WEIGHTED ARADHANA DISTRIBUTION: PROPERTIES ...joics.org/gallery/ics-1262.pdfAradhana distribution is a newly proposed lifetime model formulated by Shanker (2016) for several medical

WEIGHTED ARADHANA DISTRIBUTION:

PROPERTIES AND APPLICATIONS

1Rashid A. Ganaie,

2 V. Rajagopalan and

3Aafaq A. Rather

1,2,*Department of Statistics, Annamalai University, Annamalai nagar, Tamil Nadu

1Email: [email protected] ,

2Email: [email protected]

3Corresponding Author Email: [email protected]

Abstract

In this paper, we introduced a new generalization of Aradhana distribution called as

weighted Aradhana distribution (WAD), for modeling life-time data. This has been compared

with one parameter Aradhana distribution. The statistical properties of this distribution has

been derived and the model parameters are estimated by maximum likelihood estimation.

Finally, an application to real life-time data set is fitted and the fit has been found to be good.

Keywords: Aradhana distribution, weighted distribution, Maximum Likelihood

Estimator, Likelihood Ratio test, order statistics.

1. Introduction

The idea of weighted distributions was first given by Fisher (1934) to model the

ascertainment bias. Later Rao (1965) has developed this concept in a unified manner while

modeling the statistical data, when the standard distributions were not appropriate to record

these observations with equal probabilities. As a result, weighted models were formulated in

such situations to record the observations according to some weighted function. The weighted

distribution reduces to length biased distribution as the weight function considers only the

length of the units. The concept of length biased sampling, was first introduced by Cox

(1969) and Zelen (1974). Van Deusen in (1986) arrived at Size-biased distribution theory

independently and fit it to the distributions of diameter of breast height (DBH). Subsequently,

Lappi and Bailey (1987), used weighted distributions to analyse the HPS diameter increment

data. In fisheries, Taillie et al (1995) modeled populations of fish stocks using weights.

Generally, the size-biased distribution is when the sampling mechanism selects the units with

probability which is proportional to some measure of the unit size. There are various good

sources which provide the detailed description of weighted distributions. Different authors

have reviewed and studied the various weighted probability models and illustrated their

applications in different fields. Weighted distributions are applied in various research areas

related to biomedicine, reliability, ecology and branching processes. Para and Jan (2018)

introduced the Weighted Pareto type-II distribution as a new model for handling medical

science data and studied its statistical properties and applications. Rather and Subramanian

(2018) discussed the characterization and estimation of length biased weighted generalized

uniform distribution. Rather and Subramanian (2019), discussed the weighted sushila

distribution with its various statistical properties and applications.

Journal of Information and Computational Science

Volume 9 Issue 8 - 2019

ISSN: 1548-7741

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mailto:[email protected]

mailto:[email protected]

Aradhana distribution is a newly proposed lifetime model formulated by Shanker

(2016) for several medical applications and calculated its various mathematical and statistical

properties including its shape, moment, stochastic ordering, generating function, skewness,

kurtosis, mean deviation, Renyi entropy, order statistics, stress-strength reliability, hazard

rate function, mean residual life function Bonferroni and lorenz curves, Maximum likelihood

estimation. The new one parameteric life-time distribution called as Aradhana distribution

has better flexibility in handling lifetime data as compared to Lindley and exponential

distributions. Shankar (2016) introduced two parameter quasi Aradhana distribution and

obtained its raw moments and central moments and also discussed its mathematical and

statistical properties. Shankar (2017) has obtained a poisson Aradhana distribution and

named as discrete poisson-Aradhana distribution discussed its various statistical properties

and estimation of parameter.

2. Weighted Aradhana Distribution

The probability density function of the Aradhana distribution is given by

00 )1()22(

);( 2

2

3

, θ ; xexxf θx

(1)

and the cumulative density function of the Aradhana distribution is given by

0θ ,0

)22(

2211)(

2

; xe

θθxθxx;F θx

(2)

Suppose X is a non- negative random variable with probability density function xf .Let

xw be the non- negative weight function, then the probability density function of the

weighted random variable wX is given by

, x

xwE

xfxwxfw 0 ,

Where )(xw be a non-negative weight function and .dxxfxwxwE

In this paper, we will consider the weight function as cxxw to obtain the weighted

