In the present paper, a lifetime distribution named, “Ishita distribution” for modeling lifetime data from biomedical science and engineering has been proposed. Statistical properties of the distribution including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability have been discussed. The condition under which Ishita distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along with the conditions under which Akash distribution, introduced by Shanker,1 Lindley distribution, introduced by Lindley2 and exponential distribution are over-dispersed, equi-dispersed and under-dispersed. Method of maximum likelihood estimation and method of moments have been discussed for estimating the parameter of the proposed distribution. Finally, the goodness of fit of the proposed distribution have been discussed and illustrated with two real lifetime data sets and the fit has been compared with exponential, Lindley and Akash distributions.
Keywords: akash distribution, lindley distribution, moments, dispersion, hazard rate function, mean residual life function, mean deviations, order statistics, stressstrength reliability, estimation of parameter, goodness of fit
The analyzing and modeling real lifetime data are crucial in many applied sciences including medicine, engineering, insurance and finance, amongst others. The two important one parameter lifetime distributions namely exponential and Lindley2 are popular for modeling lifetime data from biomedical science and engineering. Recently, Shanker et al.3 have conducted a comparative and critical study on the modeling of lifetime data from biomedical science and engineering using exponential and Lindley distributions and observed that there are many lifetime data where these two distributions are not suitable due to their shapes, nature of hazard rate functions, and mean residual life, amongst others. While searching a lifetime distribution which gives better fit than exponential and Lindley, Shanker1 has introduced a lifetime distribution named Akash distribution and showed that Akash distribution gives much better fit than both exponential and Lindley distributions. Shanker et al.4 have comparative study on the modeling of lifetime data using Akash, Lindley and exponential distribution and observed that there are several situations where these lifetime distributions are not suitable either from theoretical or applied point of view. Therefore, an attempt has been made in this paper to obtain a new lifetime distribution which is flexible than Akash, Lindley and exponential distributions for modeling lifetime data in reliability and in terms of its hazard rate shapes. The new one parameter lifetime distribution is based on a two- component mixture of an exponential distribution having scale parameter θ and a gamma distribution having shape parameter 3 and scale parameter θ with their mixing proportion θ3θ3+2 .
Lindley distribution, introduced by Lindley2 has been defined by the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) as
f1(x;θ)=θ2θ+1(1+x)e−θx ;x>0, θ>0 (1.1)
F1(x;θ)=1−[1+θxθ+1]e−θx ;x>0, θ>0 (1.2)
It can be easily verified that the density (1.1) is a two-component mixture of an exponential (θ) distribution and a gamma (2,θ) with their mixing proportion θθ+1 . Recent years much works have been done on Lindley distribution , its generalization and mixture with other distributions by several authors including Ghitany et al ,5 Zakerzadeh and Dolati,6 Mazucheli and Achcar,7 Bakouch et al,8 Shanker and Mishra9,10 Shanker et al,11 Shanker and Amanuel,12 Sankaran,13 are some among others.
Akash distribution, introduced by Shanker1 has been defined by the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) as
f2(x;θ)=θ3θ2+2(1+x2)e−θx ;x>0, θ>0 (1.3)
F2(x,θ)=1−[1+θx(θx+2)θ2+2]e−θx ;x>0,θ>0 (1.4)
It can be easily verified that the Akash distribution is a two-component mixture of exponential (θ) distribution and a gamma (3,θ) distribution with mixing proportion θ2θ2+2 . Shanker14 has obtained a Poisson mixture of Akash distribution named, “Poisson-Akash distribution (PAD) and discussed its properties, estimation of parameter and applications. Shanker et al.15 have detailed study on modeling of count data from different fields of knowledge using Poisson-Akash distribution. Shanker and Shukla16 have obtained weighted Akash distribution and studied its statistical and mathematical properties, estimation of parameters and applications to model lifetime data. Shanker17 has also obtained a quasi Akash distribution, studied its mathematical and statistical properties, estimation of parameters using both maximum likelihood estimation and method of moments and applications to model lifetime data.
The new one parameter lifetime distribution has been defined by its probability density function (p.d.f.)
f3(x;θ)=θ3θ3+2(θ+x2)e−θx ;x>0, θ>0 (1.5)
We would name this probability density function as, “Ishita distribution”. It can be easily verified that the Ishita distribution is a two-component mixture of exponential (θ) distribution and a gamma (3,θ) distribution with mixing proportion θ3θ3+2 .
