GENERALIZED X-EXPONENTIAL BATHTUB SHAPED FAILURE RATE DISTRIBUTION AND ESTIMATION OF RELIABILITY OF MULTICOMPONENT STRESS-STRENGTH Текст научной статьи по специальности «Математика»

GENERALIZED X-EXPONENTIAL BATHTUB SHAPED FAILURE RATE DISTRIBUTION AND ESTIMATION OF RELIABILITY OF MULTICOMPONENT STRESS-

STRENGTH

Faryal Shabbir, Abdul Khalique

Department of Statistics National College of Business administration and Economics Lahore, Pakistan faryalshab4@gmail.com a.khalique57@gmail.com

Abstract

In an engineering setup, one is interested to know and determine the reliability of the system of different components. These components are usually subjected to different kinds of stress, and the reliability of the components needs to be estimated under stress. In this paper, we aim to estimate the reliability of a multicomponent stress-strength model assuming that the components of the system are working independently with a common life distribution. The system follows a comparatively new distribution named as; Generalized X-Exponential bathtub failure rate distribution. This paper studies the usefulness of this distribution in terms of estimating the maximum likelihood estimate of the reliability parameter and its asymptotic confidence intervals. Paper uses methods of parametric estimation and reliability estimation. Results are computed using Monte Carlo simulation for small samples. Real data set is presented to evaluate the performance of Generalized X Exponential Distribution (GXED) reliability estimator. Findings show that with the usage of proposed distribution, estimator of reliability parameter fits very well to the real-world situations

Key words: Generalized X -Exponential distribution, Multicomponent stress-strength, Reliability, ML estimation, Average variance, Confidence intervals.

I. Introduction

The X-Exponential distribution was introduced by Chacko [4], to add another model to the class of bathtub type failure rate distributions. When x is X-Exponential with parameters a and A. It has distribution function: F(x) = (l — (1 + Ax2)e(-Ax^)a with the corresponding density function: f(x) = ae-Xx(A2x2 — 2Xx + 1)(l — (1 + Ax2)e(-Xx^)a 1. Its properties and reliability applications were studied by the author. However, in order to get more flexibility to the model, Chacko and Deepthi [5] made a small change in the exponential part. The corresponding distribution is named as Generalized X-Exponential distribution. Basically, bathtub failure rate distribution's curve illustrates three phases of a product's life. First phase is known as early failure, next is a roughly prolonged intrinsic period and failure rate is approximately constant here. This stage is very important for reliability prediction of a product. And finally, there is a wear out failure phase, where failure rate increases. In the past several bathtub failure rate distributions have been studied by Kundu &Gupta, Srinivasa Rao [11] to carry out reliability testing by using single component stress

strength, as well as multi- component stress strength models. Since no substantial work has been done on reliability estimation of multicomponent stress strength by using a flexible distribution i.e., GXED, hence there was a need to study the reliability estimator of newly introduced Generalized X-

Exponential distribution having distribution function, F(x) = (l — (1 + Ax2)e-A(x2+X)) ,x> 0,A> 0 and a > 0.and the density function is:

f(x) = ae-X(x2+x)(X(l + Ax2)(2x + 1) - 2Ax)((l - (1 + Ax2)e-Ä(x2+x))a l; a>0,A>0

ae-A(x V(l+Ax2)(2x+l)-2Ax)((l-(l+Ax2)e-A(x ))

Failure rate=-—----- v \a -'-—; x> 0,a> 0,A> 0

1-(l-(1+XX2)e-A-(x2+x>)

(1)

(2)

The authors (Chacko and Deepthi) have investigated the properties and some reliability applications of the new model. Here we are interested in the reliability analysis of multicomponent system where the components are connected in parallel and function independently, with the same Generalized X -Exponential distribution GXED and stress too has the same distribution but with different parameters.

Let the random samples Y, XiX2X3i.....XK be independent, G(y) be the continuous

distribution function of Y, and F(x) be the common distribution function ofY,X1X2X3 .....XK.The

reliability in a multi component stress-strength model developed by Bhattacharyya and Johnson [2] is given by.

