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29
A Compounded Probability Model for Decreasing Hazard and its Inferential Properties
Brijesh P. Singh,Utpal Dhar Das *&Sandeep Singh
�
Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005 INDIA utpal.statmath@gmail.com
Abstract
Early failures are generally observed due to latent defects within a product caused by faulty components, faulty assembly, transportation damage and installation damage. Also early life (infant mortality) failures tend to exhibit a decreasing failure rate over time. Such type of problems can be modelled either by a complex distribution having more than one parameter or by .nite mixture of some distribution. In this article a single parameter continuous compounded distribution is proposed to model such type of problems. Some important properties of the proposed distribution such as distribution function, survival function, hazard function and cumulative hazard function, entropies, stochastic ordering are derived. The maximum likelihood estimate of the parameter is obtained which is not in closed form, thus iteration procedure is used to obtain the estimate of parameter. The moments of the proposed distribution does not exist. Some real data sets are used to see the performance of proposed distribution with comparison of some other competent distributions of decreasing hazard using Likelihood, AIC, AICc, BIC and KS statistics.
Keywords: Entropy, Hazard function, KS, MLE, Order Statistics, Quantile function.
I. Introduction
Normal, exponential, gamma and weibull distributions are the basic distributions that demon�strated in a number of theoretical results in the distributions theory. Particularly, exponential distribution is an invariable example for a number of theoretical concepts in reliability studies. It is characterized as constant hazard rate. In case of necessity for an increasing/decreasing failure rate model ordinarily the choice falls on weibull distribution. Lindley distribution is an increasing hazard rate distribution and has its own importance as a life testing distribution. The lindley distribution is one parameter distribution that is a mixture of exponential and gamma distributions and was proposed by Lindley [16]. The lindley distribution is used to explain the lifetime phenomenon such as engineering, biology, medicine, ecology and .nance. Ghitany et al. [10]. Lindley distribution has generated little attention in excess of the exponential distribution because of its decreasing mean residual life function and increasing hazard rate however expo�nential distribution has constant mean residual life function and hazard rate. Adamidis & Loukas [1] introduced a two-parameter exponential-geometric distribution with decreasing hazard rate and Barreto-Souza et al. [5] introduced a decreasing failure rate model, compounding exponential and poisson-lindley distribution (EPL) and the probability density function is given as
..x ..x)
..2(1 + .)2e(3 + . . e
fepl (x; ., .)= ; x > 0, . > 0, . > 0 (1)
(1 + 3. + .2)(1 + . . e..x)3
Another idea was proposed by Ku.s [15] and Tahmasbi & Rezaei [27]. They introduced the exponential Poisson (EP) and exponential logarithmic (EL) distributions and the pdf is given by
.. ..x
....x+.e
fep(x; ., .)= e; x > 0, . > 0, . > 0 (2)
1 . e.. 1 .(1 . p)e..x
fel (x; ., p)= ; x > 0, . > 0, p . (0, 1) (3)
. log p 1 . (1 . p)e..x
Chahkandi & Ganjali [8] introduced a class of distributions, which is exponential power series distributions (EPS), where compounding procedure follows the same way that was previously given by Adamidis & Loukas [1]. Weibull [29] a Swedish mathematician describe the weibull distribution that is usefull for increasing as well as decresing hazard and the pdf is de.ned as
fw(x; ., .)= ... x..1e..x; x > 0, . > 0, . > 0 (4)
Natural mixing of exponential populations, giving rise to a decreasing hazard rate distribution, were .rst introduce by Proschan [23]. Subsequently other distributions with decreasing hazard rates of practical interest were discussed by Cozzolino [7]. The distributions with decreasing failure rate (DFR) are discussed in the works of Lomax [18], Barlow et al. [4], Barlow & Marshall [2, 3], Marshall & Proschan [19], Dahiya & Gurland [9], Saunders & Myhre [25], Nassar [21], Gleser [12], Gurland & Sethuraman [13]. Keeping these ideas in view, in this study, an attempt has been made to develop a new lifetime distribution by compounding exponential and lindley distribution and named as compounded exponential-lindley (CEL) distribution. The distributional properties, estimation of parameters, Fisher information, entropies, stochastic ordering, quantile function, order statistics and simulation study for the proposed distribution have been discussed in detail.