Aradhana distribution. The probability density function of weighted Aradhana distribution is

given as:

0)(

; xxE

xfxxf

c

c

w (3)



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Where

0

)()( dxx;fxxE cc

)22(

)2(2)3()1()(

2

2

c

c cθccxE (4)

Substitute (1) and (4) in equation (3), we will get the required probability density function of

weighted Aradhana distribution as

θxcc

w exxcθcc

xf

2

2

3

)1()2(2)3()1((

(5)

Now, the cumulative density function (cdf) of the weighted Aradhana distribution(WAD) is

obtained as

x

ww dxxfxF0

)()(

x

θxcc

dxexxcθcc

0

2

2

3

)1()2(2)3()1((

After simplification, we will get the cumulative distribution function of weighted Aradhana

distribution (WAD) as

),2(2),3(),1()2(2)3()1((

)(2

2

θxcθγθxcθxccθcc

xFw

(6)

3. Reliability Analysis

In this section, we will discuss about the survival function, failure rate, reverse hazard rate

and the Mills ratio of the weighted Aradhana distribution (WAD).



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The survival function or the reliability function of the weighted Aradhana distribution is

given by

),2(2),3(),1())2(2)3()1((

1)(2

2


xSw

The hazard function is also known as the hazard rate, instantaneous failure rate or force of

mortality and is given by

)(1

)()(

xF

xfxh

w

w

θxcc

exxθxcθγθxcθxccθcc

2

22

3

)1(),2(2),3(),1(())2(2)3()1((

The Reverse hazard rate is given by

θx cc

r exxθxcθγθxcθxc

xh

2

2

3

)1()),2(2),3(),1((

)(

And the Mills Ratio of the weighted Aradhana distribution is

Mills Ratio θxcc

r exx

θxcθγθxcθxc

xh

23

2

)1(

),2(2),3(),1((

)(

1

4. Moments and Associated Measures

Let X denotes the random variable of weighted Aradhana distribution with parameters and

c, then the rth order moment )( rXE of weighted Aradhana distribution is obtained as



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0

)(')( dxxfxXE w

r

r

r

dxexxcθcc

x θxrcc

2

0

2

3

)1())2(2)3()1((

dxexxcθcc

θxrcc

2

0

2

3

)1())2(2)3()1((

0

2

2

3

)1())2(2)3()1((

dxexxcθcc

θxrcc

(7)))2(2)3()1((

))2(2)3()1(()(

2

2

cθcc

rcθrcrcXE

r

r

Putting r = 1 in equation (7), we will get the mean of weighted Aradhana distribution which

is given by

))2(2)3()1((

))3(2)4()2((')(

2

2

1

cθcc

cθccXE

and putting r = 2 in equation (7),we get the second moment of weighted Aradhana

distribution as

))2(2)3()1((

))4(2)5()3(()(

22

22

cθcc

cθccXE

Therefore,

2

2

2

22

2

))2(2)3()1((

))3(2)4()2((

))2(2)3()1((

))4(2)5()3((Variance

cθcc

cθcc

cθcc

cθcc

4.1 Harmonic mean

The Harmonic mean of the proposed model can be obtained as

0

)(11

. dxxfxx

EMH w



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0

2

2

3

)1(2))(c2)3()1((

1dxexx

θccx

θxcc

0

21

2

3

)1())2(2)3()1((

dxexxccc

θxcc

0

21

2

3

)21()2(2)3()1((

dxexxxcθcc

θxcc

0

1)1(

0

1)2(

0

2)1(

2

3

2))2(2)3()1((

dxexdxexdxexcθcc

θxcθxcθxcc

),1(2),2(),1())2(2)3()1((

.2

3

θxcθxcθxccθcc

MHc

4.2 Moment generating function and characteristics function of weighted Aradhana

distribution

Let X have a weighted Aradhana distribution, then the MGF of X is obtained as

0

)()()( dxxfeeEtM w

txtx

X

Using Taylor’s series

0

2

)(...!2

)(1)()( dxxf

txtxeEtM w

tx

X

0 0

)(!j

w

jj

dxxfxj

t

0

'!j

j

j

j

t

02

2

))2(2)3()1((

))2(2)3()1((

!jj

j

cθcc

jcθjcjc

j

t

0

2

2))2(2)3()1((

!))2(2)3()1((

1)(

jj

j

X jcθjcjcj

t

cθcctM

Similarly, the characteristics function of weighted Aradhana distribution can be obtained as

)()( itMt xX



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0

2

2))2(2)3()1((

!