The corresponding cumulative distribution function (c.d.f.) of (1.5) can be obtained as
F3(x,θ)=1−[1+θx(θx+2)θ3+2]e−θx ;x>0,θ>0 (1.6)
The graph of the p.d.f. and the c.d.f. of Ishita distribution for varying values of the parameter θ are shown in figures 1 and 2.
The moment generating function of Ishita distribution (1.5) can be obtained as
MX(t)=θ3θ3+2∞∫0e−(θ−t) x(θ+x2) dx
=θ3θ3+2[θθ−t+2(θ−t)3]
=θ3θ3+2[∞∑k=0(tθ)k+2θ3∞∑k=0(k+2k)(tθ)k]
=∞∑k=0θ3+(k+1)(k+2)θ3+2(tθ)k
The r th moment about origin of Ishita distributon (1.5) is given by
μr′=r![θ3+(r+1)(r+2)]θr(θ3+2) ;r=1,2,3,... (2.1)
The first four moments about origin are thus obtained as
μ1′=θ3+6θ(θ3+2) , μ2′=2(θ3+12)θ2(θ3+2)
μ3′=6(θ3+20)θ3(θ3+2) , μ4′=24(θ3+30)θ4(θ3+2)
Using relationship between moments about mean and the moments about origin, the moments about mean of Ishita distribution (1.5) can be obtained as
μ2=θ6+16θ3+12θ2(θ3+2)2
μ3=2(θ9+30θ6+36θ3+24)θ3(θ3+2)3
μ4=3(3θ12+128θ9+408θ6+576θ3+240)θ4(θ3+2)4
The coefficient of variation (C.V) , coefficient of skewness (√β1) , coefficient of kurtosis (β2) and index of dispersion (γ) of Ishita distribution (1.5) are thus obtained as
C.V=σμ1′=√θ6+16θ3+12θ3+6
√β1=μ3μ23/2=2(θ9+30θ6+36θ3+24)(θ6+16θ3+12)3/2
β2=μ4μ22=3(3θ12+128θ9+408θ6+576θ3+240)(θ6+16θ3+12)2
γ=σ2μ1′=θ6+16θ3+12θ(θ3+2)(θ3+6)
The over-dispersion, equi-dispersion, and under-dispersion of Ishita distribution has been presented in table 1 along with Akash, Lindley and exponential distributions.
Lifetime Distributions |
Over-Dispersion (μ<σ2) |
Equi-Dispersion |
Under-Dispersion |
Ishita |
θ<1.535653152 |
θ=1.535653152 |
θ>1.535653152 |
Akash |
θ<1.515400063 |
θ=1.515400063 |
θ>1.515400063 |
Lindley |
θ<1.170086487 |
θ=1.170086487 |
θ>1.170086487 |
Exponential |
θ<1 |
θ=1 |
θ>1 |
Table 1 Over-dispersion, equi-dispersion and under-dispersion of Ishita, Akash, Lindley and exponential distributions for the parameter θ
Let X be a continuous random variable with p.d.f. f(x) and c.d.f. F(x) . The hazard rate function (also known as the failure rate function) h(x) and the mean residual life function m(x) of X are respectively defined as
h(x)=limΔx→0P(X<x+Δx|X>x)Δx=f(x)1−F(x) (3.1)
and
m(x)=E[X−x|X>x] = 11−F(x)∫∞x[1−F(t)] dt (3.2)
The corresponding hazard rate function, h(x) and the mean residual life function, m(x) of the Ishita distribution (1.5) are thus obtained as
h(x)=θ3(θ+x2)θx(θx+2)+(θ3+2) (3.3)
and m(x)=θ3+2[θx(θx+2)+(θ3+2)]e−θx∞∫x[θt(θt+2)+(θ3+2)θ3+2]e−θtdt
=θ2x2+4θx+(θ3+6)θ[θx(θx+2)+(θ3+2)] (3.4)
It can be easily verified that h(0)=θ3θ3+2=f(0) and m(0)=θ3+6θ(θ3+2)=μ1′ .It is also obvious from the graphs of h(x) and m(x) that h(x) is an increasing function of (x and θ≤1) and (x≥1 and θ>1) and decreasing function for x and θ≤1 and for x≥1 and θ>1 and decreasing function for 0<x<1 and θ>1 , where as m(x) is a decreasing function of x and θ . . The graph of the hazard rate function and mean residual life function of Ishita distribution (1.5) are shown in figures 3&4.