Rsk=P [at least s of the X1X2X3j.....XKexceed Y]

= Yk (*) /-Tt1 — F(y)Y [F(y)k-i]dG(y)

(3)

WhereX1,X2,X3,.....XKidentically and independently distributed (iid) are with common

distribution function F(x) and subjected to random stress Y. The probability in (3) is called 'Reliability in a multicomponent stress -strength model' Bhattacharyya and Johnson [2]. The survival probabilities of single component stress- strength version was considered by several authors for different distributions. Some of them are: Enis and Geisser [9], Downtown [8], Awad and Gharraf [1], McCool [18], Hanagal [12], Nandi and Aich [19], Surles and Padgett [27], Kundu and Gupta [15,16], Raqab et al. [26] and Kundu and Raqab [17]. More over Kotz & Pensky [14] studied the generalizations of stress strength model.

Reliability in a multicomponent stress-strength model was developed by Bhattacharyya and Johnson [2]. Pandey & Burhan [21] computed the estimation of reliability for a multicomponent model using Burr distribution. Zimmer et al [29] studied the reliability analysis for Burr X11 distribution. Estimation of reliability in models with correlated stress and strength has been studied by Balakrishnan &Lai [3]. Rao and Kantam [24] studied the estimation of reliability in a multicomponent stress- strength model for logistic distribution, Rao [23] also developed the procedure for the estimation of reliability in multicomponent stress-strength model based on Generalized exponential distribution. Ghitany et al. [10] studied the estimation of reliability of multicomponent model using Power Lindley distribution. Burr-X11 distribution for parametric and reliability estimation in a multicomponent stress-strength environment has been analyzed by Rao et al. [25]. Dey, S. et al [6] considered Bayesian and non-Bayesian estimation of multicomponent stress-strength reliability using Kumaraswami distribution.

Dey, Raheem & Mukherjee [7] derived the form of stress-strength reliability parameter for transmuted Rayleigh distribution. Hassan [13] developed the procedure for the estimation of stress-strength model using Lindley distribution. Estimation on Reliability in a multicomponent Stress-strength model with Power Lindley distribution is carried out by Abbas Pak et al [22]. Similarly, a recent study has been conducted on the estimation of stress strength reliability for Akash distribution by Akhila. K. Varghese & V. M. Chacko [28].

The aim of this paper is to estimate the reliability in a multi component stress-strength model based on X, Y being two independent random variables, where X~GXED, (a1,A) and Y~GXED (a2,A). We use parametric estimation and estimation reliability. Suppose a system with k identical components, functions if at least s (1 < s < k) components operate simultaneously. In its operating environment, the system is subjected to stress Y which is a random variable with distribution function G (.). The strengths of the components, that is the minimum stresses causing failure, are independently and identically distributed random variables with distribution function F(.).The reliability of the system can be obtained by (3). An attempt has been made here to study the estimation of reliability in a multicomponent stress-strength model with reference to two parameter GXED.

The remainder of the paper is organized as follows. In section 2, research methodology and procedure for expression of Rsk. The asymptotic distribution and confidence interval of (3) are calculated using MLE. The results of small sample comparisons derived from Monte Carlo simulations and analysis of real data sets are described in section 3. Findings are discussed in section 4.

2. Maximum Likelihood Estimator of Rs k

Let X~GXED (a1,A) and Y~GXED (a2,A) be independently distributed with unknown shape parameters (a2,A)while common scale parameter^. Using (3) the reliability in multicomponent stress-strength for two- parameter GXED distribution is as follows:

k

Rs,k =XCi)f [1- p(y)]i [p(y)k-i]dG(y)

i=s 0

F(y) = (l - (1 + Ax2)e-X(x2+X))a; x>0,a>0,A>0 1 - F(y) = 1 - (1 - (1 + Ax2)e-x(x2+x))a

dG(y) = ae -X(y2+y)(A(1 + Ay2)(2y + 1) - 2Ay)((1 - (1 + Ay2)e-A(y2+V))a-1 dy

where t = 1 - (1 - (1 + Ax2)e-x(y2+y))" and v=f-After simplification we get

Rs,k =YJki sQ) vB(i + 1,k-i + v) (4)

The probability in (4) is termed reliability in a multicomponent stress-strength model. It is important to mention here that MLE of Rskdepends on that of a1&a2. Hence, we need to calculate MLE of the latter to derive that of the former. Similarly, to find the MLE of a1&a2 and we need to

find the MLE of X as well. Here we assume that X1:X2, X3......Xn is a random sample from GXED

(a1, X) and Y1,Y,2Y3,.......Ym is a random sample from GXED (a2, X).