)(

))2(2)3()1((

1)(

jj

j

x jcθjcjcj

it

cθccitM

5. Order Statistics

Let X(1), X(2), .…,X(n) be the order statistics of a random sample X1, X2,…,Xn drawn

from the continuous population with probability density function fx(x) and cumulative

density function with Fx(x), then the probability density function of rth order statistics X(r) is

given by

(8) )(1)()()!()!1(

!)(

1

)(

rn

X

r

XXrX xFxFxfrnr

nxf

Using the equations (5) and (6) in equation (8), the probability density function of rth order

statistics X(r) of weighted Aradhana distribution is given by

θxc

c

rX exxcθccrnr

nxf 2

2

3

)( )1())2(2)3()1(()!()!1(

!)(

1

2

2

),2(2),3(),1())2(2)3()1((

r


rn

θxcθxcθxccθcc

),2(2),3(),1(

))2(2)3()1((1

2

2

Therefore, the probability density function of higher order statistics X(n) can be obtained as

θxc

c

nX exxcθcc

nxf 2

2

3

)( )1())2(2)3()1((

)(

1

2

2

),2(2),3(),1())2(2)3()1((

n

θxcθxcθxccθcc

And therefore the probability density function of Ist order statistics X(1) can be obtained as



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θxc

c

X exxcθcc

nxf 2

2

3

)1( )1())2(2)3()1((

)(

1

2

2

),2(2),3(),1())2(2)3()1((

1

n

θxcθxcθxccθcc

6. Likelihood Ratio Test

Let X1, X2,... , Xn be a random sample from the weighted Aradhana distribution. To test the

hypothesis

),;()(: against );()( : 10 cxfxfHxfxfH w

In order to test whether the random sample of size n comes from the Aradhana distribution or

weighted Aradhana distribution, the following test statistic is used

n

i

w

xf

cxf

L

L

10

1

);(

),;(

n

i

c

i

c

cθcc

x

12

2

))2(2)3()1((

)22(

n

i

c

i

nc

xcθcc 1

2

2

)2(2)3()1((

)22(

Thus, we reject the null hypothesis if

n

i

c

i

nc

kxcθcc 1

2

2

))2(2)3()1((

)22(

nn

ic

c

i

cθcckx

12

2*

)22(

))2(2)3()1(( or,

nn

ic

c

i

cθcckkkx

12

2***

)22(

))2(2)3()1(( where,

Thus, for large sample size n, 2 log ∆ is distributed as chi-square distribution with one degree

of freedom and also p-value is obtained from the chi-square distribution.Thus we reject the

null hypothesis, when the probability value is given by



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n

i

c

i

n

i

c

i xxp11

*** and cesignifican of level specified than theless is where,

Is the observed value of the statistic ∆*.

7. Maximum Likelihood Estimator and Fisher Information Matrix

This is one of the most useful method for estimating the different parameters of the

distribution. Let X1, X2,….Xn be the random sample of size n drawn from the weighted

Aradhana distribution, then the likelihood function of weighted Aradhana distribution is

given as:

n

i

w cxfcxL1

),;(),;(

n

i

θx

i

c

i

c

iexxcθcc1

2

2

3

)1())2(2)3()1((

n

i

θx

i

c

in

cn

exxcθcc

cxL1

2

2

)3(

)1())2(2)3()1((

),;(

The log likelihood function is

n

i

ixccθccncnL1

2 log))2(2)3()1((loglog)3(log

n

i

n

i

ii xx1 1

)1log(2 (9)

The maximum likelihood estimates of θ and c can be obtained by differentiating equation (9)

with respect to θ and c and must satisfy the normal equation

0))2(2)3()1((

))2(2)1(2()3(log

12

n

i

ixcθcc

ccθn

cnL

n

i

ixcnnc

L

1

0log)1(loglog

Where (.) is the digamma function.