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
The following results due to Shaked and Shanthikumar18 are well known for establishing stochastic ordering of distributions
X≤lrY⇒X≤hrY⇒X≤mrlY (4.1)
⇓X≤stY
The Ishita distribution is ordered with respect to the strongest ‘likelihood ratio’ ordering as shown in the following theorem:
Theorem
Let X ∼ Ishita distributon (θ1) and Y ∼ Ishita distribution (θ2) . If θ1>θ2 , then X≤lrY and hence X≤hrY , X≤mrlY and X≤stY .
Proof
We have
fX(x,θ)fY(x,θ)=θ13(θ23+2)θ23(θ13+2)(θ1+x2θ2+x2)e−(θ1−θ2)x ; x>0
Now
lnfX(x,θ)fY(x,θ)=ln[θ13(θ23+2)θ23(θ13+2)]+ln(θ1+x2θ2+x2)−(θ1−θ2)x
This gives ddxlnfX(x,θ)fY(x,θ)=− 2(θ1−θ2)(θ1+x2)(θ2+x2) − (θ1−θ2)
Thus for θ1>θ2 , ddxlnfX(x,θ)fY(x,θ)<0 . This means that X≤lrY and hence X≤hrY , X≤mrlY and X≤stY .Generally the amount of scatter in a population is measured to some extent by the totality of deviations usually from their mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined as
δ1(X)=∞∫0|x−μ| f(x)dx and δ2(X)=∞∫0|x−M| f(x)dx , respectively, where μ=E(X) and M=Median (X) . The measures δ1(X) and δ2(X) can be calculated using the following relationships
δ1(X)=μ∫0(μ−x)f(x)dx+∞∫μ(x−μ)f(x)dxa
=μF(μ)−μ∫0x f(x)dx−μ[1−F(μ)]+∞∫μx f(x)dx
=2μF(μ)−2μ+2∞∫μx f(x)dx
=2μF(μ)−2μ∫0x f(x)dx (5.1)
and
δ2(X)=M∫0(M−x)f(x)dx+∞∫M(x−M)f(x)dx
=M F(M)−M∫0x f(x)dx−M[1−F(M)]+∞∫Mx f(x)dx
=−μ+2∞∫Mx f(x)dx
=μ−2M∫0x f(x)dx (5.2)
Using p.d.f. (1.5) and expression for the mean of Ishita distribution, we get
μ∫0x f3(x,θ)dx=μ−{θ4μ+θ2(μ3+1)+3θ2μ2+6 (θ μ+1)}e−θ μθ(θ3+2) (5.3)
M∫0x f3(x,θ)dx=μ−{θ4M+θ3(M3+1)+3 θ2M2+6 (θM+1)}e−θ Mθ(θ3+2) (5.4)
Using expressions from (3.1), (3.2), (3.3), and (3.4), the mean deviation about mean, δ1(X) and the mean deviation about median, δ2(X) of Ishita distribution are obtained as
δ1(X)=2{θ2μ2+4θ μ+(θ3+6)}e−θ μθ(θ3+2) (5.5)
δ2(X)=2{θ4M+θ3(M3+1)+3 θ2M2+6 (θM+1)}e−θ Mθ(θ3+2)−μ (5.6)
Let X1, X2, ..., Xn be a random sample of size n from Ishita distribution (3.5). Let X(1)<X(2)< ... <X(n) denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k th order statistic, say Y=X(k) are given by
fY(y)=n!(k−1)! (n−k)! [F(y)]k−1[1−F(y)]n−kf(y)
=n!(k−1)! (n−k)! n−k∑l=0(n−kl)(−1)l[F(y)]k+l−1f(y)
and
FY(y)=n∑j=k(nj) [F(y)]j[1−F(y)]n−j
=n∑j=kn−j∑l=0(nj)(n−jl) (−1)l[F(y)]j+l ,
respectively, for k=1,2,3,...,n .Thus, the p.d.f. and the c.d.f of k th order statistic of Ishita distribution (3.5) are given by
fY(y)=n!θ3(θ+x2)e−θx(θ3+2)(k−1)! (n−k)! n−k∑l=0(n−kl)×[1−θx(θx+2)+(θ3+2)θ3+2e−θx]k+l−1
and
FY(y)=n∑j=kn−j∑l=0(nj)(n−jl) (−1)l[1−θx(θx+2)+(θ3+2)θ3+2e−θx]j+l
The Bonferroni and Lorenz curves Bonferroni19 and Bonferroni and Gini indices 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
B(p)=1pμq∫0x f(x) dx=1pμ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1pμ[μ−∞∫qx f(x) dx] (7.1)
and L(p)=1μq∫0x f(x) dx=1μ[∞∫0x f(x)dx−∞∫qx f(x) dx]=1μ[μ−∞∫qx f(x) dx] (7.2)
respectively or equivalently
< B(p)=1pμp∫0F−1(x) dx (7.3)
and L(p)=1μp∫0F−1(x) dx (7.4)
respectively, where μ=E(X) and q=F−1(p) .