II. Proposed Distribution
Let X1, X2, ..., Xn be a random sample from following exponential distribution with scale parame�ter . > 0 and the probability density function (pdf) is in the form
f (x|.)= .e..x; x > 0, . > 0 (5)
The parameter . > 0 of the above distribution takes continuous value and hazard of the distribution is constant. Now we assume the parameter . is a random variable follows lindley distribution with pdf given as
.2
...;
.(.; .)= (1 + .)e. > 0, . > 0 (6)
(. + 1)
Now the pdf of the proposed distribution CEL is given by
..2 .
00
..(x+.)d.
g(x; .)= f (x|.).(.; .)d. =(. + .2)e
(. + 1)
00 .2 (x + . + 2)
= ; x > 0, . > 0 (7)
(. + 1)(x + .)3
Figure 1: Probability density function of CEL distribution
and the cumulative distribution function (cdf) of CEL is obtained as x [x(. + 1)+ .(. + 2)]
G(x; .)= ; x > 0, . > 0 (8)
(. + 1)(x + .)2
Figure 2: Cumulative distribution function of CEL distribution
From the .gure 1 and 2, it is clear that the distribution is early failure distribution for smaller value of .. The survival or reliability function S(x) of CEL having pdf (7), is given as
.2(x + . + 1)
S(x)= (9)
(. + 1)(x + .)2
Figure 3: Survival function of CEL distribution Figure 4: Hazard rate function of CEL distribution
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
The hazard function is de.ned as
h(x) = g(x) 1 . G(x) = g(x) S(x) = (x + . + 2) (x + .)(x + . + 1) (10)
According to Glaser [11], g(t) is density function, g'(t) is the .rst order derivative and '(t)
.(t)= . gg(t) . If .'(t) > 0 . t > 0, then the distribution has increasing failure rate (IFR) and if .'(t) < 0 . t > 0, then the distribution has decreasing failure rate (DFR). For the proposed CEL distribution 2t + 3. + 3
.(t)= (11)
(t + .)(t + . + 2) Differentiating .(t) with respect to t we get 2 42(. . 1)
.'(t)= .. .
(t + .)(t + . + 2)(t + .)2(t + . + 2)(t + .)(t + . + 2)2 2(. . 1)
. 2 (12)
[(t + .)(t + . + 2)]
Now from the equation (12) we have .'(t) < 0 for all t > 0, hence distribution has DFR. Also the hazard function of the CEL distribution is
(x + . + 2) 21
h(x)= = .
(x + .)(x + . + 1)(x + .)(x + . + 1)
After differentiating (10) with respect to x we get
h'(x)= . 21
(x + .)2 +(x + . + 1)2 lim h'(x)= . 21 < 0 . . > 0 (13)
.2 +
x>0 (. + 1)2
Therefore h'(0) < 0 . . > 0, Hence CEL distribution is a distribution of monotonic decreasing hazard with increasing time. Now Cumulative hazard function H(t) is de.ned as
0t .. .. .2 .
. + 1 t + .
H(t)= h(x)dx = log (14)
t + . + 1 .
0
Theorem 1. The moments of the CEL(.) distribution does not exists.
Proof: Suppose the random variable X comes from CEL(.) then the rth moment is given by
0. 0.
.2 x + . + 2
rr
E(Xr)= xg(x)dx = xdx
. + 1 (x + .)3 00
Now
0. 0.
1 xr2 xr
dx + dx
2 3
. + 11 + x .(. + 1) 1 + x 0 .0 .
Let x = z; dx = .dz; x > 0, z > 0, and x > ., z > . above integral become
.
0. 0.
.r+1 zr 2.r+1 zr dz + dz
. + 1 (1 + z)2 .(. + 1)(1 + z)3 00
.
