A NEW EXTENSION OF DEVYA DISTRIBUTION WITH PROPERTIES AND APPLICATIONS IN REAL LIFE DATA M. Sajish Kumar 1 and C. Subramanian 2 1,2 Department of Statistics, Annamalai University, Annamalai nagar, Tamil Nadu, India 1 Corresponding Author: [email protected]Abstract In this paper, we proposed a new probability model called as the Length Biased Devya distribution. The statistical properties of this distribution including, the mean, variance, harmonic mean and probability generating functions have been studied. The maximum likelihood method has been used to estimate the parameter. Finally a real life data has been analysed for illustration purpose. Keywords: Devya distribution, weighted distribution, Reliability Analysis, Order Statistics, Maximum Likelihood Estimation. 1. INTRODUCTION Weighted distribution provides an approach to dealing with model specification and data interpretation problems. Fisher (1934) and Rao (1965) introduced and unified the concept of weighted distribution. When the probability of observing a positive-valued random variable is proportional to the value of the variable the resultant is length-biased distribution. A table for some basic distributions and their length-biased forms is given by Patil and Rao (1978) such as Lognormal, Gamma, Pareto, Beta distribution. Modi and Gill (2015), obtained length biased weighted Maxwell distribution and its applications. Shenbagaraja, et al. (2019), discussed on length biased Garima distribution with several properties and its applications. Mudasir and Ahmad (2018), discussed on characterization and estimation of the length biased Nakagami distribution. Recently, Rather and Subramanian (2019) discussed on length biased erlang truncated exponential distribution with life time data which shows more flexibility than classical distributions. A new one parametric lifetime distribution named Devya distribution was given by Shanker (2016). Its statistical and mathematical properties including moments, order statistics, Bonferroni and Lorenz curves have been discussed. In this paper, we will discuss the length biased version of Devya distribution with its properties and applications. Journal of Information and Computational Science Volume 9 Issue 11 - 2019 ISSN: 1548-7741 www.joics.org 1188
Transcript
A NEW EXTENSION OF DEVYA DISTRIBUTION WITH
PROPERTIES AND APPLICATIONS IN REAL LIFE DATA
M. Sajish Kumar1 and C. Subramanian2
1,2Department of Statistics, Annamalai University, Annamalai nagar, Tamil Nadu, India
The probability density function (pdf) of the Devya Distribution is given by
)1(0,0;1)2462(
),( 432
234
5
xexxxxxf x
and the cumulative density function (cdf) of the Devya Distribution is given by
)2(,0;
)2462(
24)2(6)234()(11),(
234
222332344
xe
xxxxxxxxxxxF
x
and its mean is
)3()2462(
1202462)(
234
234
XE
Suppose X is a non- negative random variable with probability density function )(xf .Let
)(xw be the nonnegative weight function, then the probability density function of the
weighted random variable Xw is given
)4(0;))((
)()()( x
xwE
xfxwxf w
Where w(x) be a non-negative weight function and )()()( xfxwXE .
For different weighted models, we have different choices of the weight function w(x).when cx the resulting distribution is termed as weighted distribution. In this paper, we have to find
the Length biased version of Devya distribution, so we will take c = 1 in weightscxxw )( ,
in order to get the Length biased Devya distribution and its probability density function (pdf)
is given by:
)5()(
)()(
xE
xxfxf l
Substitute, the values of equation (1) and (3) in equation (5), we will get the probability
density function of Length biased Devya distribution.
)6(0,0;1)1202462(
),( 432
234
6
xexxxxxxf x
l
Journal of Information and Computational Science
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Now, the cumulative distribution function of the Length biased Devya distribution is obtained
as
x
ll dxxfxF0
)(),(
x
x
l xdxexxxxxxF0
432
234
6
0,0;1)1202462(
),(
After simplification, we will get the cumulative distribution function of Length biased Devya
distribution as
)7(
;61
;51
;41
;31
;2
)1202462()(
43
2
234
4
xx
xxx
xFl
3. RELIABILITY ANALYSIS
In this section, we will discuss about the survival function, failure rate, reverse hazard rate
and Mills ratio of the length biased Devya distribution.