The loglikelihood function LLF of these samples is expressed as:

L(a1,a2,X) = mlna1 + nlna2 — (m + n)X(x2 + xt + yj + yj) +

m ln£(X(1 + Xx2)(2xL + 1) — 2Xxl ) + (a1 — 1)UHl — (1+ Xx?)e-A(x2 +x))) + n ln£(X(l + Xyj2)(2y1 + l) — 2Xyj) + (a2 — 1)E(ln(1 — (1 + Xy?)e-^2

(5)

Thus, the MLE of X is the solution of

дЮдЦд^Л) f 2 , , ,ym ((2xl+l)(l+2Äx^)—2xl)

6X = 0^ ¿i = l(*i + x0 +^i=l(Ä(1+ÄX2y2Xl+1)-2ÄXi) + (ai

1 Лут (l+Äx^e^t+^ixf+xQ-e^+^x? (л,2 + лЛ

((2У]+ф+2ЛУ1)-2У]) (i+^jyW+^J+^-e^+^f _

+ZJ = 1(Ä(l+Äyft)(2yj+l)-2Äyj)+ (a2 1)ZJ = 1 (iHi+xyjyK>f+^ -0

Similarly, the MLE of a1 can be obtained as the solution of

dlogL(ai,a2,X) _ n m

(6)

= 0 ^m + Im=l log(l - (1 + Xx2)e-Ä(xi+Xi)) = 0

da1 ~ a1 ' i 1

(7)

Also, for a2

ОЮдЦа^» _ n -^JL + Zn=i log (l -(l + Xyf)e—X(yJ+yj)) =

бъ =0 + Z?=l log(1-(1+ ty)'-™ +yJJ ) = 0

(8)

From (7) and (8) we obtain:

a^(X) =--,-—y- and a2(X) =

z£1l°g(l-(l+Äxt)e-*(xt+xl)) 2 ll1\og(l-(l+Äxt)e-i(v^+yl))

(9)

Putting the values of a^(X) and a2(X)into equation (6), we got a function of X which is nonlinear.

sh(X) = X (10)

Ux[ + 2Xxl±1 n 4Xyj + 2Xyj + 1

l=1 2Xxf + 2xf + 1 + 1=1 2Xyf + 2yf + 1 zm=1(x?+xi) + zn=1(yj+yj)+ , m ^„=1 Xj{1 + Xxx + X.x:2). e-A(xj+Xi^)

32

Y^i log (l - (1 + Xxf)e-X(xi+xl)) i=l (l-(1 + Xx2)e-Ä(xt+xl))

yj(l + Xyj3+Xyj2).e-Ä(y2+yJ)

ft y i \ -t. i yu y , i yu у ,

+--£П_ _il_il_

Y,J=llog(l-(l + Xyf)e-Ä(yJ+yj)) (1-(1 + Xyf)e-Ä(yi+yj))

__n_gm xl(l+Äx?+Äx2).e-Ä(xi +Xl))

Yj=1log(l-(l+lx2)e-X(xï+xl)) i=l (l-(l+xx2)e<x2+xl)) ( )

Here X"is a fixed-point solution of nonlinear equation (10). It can be obtained using a simple iterative procedure:

—n

hm = A(j + 1) (12)

Where Aj is the jth iteration of A1 .During the simulation process, when the difference between Aj and

A(j + 1) becomes sufficiently small; then we stop the iterative process. Once we obtain A , the parameters a" and a2can be obtained from (9) as respectively. To obtain the asymptotic confidence interval for Rsk we proceed as follows.

2.1 Asymptotic Variance and Confidence Intervals

22 V(a") = [E(-d2L/da2)]-1 = ^ and V(a") = [E(-d2L/da2^)]-1 = f- (13)

The asymptotic variance AV of an estimate of Rs,k which is a function of two independent statistics a", a" is established by Rao (1973):

AV(R",k) = V(a")(^)2 + V(a")(^)2 (14)

Thus from (14), asymptotic variance in Rsk can be obtained for GXED.