0
xm.1
using Beta integral of second kind i.e dx = B(m, n) ; m > 0; n > 0, we get
(1+x)m+n 0
.r+12.r+1 E(Xr)= B(r + 1, 1 . r)+ B(r + 1, 2 . r)
. + 1 .(. + 1)
.r+1
2
= B(r + 1, 1 . r)+ B(r + 1, 2 . r)(15)
. + 1.
Here range is .1 < r < 1. But range of r should be r . 1. Hence E(Xr) does not exists. Therefore
mean, variance, SD as well as higher order moments does not edxists for CEL(.).
Theorem 2. The moment generating function of CEL(.) does not exists.
Proof: Let X be the random variable from NWEL(.) distribution then the moment generating
function (mgf) is given by
0. 0.
.2
tx tx x + . + 2
E(etx)= eg(x)dx = edx
. + 1 (x + .)3 00
..
0. 0.
.2 etx 2etx = . dx + dx . (16)
. + 1 (x + .)2 (x + .)3 00
Now
0. . 0.
tx tx tx
eee
dx =+ t dx
(x + .)2 .(x + .)(x + .)
0
00
..
0
etx
1
=+ lim.t dx . (17)
. >. (x + .)
0 Now applying L�Hospital rules we get
tx tetx
e
lim = lim = .
x>. (x + .) x>. 1
Hence integrand is divergent, as well as the function is not integrable over R we conclude thta E(etx) does not exists. The characteristic function of CEL distribution is de.ned as
0.
.
12
.x(t)= eitx g(x)dx = . (.1)k (k + 1)!1 +(k + 2)(18)
. + 1 (it)k+1.
k=0
0
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
III. Entropies
An entropy is a measure of randomness occured in any system. Entropy is an important property of probability distributions and it measures the uncertainty in a probability distribution.
I. Renyi Entropye
An entropy is a measure of variation of the uncertainty, Renyi [24] gave an expression of the Entropy function de.ned by
..
0.
e(.)= 1 log . g. (x)dx .
1 . .
0
where 0 < . < 1, Substituting the value of g(x) from (7)
..
1 .0. .2 (x + . + 2) . .
e(.)= log dx
1 . . (. + 1)(x + .)3
0
..
. 0. ..
1 . .2 12 .
= log dx
1 . .. + 1 (x + .)2 +(x + .)3
0
n
(nk)akbn.k we get Now applying Binomial expansion (a + b)n
=
.
k=0
..
0.
.. 2k ..k
1 .2 1 2
.
.
.
dx
.
log
(x + .)3
1 . .
. + 1
kx + .
k=0
0
after simli.cation we get the Renyi entropy as
.
.2
2..k1
. .
.
log log (19)
e(.)=
+
(3. . k . 1).(3..k.1)
1 . .
. + 1
1 . .
k
k=0
where 0 < . < 1, . > 0, x > 0
II. Tsallis Entropy This is introduced by Tsallis [28] as a basis for generalizing the standard statistical mechanics
S. = 1 1 . . . .1 . 0. g.(x)dx . .
= 1 1 . . . .1 . 0 .2 (. + 1) . 0. 0 (x + . + 2) (x + .)3 . dx . .
n
(nk)akbn.k and simplifying we get Tsallis Entropy Now applying Binomial expansion (a + b)n
=
.
k=0
as in (20).
.
.2
2..k
.
1
.
.
1 . (20)
e(.)=
1 . .
. + 1
k
(3. . k . 1).(3..k.1)
k=0
235
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
IV. Quantile Function
The quantile function for CEL distribution is de.ned in the form xq = Q(u)= G.1(u) where Q(u) is the quantile function of G(x) in the range 0 < u < 1. Taking G(x) is the cdf of CEL distribution and inverting it as above will give us the quantile function as follows
x [x(. + 1)+ .(. + 2)]
G(x)= = u (21)
(. + 1)(x + .)2 Simplifying equation (21) above gives the following:
x 2 x.(. + 2)
+= u
x + . (x + .)2
x
Now let = z we get from above
x+.