The survival function or the reliability function of the Length biased Devya
distribution is given by
Journal of Information and Computational Science
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ISSN: 1548-7741
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)(1)( xFxS l
xx
xxx
xS
;61
;51
;41
;31
;2
)1202462(1)(
43
2
234
4
The hazard function is also known as the hazard rate, instantaneous failure rate or force of
mortality and is given by
)(1
)()(
xF
xfxh
l
l
xx
xxx
exxxxxxh
x
;61
;51
;41
;31
;2
)1202462(
1)(
43
2
4234
4324
and Reverse hazard rate and Mills Ratio of the Length biased Devya distribution is given by
)(
)()(
xF
xfxh
l
l
r
xxxxx
exxxxxxh
x
r
;61
;51
;41
;31
;2
1)(
432
4324
)(
1
xhMillsRatio
r
xexxxxx
xxxxx
)1(
;61
;51
;41
;31
;2
4324
432
3.1 Harmonic Mean
The Harmonic mean of the Length Biased Devya distribution model can be obtained as
xEMH
1.
Journal of Information and Computational Science
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dxxfx
l )(1
0
dxexxxxxx
x
0
432
234
6
1)1202462(
1
0
432
234
6
1)1202462(
dxexxxx x
0
4
0
3
0
2
00
234
6
)1202462(
dxexdxex
dxexdxxedxe
xx
xxx
on simplification, we get
)1202462(
)2462(..
234
234
MH
4. MOMENTS AND ASSOCIATED MEASURES
Let X denotes the random variable of Length biased Devya distribution then the thr order
moment )( rXE of Length biased Devya distribution is obtained as
dxexxxxxx xr
432
234
6
0
1)1202462(
)8(1202462
)!5()!4()!3()!2()!1(234
234
r
rrrrr
Putting r =1 in equation (8), we will get the mean of Length biased Devya distribution which
is given by
1202462
7201202462234
234
1
dxxfxXE l
r
r
r )()(0
'
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and putting r = 2 in equation (8), we get the second moment of Length biased Devya
distribution as
1202462
50407201202462342
234
2
therefore,
2
12 )(Variance
2
234
234
2342
234
1202462
7201202462
1202462
5040720120246
2
234
234
2342
234
1202462
7201202462
1202462
5040720120246).(.
DS
Coefficient of Variation (C.V) ='
1
7201202462
720120246212024625040720120246234
2234234234
Coefficient of Dispersion (γ) = '
1
2
12024627201202462
720120246212024625040720120246234234
2234234234
5. MOMENT GENERATING FUNCTION AND CHARACTERISTIC
FUNCTION
Let X have a Length biased Devya distribution, then the MGF of X is obtained as
0
)()()( dxxfeeEtM w
txtx
X
Using Taylor’s expansion
Journal of Information and Computational Science
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0
2
)(!2
)(1)()( dxxf
txtxeEtM l
tx
X
0 0
)(!
dxxfxj
tl
j
j
j
'!0
j
j
j
j
t
)9(1202462
)!5()!4()!3()!2()!1(
! 234
234
0
j
j
j jjjjj
j
t
Similarly, the characteristic function of Length biased Devya distribution can be obtained as
)()( itMt XX
0234
234
)10(!1202462
)!5()!4()!3()!2()!1(
j
j
j j
itjjjjj
6. 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 pdf of rthorder statistics X(r) is given by
rn
X
r
XXrX xFxFxfrnr
nxf
)(1)()(
)!()!1(
!)(
1
)( (11)
Using the equations (6) and (7) in equation (11), the probability density function of rth order
statistics X(r) of Length biased Devya distribution is given by
rn
r
x
rX
xxxxx
xxxxx
exxxxxnr
nxf
;61
;51
;41
;31
;2)1202462(
1
;61
;51
;41
;31
;2)1202462(
1)1202462()!1()!1(
!)(
432234
6
1
432234
6
432
234
6
)(
Therefore, the probability density function of higher order statistics X(n) can be obtained as
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1
432234
6
432
234
6
)(
;61
;51
;41
;31
;2)1202462(
1)1202462(
)(
n
x
nX
xxxxx
exxxxxnxf
and the pdf of 1st order statistic X(1)can be obtained as
1
432234
6
432
234
6
)1(
;61
;51
;41
;31
;2)1202462(
1
1)1202462(
)(
n
x
X
xxxxx
exxxxxxf
7. 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.