The Bonferroni and Gini indices are thus defined as
B=1−1∫0B(p) dp (7.5)
And G=1−21∫0L(p) dp (7.6)
respectively.
Using p.d.f. (1.5), we get
∞∫qx f3(x,θ) dx={θ4q+θ3(q3+1)+3θ2q2+6(θq+1)}e−θqθ(θ3+2) (7.7)
Now using equation (5.7) in (5.1) and (5.2), we get
B(p)=1p[1−{θ4q+θ3(q3+1)+3θ2q2+6(θq+1)}e−θqθ3+6] (7.8)
and L(p)=1−{θ4q+θ3(q3+1)+3θ2q2+6(θq+1)}e−θqθ3+6 (7.9)
Now using equations (5.8) and (5.9) in (5.5) and (5.6), the Bonferroni and Gini indices of Ishita distribution (1.5) are obtained as
B=1−{θ4q+θ3(q3+1)+3θ2q2+6(θq+1)}e−θqθ3+6 (7.10)
G=2{θ4q+θ3(q3+1)+3θ2q2+6(θq+1)}e−θqθ2+6 − 1 (7.11)
An entropy of a random variable X is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy.20 If X is a continuous random variable having probability density function f(.) , then Renyi entropy is defined as
TR(γ)=11−γlog{∫fγ(x)dx}
where γ>0 and γ≠1 .
Thus, the Renyi entropy for the Ishita distribution (3.5) can be obtained as
TR(γ)=11−γlog[∞∫0θ3γ(θ3+2)γ(θ+x2)γe−θ γ xdx]
=11−γlog[∞∫0θ3γ(θ3+2)γθγ(1+x2θ)γe−θ γ xdx]
=11−γlog[∞∫0θ4γ(θ3+2)γ∞∑j=0(γj)(x2θ)je−θ γ xdx]
=11−γlog[∞∫0θ4γ−j(θ3+2)γ∞∑j=0(γj) x2je−θ γ xdx]
=11−γlog[∞∑j=0(γj)θ4γ−j(θ3+2)γ∞∫0e−θ γ xx2j+1−1dx]
=11−γlog[∞∑j=0(γj)θ4γ−j(θ3+2)γΓ(2j+1)(θγ)2j+1]
=11−γlog[∞∑j=0(γj)θ4γ−3j−1(θ3+2)γΓ(2j+1)(γ)2j+1]
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 statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.
Let X and Y be independent strength and stress random variables having Ishita distribution (3.5) with parameter θ1 and θ2 respectively. Then the stress-strength reliability R of Ishita distribution can be obtained as
R=P(Y<X)=∞∫0P(Y<X|X=x)fX(x)dx
=∞∫0f3(x,θ1) F3(x,θ2)dx
=1−θ13[24θ22+12θ2(θ1+θ2)+2(θ23+θ1θ22+2)(θ1+θ2)2+2θ1θ2(θ1+θ2)3+θ1(θ23+2)(θ1+θ2)4](θ13+2)(θ23+2)(θ1+θ2)5 .
Maximum likelihood estimate (MLE)
Let
(x1, x2, x3, ... ,xn)
be a random sample from Ishita distribution (1.5). The likelihood function,
L
of (1.5) is given by
L=(θ3θ3+2)nn∏i=1(θ+xi2) e−n θ ˉx
The natural log likelihood function is thus obtained as
lnL=nln(θ3θ3+2)+n∑i=1ln(θ+xi2)−n θ ˉx
Now dlnLdθ=6nθ(θ3+2)+n∑i=11θ+xi2−n ˉx
where ˉx is the sample mean.
The maximum likelihood estimate, ˆθ of θ is the solution of the equation dlnLdθ=0 and it can be obtained by solving the following non-linear equation
6nθ(θ3+2)+n∑i=11θ+xi2−n ˉx=0 .
Method of moment estimate (MOME)
Equating the population mean of the Ishita distribution (1.5) to the corresponding sample mean, the method of moment estimate (MOME) ˜θ of θ is the solution of the following non-linear equation
ˉx θ4−θ3+2θˉx−6=0 , where ˉx is the sample mean.