. + 2
z2 + z(1 . z)= u
. + 1 z2 . z(. + 2)+ u(. + 1)= 0 (22)
This is a quadratic equation and after solving we get the solution for x as
x (. + 2) �(. + 2)2 . 4u(. + 1)
z ==
x + . 2 2
Q(u)= . . 1 (23).. �(. + 2)2 . 4u(. + 1)
where u is a uniform variate on the unit interval (0,1).
The median of X from the CEL distribution is simply obtained by setting u = 0.5 and this
substitution of u = 0.5 in the above equation (23) gives.
2
Median = . . 1 (24)
.. +(. + 1)2 + 1 Bowley�s measure of skewness based on quartiles is de.ned as: Q( 34 ) . 2Q( 12 )+ Q( 14 )
SK = (25)
Q( 34 ) . Q( 14 ) and [20] presented the Moors� kurtosis based on octiles by Q( 78 ) . Q( 58 ) . Q( 83 )+ Q( 18 )
KT = (26)
Q( 68 ) . Q( 18 ) where Q(.) is calculated by using the quantile function from equation (23).
V. Stochastic Orderings
Stochastic ordering of a continuous random variable is an important tool to judging their comparative behaviour. A random variable X is said to be smaller than a random variable Y.
(i)
Stochastic order X .st Y if FX (x) . FY(x) for all x.
(ii)
Hazard rate order X .hr Y if hX (x) . hY(x) for all x.
(iii) Mean residual life order X .mrl Y if mX (x) . mY(x) for all x.
(iv) Likelihood ratio order X .lr Y if fX (x) decreases in x.
fY (x)
The following results by Shaked & Shanthikumar [26] are well known for introducing stochastic ordering of distributions
==
X .lr Y . X .hr Y . X .mrl Y
i.eX .st Y
with the help of following theorem we claim that CEL distribution is ordered with respect to strongest likelihood ratio ordering
Theorem 3. Let X . CEL(.1) distribution and Y . CEL(.2) distribution. If .1 > .2 then X .lr Y and therefore X .hr Y,X .mrl YandX .st Y.
Proof: We have
3
fX (x) .12(.2 + 1) x + .1 + 2 x + .2
= ; x > 0
fY(x) .22(.1 + 1) x + .2 + 2 x + .1
Now taking log both side we get .2
fX (x) 1 (.2 + 1) x + .1 + 2 x + .2
log = log + log + 3 log
fY(x) .22(.1 + 1) x + .2 + 2 x + .1
By differentiating both side we get d fX (x) .2 . .13(.2 . .1)
log =+
dx fY(x)(2 + .1 + x)(2 + .2 + x)(x + .1)(x + .2)
d
Thus for .1 > .2, dx log fX (x) < 0.This means that X .lr Y and hence X .hr Y, X .mrl Y and
fY (x) X .st Y.
VI. Distribution of order statistics
Let X1, X2, ..., Xm be a random sample of size m from CEL distribution and let X1;m . X2;m . ... . Xm;m represent the corresponding order statistics. The pdf of Xm;m i.e rth order statistics is given by
m! m.r
g(r:m)(x)= Gr.1(x)[1 . G(x)]g(x)
(r . 1)!(m . r)! m.r
m . r
(.1)lGr+l.1(x)g(x)
.
= Z
(27)
l
l=0
where Z = m! and g(x) and G(x) are pdf and cdf of CEL distribution de.ned in (7) and
(r.1)!(m.r)!
(8) respectively.
Substituting for G(x) and g(x) in (27) and applying the general binomial expansion, we have
m.rr+l.1
.2
x [x(. + 1)+ .(. + 2)] (x + . + 2)
m . r
(.1)l
.
g(r:m)(x)= Z
(. + 1)(x + .)2 (. + 1)(x + .)3
l m.r (r+l.1) l=0
2r+2l.k.2(x + . + 2)
m . rr + l . 1 x
.
.
= Z
Cl;k
(28)
(x + .)2r+2l+1
lk
k+1 k .2 .+2
l=0 k=0
where Cl;k =(.1)l.
(.+1).
Hence, the pdf of the minimum order statistic X(1) and maximum order statistic X(n) of the CEL distribution are respectively given by, respectively given by
m.1 l
m . 1 lx2l.k(x + . + 2)g(1:m)(x)= Z
.