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Because of the complicated form of the likelihood equations, algebraically it is very difficult

to solve the system of non- linear equations. Therefore we use R and wolfram mathematics

for estimating the required parameters. To obtain confidence interval we use the asymptotic

normality tests. We have that, if )ˆ,ˆ(ˆ c denotes the Maximum likelihood estimates of

),( c , we can state the result as follows:

))(,0()ˆ( 1

2 INn

Where I(λ) is Fisher’s Information Matrix, i.e.,

2

22

2

2

2

log

log

log

log

1)(

c

L E

c

LE

c

L E

LE

nI

Where

22

2

22

2

))2(2)3()1((

))2(2)1(2(

))2(2)1(2()1(2))(2(2)3()1((

)3(log

cθcc

ccθ

ccθccθcc

ncnL

E

)1('log

2

2

cn

c

LE

n

cθ

LE

log2

function. digamma of derivativeorder first theis (.)' where

cθ

) λ(I) I λ

and for intervals confidence asymptotic

obtain toused becan thisandˆby ( estimate we,unknown being since 11

8. Bonferroni And Lorenz Curves

The Bonferroni and the Lorenz curves are not only used in economics in order to study the

income and poverty, but it is also being used in other fields like reliability, medicine,

insurance and demography. The Bonferroni and Lorenz curves are given by

q

dxxxfp

pB01

)('

1)(



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q

dxxxfpL01

)('

1)( and

)2(2)3()1((

))3(2)4()2((' where

2

2

1

cθcc

cθcc

and q=F-1(p)

q

θxcc

dxexxcθcccθccp

cθccpB

0

21

2

3

2

2

)1()2(2)3()1(())3(2)4()2((

))2(2)3()1(()(

After simplification, we get

),3(2),4(),2())3(2)4()2((

)(2

4

qcqcqccθccp

pBc

And

),3(2),4(),2())3(2)4()2((

)()(2

4

qcqcqccθcc

ppBpLc

9. Entropies

The concept of entropy is important in different areas such as probability and

statistics, physics, communication theory and economics. Entropies quantify the diversity,

uncertainty, or randomness of a system. Entropy of a random variable X is a measure of

variation of the uncertainty.

9.1: Renyi Entropy

The Renyi entropy is important in ecology and statistics as index of diversity. The Renyi

entropy is also important in quantum information, where it can be used as a measure of

entanglement. For a given probability distribution, Renyi entropy is given by

dxxfe )(log

1

1)(

1 and 0 where,

0

2

2

3

)1())2(2)3()1((

log1

1)( dxexx

cθcce θxc

c

0

2

2

3

)1())2(2)3()1((

log1

1)( dxexx

cθcce βθxβc

c



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(10) )1())2(2)3()1((

log1

1)( 2

0

2

3

dxxexcθcc

e βθxβcc

Using binomial expansion in equation (10), we get

002

3

)())2(2)3()1((

log1

1)( dxexx

jcθcce βθxβcj

j

c

dxxejcθcc

e jβcβθx

j

c

0

1)1(

02

3

))2(2)3()1((log

1

1)(

10

2

3

)(

)1(

))2(2)3()1((log

1

1)(

jcj

c jc

jcθcce

9.2: Tsallis Entropy

A generalization of Boltzmann-Gibbs (B-G) statistical mechanics initiated by Tsallis has

focused a great deal to attention. This generalization of B-G statistics was proposed firstly by

introducing the mathematical expression of Tsallis entropy (Tsallis, 1988) for a continuous

random variable is defined as follows

dxxfS )(11

1

0

dxexxcθcc

S θxcc

0

2

2

3

)1())2(2)3()1((

11

1

(11) )1())2(2)3()1((

11

1

0

2

2

3

dxxexcθcc

S λθxλcc

Using binomial expansion in equation (11), we get

0 0

2

3

)())2(2)3()1((

11

1

j

λθxλcjc

dxexxjcθcc

S

0 0

1)1(

2

3

))2(2)3()1((1

1

1

j

λθxjλcc

dxexjcθcc

S



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012

3

)(

)1(

))2(2)3()1((1

1

1

jjλc

c jc

jcθccS

10. Data Analysis

The weighted Aradhana distribution has been fitted to a number of life-time data set

from biomedical science and Engineering. In this section, we present a real life-time data set

to show that weighted

Aradhana distribution can be better than one parametric Aradhana distributions. We consider

a data set, which represents the life-time data of relief times (minutes) of 20 patients

receiving an analgesic reported by Gross and Clark (1975). The data set is given as follows:

