

[5] Lindley, D.V. (1958): Fiducial distributions and Bayes’ theorem, Journal of the Royal Statistical Society, Series B, 20, 102- 107.

[6] Renyi, A. (1961): On measures of entropy and information, in proceedings of

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– 561, Berkeley, university of California press

[7] Shaked, M. and Shanthikumar, J.G.(1994): Stochastic Orders and Their

Applications, Academic Press, New York.

[8] Shanker, R. (2015 a): Shanker Distribution and Its Applications, International Journal of Statistics and Applications, 5 (6), 338 – 348

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[11] Shanker, R. (2017): The discrete Poisson-Aradhana distribution, Turkiye Klinikleri Journal of Biostatistics, 9(1), 12 - 22

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