.
Cl;k
(29)
(x + .)2l+3
lk
l=0 k=0
VII. Estimation of the Parameter of CEL Distribution
Suppose X =(X1, X2, X3, ..., Xn) be an independently and identically distributed (iid) random variables of size n with pdf (7) from CEL(.). Then, the likelihood function based on observed sample X =(x1, x2, x3, ..., xn) is de.ned as
n
.2
n
xi + . + 2
.
L(.; x)=
(30)
(xi + .)3
. + 1
i=0
The log-likelihood function corresponding to (30) is given by
n
.
log L = 2n log . . n log(. + 1)+ {log(xi + . + 2) . 3 log(xi + .)} (31)
i=0
Hence, the log-likelihood equation for estimating . is
n
2n 13
n
.
i=0
= 0 (32)
.
.
(. + 1)+ (xi + . + 2) (xi + .)
.
Above equation is not solvable analytically for .. Thus numerical iteration technique is used to get its numerical solution. Fisher Information matrix can be estimated by
..2
I(..)= log L
..2 .=..
.2
n
2n 31
n
.
i=0
log L = . (33)
.
+
(. + 1)2 +
..2 .2
(xi + .)2 (xi + . + 2)2
For large samples, we can obtain the con.dence intervals based on Fisher information matrix I.1(..) which provides the estimated asymptotic variance for the parameter .. Thus, a two-sided
v
100(1 . .)% con.dence interval of . and it is de.ned as ..� Z./2 var ... Where Z./2 denotes the upper .-th percentile of the standard normal distribution.
VIII. Simulation study
In this section we evaluate the performance of the MLEs of the model parameter for the CEL distribution. We generate random variables from CEL(.) and then obtain m.l.e. of the parameter ., Now for . = 1.5, 2, 2.5, 3 we generate the sample size 20, 30, 50, 90, 150, 200. The program is replicated N= 2,500 times to get the maximum likelihood estimate of .. The simulation results are reported in Table (1).
Table 1: Simulation results for different values of .
n Bias MSE Var. Est.
20 0.07273 0.25756 0.26034 1.57273 30 0.06864 0.16356 0.16447 1.56864 50 0.03938 0.09339 0.09211 1.53938
.=1.5
90 0.01632 0.04662 0.04854 1.51632
150 0.01270 0.02856 0.02858 1.51270
200 0.01126 0.02597 0.02152 1.51126
20 0.11554 0.53363 0.50319 2.11554 30 0.07354 0.29175 0.30196 2.07354 50 0.04340 0.16161 0.16965 2.04340
.=2 90 0.01612 0.08701 0.08982 2.01612
150 0.01415 0.05453 0.05321 2.01415
200 0.00896 0.03761 0.03949 2.00896
20 0.15021 0.82604 0.81882 2.65021 30 0.11545 0.50885 0.50463 2.61544 50 0.06234 0.27557 0.27921 2.56234
.=2.5
90 0.02329 0.14411 0.14678 2.52329
150 0.02114 0.08766 0.08675 2.52114
200 -0.00545 0.06452 0.06337 2.49455
20 0.21267 1.23999 1.25054 3.21267 30 0.16488 0.82261 0.77061 3.16488 50 0.09941 0.40727 0.42342 3.09941
.=2.5
90 0.06733 0.22374 0.22481 3.06733
150 0.03938 0.12947 0.13061 3.03938
200 0.03598 0.09335 0.09728 3.03598
It is clearly observed from the Table (1) that the values of bias and mean square error (MSE) of the parameter estimates decreases as the sample size n increases. It indicates the consistency of the estimator.