1.1 1.4 1.3 1.7 1.9 1.8 1.6 2.2 1.7 2.7

4.1 1.8 1.5 1.2 1.4 3.0 1.7 2.3 1.6 2.0

In order to compare the distributions, we consider the criteria like Bayesian

information criterion (BIC), Akaike Information Criterion (AIC), Akaike Information

Criterion Corrected (AICC) and -2 logL. The better distribution is which corresponds to lesser

values of AIC, BIC, AICC and – 2 log L. For calculating AIC, BIC, AICC and -2 logL can be

evaluated by using the formulas as follows

AIC = 2K - 2logL, BIC = klogn - 2logL, )1(

)1(2

kn

kkAICAICC

Where k is the number of parameters, n is the sample size and -2 logL is the maximized value

of log likelihood function.

Table. 1: MLE’s, S.E, -2 logL, AIC, BIC and AICC of the fitted distribution of the given

data.

It can be easily seen that the weighted Aradhana distribution have the lesser AIC, BIC, AICC

and -2 logL values as compared to Aradhana distribution. Hence, we can conclude that the

weighted Aradhana distribution leads to better fit than the Aradhana distribution.

Distribution MLE S.E -2 logL AIC BIC AICC

Aradhana 𝜃=1.1231938 𝜃=0.1526142 56.37 58.37 59.365 58.6

Weighted

Aradhana

�̂�=7.984101

𝜃=5.514582

�̂�=2.957678

𝜃=1.610589

11.81 15.81 17.80 16.48



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on(AD)Distributi Aradhana and (WAD)on Distributi Aradhana Weightedof :FittingFig.4

11. Conclusion

In the present study, we have introduced a new generalization of the Aradhana

distribution termed as Weighted Aradhana distribution and has two parameters. The subject

distribution is generated by using the weighting technique and the parameters have been

obtained by using maximum likelihood technique. Some mathematical properties along with

reliability measures are discussed. The new distribution with its applications in real lifetime

data has been demonstrated. The results are compared with Aradhana distribution and has

been found that weighted Aradhana distribution provides better fit than Aradhana

distribution.

Reference

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Survey Sampling, Johnson, N. L. and Smith, H., Jr .(eds.) New York Wiley-

Interscience, 506-527.

2) Fisher, R.A., (1934), The effects of methods of ascertainment upon the estimation of

frequencies. Annals of Eugenics, 6, 13-25.

3) Gross, A.J. and Clark, V.A., (1975), Survival Distributions: Reliability Applications in

the Biometrical Sciences, John Wiley, New York.

4) Lappi, J. and Bailey, R.L., (1987), Estimation of diameter increment function or other

tree relations using angle-count samples. Forest Science, 33, 725-739.

5) Para, B. A., & Jan T. R., (2018), On Three Parameter Weighted Pareto Type II

Distribution: Properties and Applications in Medical Sciences. Applied Mathematics

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6) Rao, C. R., (1965), On discrete distributions arising out of method of ascertainment,

in classical and Contagious Discrete, G.P. Patiled; Pergamum Press and Statistical

publishing Society, Calcutta. 320-332.

7) Rather, A. A. and Subramanian, C. (2019), On Weighted Sushila Distribution with

Properties and its Applications, International Journal of Scientific Research in

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Scientific Research in mathematical and Statistical sciences, vol-5,issue-5,pp.72-76.

9) Shankar. R, (2017),The Discrete Poisson-Aradhana distribution,Turkiye Klinikleri J

Biostat 2017; vol-9,issue-1, pp 12-22.

10) Shanker, R & Shukla, K. K., (2018), A Quasi Aradhana distribution with Properties

and Applications, International Journal of Statistics and systems, vol-13, issue-1, pp

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12) Taillie, C., Patil, G.P. and Hennemuth, R. (1995), Modelling and analysis of

recruitment distributions, Ecological and Environmental Statistics, 2(4), 315-329.

13) Van Deusen, P. C. (1986), Fitting assumed distributions to horizontal point sample

diameters. 32,166-148.

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706.



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WEIGHTED ARADHANA DISTRIBUTION: PROPERTIES ...joics.org/gallery/ics-1262.pdfAradhana distribution is a newly proposed lifetime model formulated by Shanker (2016) for several medical