The goodness of fit of Ishita distribution has been done on several lifetime data sets. In this section, we present the goodness of fit of Ishita distribution using maximum likelihood estimate of the parameter on two data sets and the fit has been compared with Akash, Lindley and exponential distributions. For testing the goodness of fit of Ishita distribution over exponential, Lindley and Akash distributions, following two data sets have been considered.
Data set 1: The second data set is the strength data of glass of the aircraft window reported by Fuller et al.21
18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52,
25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73,
33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045,
45.29, 45.381
Data Set 2: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader and Priest22
1.312 |
1.314 |
1.479 |
1.552 |
1.7 |
1.803 |
1.861 |
1.865 |
1.944 |
1.958 |
1.966 |
1.997 |
2.006 |
2.021 |
2.027 |
2.055 |
2.063 |
2.098 |
2.14 |
2.179 |
2.224 |
2.24 |
2.253 |
2.27 |
2.272 |
|
2.274 |
2.301 |
2.301 |
2.359 |
2.382 |
2.382 |
2.426 |
2.434 |
2.435 |
2.478 |
2.49 |
2.511 |
|
2.514 |
2.535 |
2.554 |
2.566 |
2.57 |
2.586 |
2.629 |
2.633 |
2.642 |
2.648 |
2.684 |
2.697 |
|
2.726 |
2.77 |
2.773 |
2.8 |
2.809 |
2.818 |
2.821 |
2.848 |
2.88 |
2.954 |
3.012 |
3.067 |
|
3.084 |
3.09 |
3.096 |
3.128 |
3.233 |
3.433 |
3.585 |
3.585 |
In order to compare Ishita, Akash, Lindley and exponential distributions, values of −2lnL , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) for two real data sets have been computed and presented in table 2. The formulae for computing AIC, AICC, BIC, and K-S Statistic are as follows:
AIC=−2lnL+2k , AICC=AIC+2k(k+1)(n−k−1) , BIC=−2lnL+kln n and
K-S =Supx|Fn(x)−F0(x)| , where k = the number of parameters, n = the sample size and Fn(x) is the empirical distribution function.
The best distribution corresponds to lower values of −2lnL , AIC, AICC, BIC, and K-S statistic.
It can be easily seen from above table that Ishita distribution gives better fit than exponential, Lindley and Akash distribution and hence Ishita distribution should be preferred to exponential, Lindley and Akash distributions for modeling lifetime data from biomedical science and engineering.
|
Distributions |
MLE of ˆθ |
S.E(ˆθ) |
−2lnL |
AIC |
AICC |
BIC |
K-S |
Data 1 |
Ishita |
0.0973 |
0.0100 |
240.48 |
242.48 |
242.62 |
243.91 |
0.297 |
Akash |
0.0971 |
0.0101 |
240.68 |
242.68 |
242.82 |
244.11 |
0.298 |
|
Lindley |
0.0630 |
0.0080 |
253.98 |
255.98 |
256.12 |
257.41 |
0.365 |
|
Exponential |
0.0324 |
0.0058 |
274.52 |
276.52 |
276.66 |
277.95 |
0.458 |
|
Data 2 |
Ishita |
0.9315 |
0.0560 |
223.14 |
225.14 |
225.20 |
227.37 |
0.331 |
Akash |
0.9647 |
0.0646 |
224.27 |
226.27 |
226.33 |
228.50 |
0.362 |
|
Lindley |
0.6545 |
0.0580 |
238.38 |
240.38 |
240.44 |
242.61 |
0.401 |
|
Exponential |
0.4079 |
0.0491 |
261.73 |
263.73 |
263.79 |
265.96 |
0.448 |
Table 2 MLE,s of ˆθ , S.E. (ˆθ) , -2ln L, AIC, AICC, BIC, and K-S Statistic of the fitted distributions of data set 1 and 2
A lifetime distribution named, “Ishita distributions” for modeling lifetime data from biomedical science and engineering has been proposed and its various statistical and mathematical properties including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability have been studied. The conditions of over-dispersed, equi-dispersed, and under-dispersed of Ishita distribution has been presented along with Akash, Lindley and exponential distributions. The estimation of parameter has been discussed using both maximum likelihood estimation and method of moments. The goodness of fit of Ishita distribution has been discussed and illustrated with two real lifetime data sets and it has been shown that it gives better fit than exponential, Lindley and Akash distributions.
NOTE: The paper is named Ishita distribution in the name of Ishita Shukla, a lovely daughter of second author Dr. Kamlesh Kumar Shukla, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea.
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