IX. Goodness of fit
The application of goodness of .t of proposed CEL distribution has been discussed with two real data sets. First data set presents the results of a life-test experiment in which specimens of a type of electrical insulating .uid were subject to a constant voltage stress (34 KV/minutes), this data set is reported by Nelson [22] and other data is represents 30 failure times of the air conditioning system of an airplane has been reported in a paper by Linhart & Zucchini [17] and has also been analyzed by Barreto-Souza & Bakouch [6] and so on. For comparing the suitability of the model, we have considered following criterion�s; namely AIC (Akaike Information Criterion), BIC (Bayesian information criterion), AICc (Corrected Akaike information criterion) and KS statistics with associated p-value of the .tted distributions are presented in Table (2) and Table (3).The AIC, BIC, AICc and KS Statistics are computed using the following formulae
AIC = .2loglik + 2k, BIC = .2loglik + k log n 2k2 + 2k
AICc = AIC + , D = sup |Fn(x) . F0(x)|
n . k . 1
x
where k= the number of parameters, n= the sample size, and the Fn(x)=empirical distribution function and F0(x) is the theoretical cumulativedistribution function.
Table 2: MLE�s, -2ln L, AIC, KS and p-values of the .tted distributions for the 1st dataset.
Distribution Estimate -2LL AIC BIC AICc KS p-value
CEL(.) 7.0385 137.98 139.98 140.92 140.21 0.1131 0.9458
EPL(., .) (0.0334, 0.5521) 136.18 140.18 142.06 140.93 0.1500 0.7312
EL(., p) (0.0393, 0.0982) 135.98 139.98 141.87 140.73 0.1382 0.8137
EP(., .) (0.0409, 2.2112) 136.89 140.89 142.78 141.64 0.1611 0.6497
Weibull(., .) (0.0818, 0.7708) 136.77 140.77 142.66 141.52 0.1613 0.6482
Gamma(., .) (0.0480, 0.6897) 137.23 141.23 143.12 141.98 0.1846 0.4802
Figure 5: Fitted pdfs of 1st data set
Figure 6: Fitted cdfs and ecdf of 1st data set Figure 7: p-p and q-q plot for the 1st data set.
Here we notice that all of the considered models .t the data at 5% level of signi.cance but the proposed distribution has minimum KS and maximum p-value among all the .tted models. Therefore, we may say that proposed CEL distribution is the most acceptable model for the present data set among the other considered models. For better visualization of the .tted models the estimated pdfs, cdfs, pp and qq plots are shown in Figure 5, Figure 6, Figure 7 for the .rst data set.
Table 3: MLE�s, -2ln L, AIC, KS and p-values of the .tted distributions for the 2nd dataset.
Distribution Estimate -2LL AIC BIC AICc KS p-value
CEL(.) 30.267 307.17 309.17 310.57 309.31 0.1061 0.8695
EPL(., .) (0.0101, 0.9193) 302.87 306.87 309.68 307.32 0.1282 0.7076
EL(., p) (0.0111, 0.1932) 302.83 306.83 309.63 307.28 0.1291 0.6986
EP(., .) (0.0105, 1.8243) 303.22 307.22 310.02 307.66 0.1468 0.5375
Weibull(., .) (0.0183, 0.8536) 307.87 310.68 308.32 303.87 0.1534 0.4806
Gamma(., .) (0.0136, 0.8119) 304.33 308.33 311.13 308.78 0.1694 0.3556
Figure 8: Fitted pdfs of 2nd data set
241
Figure 9: Fitted cdfs of 2nd data set
For the second data set also all the considered models .t well. Here also, the value of KS statistics is the minimum for CEL distribution with the maximum p-value. From the above discussion on two real data sets we see that all the considered six decreasing failure models .t to the two data sets. The .tted models the estimated pdfs, cdfs, pp and qq plots are shown in Figure 8, Figure 9, Figure 10 for the second data set.
X. Applications on infant mortality data
Since the CEL distribution is an early failure distribution then this may be suitable for the data of infant deaths. In this study an attemt has been made to apply CEL distribution for the data of infant deaths taken from the fourth round of National Family Health Survey (NFHS-4) for the most poupulous state of India i.e. Uttar Pradesh conducted in 2015-16 (IIPS and ICF, 2017)[14]. The data on infant deaths for four categories have been extacted and CEL distribution with other compitent distributions considered here have been applied. The .tting, estimate of parameters, KS distance and its p-value are provided in table 4-7. The p-value reveals that the CEL distribution is most appropriate among all considered distributions.
