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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.
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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
θxcθγθxcθxccθcc
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
θxcθγθxcθxccθcc
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
Journal of Information and Computational Science
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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.
Journal of Information and Computational Science
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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.
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