7.1 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)(
; where β > 0, and β ≠1
0
432
234
6
1)1202462(1
1)( dxexxxxxLoge x
iiiii
0
432
234
6
112024621
1)( dxexxxxxLoge x
iiiii
)12()1(
12024621
1)(
0
432
234
6
dxexxxxxLoge x
iiii
Using binomial expansion in equation (12), we get
Journal of Information and Computational Science
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ISSN: 1548-7741
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dxex
l
k
k
j
j
i
ie xlkji
i j k l
0
11
0 0 0 0234
6
1202462log
1
1)(
10 0 0 0
234
6 1
1202462log
1
1)(
lkji
i j k l
lkji
l
k
k
j
j
i
ie
7.2 Tsallis Entropy
A generalization of Boltzmann-Gibbs (B-G) statistical mechanics initiated by Tsallis has
focussed 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
0
)(11
1dxxfS
0
432
234
6
1)1202462(
11
1dxexxxxxS x
iiiii
0
432
234
6
11202462
11
1dxexxxxxS x
iiiii
)13()1(
12024621
1
1
0
432
234
6
dxexxxxxS x
iiii
Using binomial expansion to equation (13), we get
dxex
l
k
k
j
j
i
iS xlkji
i j k l
0
11
0 0 0 0234
6
12024621
1
1
10 0 0 0
234
6 1
12024621
1
1lkji
i j k l
klji
l
k
k
j
j
i
iS
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8. BONFERRONI AND LORENZ CURVES
The Bonferroni and Lorenz curves (Bonferroni, 1930) have applications not only in
economics to study income and poverty, but also in other fields like reliability, demography,
insurance and medicine. The Bonferroni and Lorenz curves are defined as
q
dxxxfp
pB01
)('
1)(
and
q
dxxxfpL01
)('
1)(
where 1202462
7201202462234
234/
1
and
)(1 pFq
dxexxxxxp
pB x
q
0
432
234
1
6
1)1202462(
)(
dxexdxexdxexdxexdxexp
pB x
q
x
q
x
q
x
q
x
q
0
5
0
4
0
3
0
2
0
234
1
6
)1202462()(
dxetdxet
dxetdxetdxet
ppB
x
q
x
q
x
q
x
q
x
q
0
5
6
0
4
5
0
3
4
0
2
3
0
2
234
1
6
11
111
)1202462()(
dxetdxet
dxetdxetdxet
ppB
x
q
x
q
x
q
x
q
x
q
0
16
6
0
15
5
0
14
4
0
13
3
0
12
2
234
1
6
11
111
)1202462()(
);!5(
1);!4(
1);!3(
1;!2(
1;1(
)7201202462()(
432234
5
qqqqqp
pB
L(p)=p(B(p)
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);!5(
1);!4(
1);!3(
1;!2(
1;1(
)7201202462()(
432234
5
qqqqqpL
9. MAXIMUM LIKELIHOOD ESTIMATOR
Let ).........,,,( 321 nxxxx be a random sample of size from Devya distribution. The likelihood
function, of is given by
)14(1()1202462(
);(1
432
234
6
n
i
x
iiiii
n
iexxxxxxL
The natural log likelihood function thus obtained as
)15(1ln
)1202462(ln);(ln
0
432
234
6
i
i
iiiii xxxxxxnxL
The maximum likelihood estimates of θ can be obtained by differentiating equation (15) with
respect to θ and must satisfy the normal equation
XnxxxxxnL
iiiii
4322346 1lnln)1202462ln((lnlog
)16(0)1202462(
2412646log234
23
Xnn
L
Because of the complicated form of likelihood equations (16), algebraically it is very
difficult to solve the system of nonlinear equations. Therefore, we use R and wolfram
mathematica for estimating the required parameters.
10. LIKELIHOOD RATIO TEST
Let X1, X2, ... , Xn be a random sample from the LBD distribution. To test the hypothesis
);()(:against);()(: 1 xfxfHxfxfH lo
In order to test whether the random sample of size n comes from the Devya distribution or
LBD distribution, the following test statistic is used
n
i
l
xf
xf
L
L
10
1
);(
);(
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i
n
i
x
1234
234
)1202462(
)2462((
n
i
i
n
x1
234
234
)1202462(
)2462(
We reject the null hypothesis if
n
i
i
n
kx1
234
234
)1202462(
)2462(
n
n
i
i kx
)2462(
)1202462(or,
234
234
1
*
n
n
i
i kkkx
)2462(
)1202462(where,
234
234**
1
*
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