Table 4: Comperison of Goodness of .t for CEL, EPL, EP, EL, Weibull and Gamma on infant mortality data (Infant deaths of mothers aged 20-25)
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
Age at Infant Death Observed frequencies Expected frequencies of CEL Expected frequencies of EPL Expected frequencies of EP Expected frequencies of EL Expected frequencies of Weibull Expected frequencies of Gamma
0-1 104 90.61 83.12 80.71 76.30 72.18 143.00
1-2 17 30.67 33.66 36.74 34.29 36.55 16.75
2-3 2 15.03 17.46 18.91 19.43 21.51 4.39
3-4 10 98.83 10.35 10.69 12.09 13.18 1.28
4-5 5 5.79 6.66 6.49 7.90 8.26 0.39
5-6 7 4.07 4.53 4.16 5.32 5.26 0.12
6-7 7 3.02 3.19 2.78 3.64 3.39 0.04
7-8 2 2.33 2.31 1.92 2.52 2.21 0.01
8-9 3 1.85 1.71 1.36 1.76 1.45 0.00
9-10 4 1.50 1.28 0.98 1.24 0.95 0.00
10-11 2 1.24 0.97 0.71 0.87 0.63 0.00
11-12 3 1.05 0.74 0.53 0.62 0.42 0.00
Total 166 166.00 166.00 166.00 166.00 166.00 166.00
Estimates of parameter . = 1.4410 . = 0.6102 . = 0.2399 . = 2.4852 . = 0.2700 p = 0.2378 . = 0.3399 . = 0.8961 . = 0.5304 . = 0.4745 . = 0.9554
K-S Distance 0.0807 0.1257 0.1403 0.1668 0.1917 0.2478
p-value 0.2206 0.0094 0.0025 0.0002 0.0000 0.0000
Table 5: Comperison of Goodness of .t for CEL, EPL, EP, EL, Weibull and Gamma on infant mortality data (Infant deaths of mothers aged 25-30)
Age at Infant Death Observed frequencies Expected frequencies of CEL Expected frequencies of EPL Expected frequencies of EP Expected frequencies of EL Expected frequencies of Weibull Expected frequencies of Gamma
0-1 94 86.94 84.05 84.79 76.14 75.60 104.47
1-2 17 22.52 24.63 27.91 28.19 30.36 22.28
2-3 8 9.93 11.02 11.38 14.02 14.74 6.86
3-4 3 5.51 6.01 5.43 7.72 7.55 2.24
4-5 3 3.48 3.67 2.91 4.45 3.99 0.75
5-6 0 2.40 2.41 1.69 2.64 2.15 0.26
6-7 3 1.74 1.66 1.05 1.58 1.18 0.09
7-8 2 1.33 1.18 0.68 0.96 0.66 0.03
8-9 4 1.04 0.86 0.46 0.58 0.37 0.01
9-10 1 0.84 0.64 0.31 0.36 0.21 0.00
10-11 2 0.69 0.48 0.22 0.22 0.12 0.00
11-12 0 0.57 0.37 0.16 0.13 0.07 0.00
Total 137 137.00 137.00 137.00 137.00 137.00 137.00
Estimates of parameter . = 1.0624 . = 0.3689 . = 0.2355 . = 3.4829 . = 0.3033 p = 0.2501 . = 0.4879 . = 0.8868 . = 0.7795 . = 0.7081 . = 0.9833
K-S Distance 0.0515 0.0726 0.0672 0.1304 0.1342 0.1150
p-value 0.8509 0.4507 0.5508 0.0171 0.0128 0.0492
Table 6: Comperison of Goodness of .t for CEL, EPL, EP, EL, Weibull and Gamma on infant mortality data (Infant deaths in year 2003)
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
Age at Infant Death Observed frequencies Expected frequencies of CEL Expected frequencies of EPL Expected frequencies of EP Expected frequencies of EL Expected frequencies of Weibull Expected frequencies of Gamma
0-1 76 66.35 64.10 60.46 55.41 55.08 98.89
1-2 9 18.33 19.84 23.23 20.33 21.54 7.25
2-3 3 8.28 9.03 10.54 11.09 11.82 1.43
3-4 2 4.65 4.99 5.42 6.87 7.05 0.32
4-5 1 2.97 3.09 3.06 4.54 4.41 0.07
5-6 2 2.05 2.07 1.86 3.11 2.84 0.02
6-7 3 1.50 1.46 1.19 2.18 1.87 0.00
7-8 1 1.14 1.07 0.80 1.55 1.25 0.00
8-9 3 0.90 0.81 0.55 1.11 0.85 0.00
9-10 2 0.72 0.63 0.39 0.81 0.59 0.00
10-11 3 0.60 0.50 0.28 0.59 0.41 0.00
11-12 3 0.50 0.40 0.20 0.43 0.29 0.00
Total 108 108.00 108.00 108.00 108.00 108.00 108.00
Estimates of parameter . = 1.1394 . = 0.2528 . = 0.1512 . = 3.1272 . = 0.2785 p = 0.1183 . = 0.3040 . = 0.7880 . = 0.6433 . = 0.7081 . = 0.9833
K-S Distance 0.0894 0.1102 0.1439 0.1907 0.1937 0.2119
p-value 0.3392 0.1361 0.0204 0.0006 0.0005 0.0000
Table 7: Comperison of Goodness of .t for CEL, EPL, EP, EL, Weibull and Gamma on infant mortality data (Infant death in year 2004)
Age at Infant Death Observed frequencies Expected frequencies of CEL Expected frequencies of EPL Expected frequencies of EP Expected frequencies of EL Expected frequencies of Weibull Expected frequencies of Gamma
0-1 54 46.83 42.23 42.22 38.71 36.31 75.22
1-2 15 17.27 18.87 20.22 19.41 20.56 10.87
2-3 3 8.79 11.02 10.77 11.40 12.59 3.25
3-4 2 5.28 6.01 6.28 7.21 7.89 1.08
4-5 1 3.51 3.67 4.11 4.74 5.00 0.37
5-6 2 2.49 2.41 2.82 3.19 3.20 0.13
6-7 4 1.86 1.66 2.00 2.18 2.06 0.05
7-8 2 1.44 1.18 1.45 1.51 1.33 0.02
8-9 2 1.15 0.86 1.07 1.05 0.87 0.01
9-10 2 0.94 0.64 0.08 0.73 0.57 0.00
10-11 2 0.78 0.48 0.60 0.56 0.37 0.00
11-12 2 0.66 0.37 0.46 0.36 0.24 0.00
Total 91 91.00 91.00 91.00 91.00 91.00 91.00
Estimates of parameter . = 1.6062 . = 0.7679 . = 0.2496 . = 2.6031 . = 0.2382 p = 0.3311 . = 0.3477 . = 0.9427 . = 0.4853 . = 0.5032 . = 1.0726
K-S Distance 0.0788 0.1293 0.1294 0.1680 0.1943 0.2332
p-value 0.6064 0.0876 0.0871 0.0102 0.0017 0.0000
Brijesh P. Singh, Utpal Dhar Das & Sandeep Singh RT&A, No 2 (62)
A Compounded Probability Model for Decreasing Hazard.. Volume 16, June 2021
XI. Conclusions
A single parameter lifetime distribution CEL(.) has been introduced. The CEL(.) distribution is mean free distribution and has decreasing hazard. The moment generating function, rth oreder moments does not exists thus mean, variance, cumulant generating function, mean deviation about mean and median, Bonferroni, Gini index, mean residual life function (MRLF) also does not exists. The beauty of CEL distribution is that, this is a single parameter decreasing hazard distribution and explains the phenomenon better than other two parameter models. Although the moments do not exist, but Figure 1 indicates that, the distribution is highly positively skewed distribution. As the value of . is increasing the density of the distribution becomes .atten. Hence, we can easily conclude that the proposed CEL distribution may be considered as a suitable model for the case of decreasing failure rate scenario with a hope to get better model in various disciplines such as medical, engineering, and social sciences.
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