We obtain Rsk and their derivatives for (s, k) = (1, 3) and (2, 4) separately:

3v2 + 9v + 6 a 12(v2+3v + 2)

^ (v + 1)(v + 2)(v + 3) 24 (v + 1)(v + 2)(v + 3)(v + 4)

dR"3 _ 3v(v4 + 6v3 + 13v2 + 12v + 4) da1 = aJ(vTl)(vT2)(vT3)]2

dR"3 _ -3v(v4 + 6v3 + 13v2 + 12v + 4) da2 = a![(v+1:}(vT:Z}(vT:3y]2

dR"4 12v(2v5+ 19v4 + 68v3 + 115v2 +92v + 28)

and

da1 a1[(v + 1)(v + 2)(v + 3)(v + 4)]2

dR2fl _ -12(2vs + 19v4 + 68v3 + 115v2 + 92v + 28) da2 = a1[(v + 1)(v + 2)(v + 3)(v + 4)]2

Therefore as andm^œ, (R*k - Rsk)/AV(R2sk) N (0,1)

9v2(v4 + 6v3 + 13v2 + 12v + 4)2(1/m + 1/n)

AVtfu) = and AV (R2 4) =

1,3)~ [(v + 1)(v + 2)(v + 3)]4

144v2(2v5 + 19v4 + 68v3 + 115v2 + 92v + 28)2(1/m + 1/n) [(v + 1)(v + 2)(v + 3)(v + 4)]4

Where R"jk+ 1.96^AV(Rsk)is the asymptotic 95%confidence interval (C.I) of system reliability Rs k and asymptotic 95% C.I for Rh3 is given by:

A _ 3v(v4 + 6v3 + 13v2 + 12v + 4)^1/m+ 1/n R1,3 +196

and the asymptotic 95% confidence interval (C.I) for R24 is given by:

5 4 3 2

„ _ 12v(2v5 + 19v4 + 68v3 + 115v2 + 92v + 28)^1/m + 1/n R2,4 + 1-96 [(v + 1)(v + 2)(v + 3)(v + 4)]2

3. Simulation Study

3.1 Results

5000 random samples are generated each of size 10(5)30 from stress and strength populations for different values of a1 and a2: (2.0,2.5), (2.0,3.0), (2.0,3.5), (3.0,2.0), (3.0,2.5), (3.0,3.0). The MLE of scale parameter A is estimated by the iterative method and using A the shape parameters a1 and a2 are estimated from eq (8).

These ML estimators of a1 anda2are then substituted in V to obtain the multicomponent reliability for (s, k) = (1,3) and (2,4). The average bias and average MSE of reliability estimate over 5000 replications are presented in Table 1 and Table 2. Average length of confidence interval and coverage probability of the simulated 95% CIs of Rsk are given in Table 3 and Table 4. The true values of reliability in multicomponent stress -strength with given combinations of a", a" for (s, k) = (1,3) are 0.7058824, 0.6666667, 0.6315789, 0.8181074, 0.7826768, 0.75, 0.7142857 and for (s, k) = (2,4) are 0.5378151, 0.4848485, 0.4393593, 0.7011849, 0.6477772, 0.6, 0.5494505.

Here it is seen that the true value of reliability in multicomponent stress-strength decreases as a2 is increased for a fixed value of a1,whereas reliability in multicomponent stress-strength also decreases as a1is increased for a fixed value ofa2 . Thus, the true value of reliability increases as v decreases and vice versa.

Table 1: Average bias of the simulated estimates of Rsk(a1, a2)

s, k n, m 2.0,2.5 2.0,3.0 2.0,3.5 3.0,2.0 3.0,2.5 3.0,3.0

1,3 10,10 -.006581 -.0016484 -.007107 -.010848 -.017399 -.061099

15,15 -.006425 -.0041552 -.003801 -.005277 -.005233 -.056924

20,20 -.005644 -.0009021 -.002751 -.003377 -.003877 -.055159

25,25 -.004301 -.0035697 -.002016 -.003189 -.003199 -.055473

30,30 -.003075 -.0032642 -.001574 -.003726 -.002866 -.055240

2,4 10,10 -.003129 -.0009622 0.000633 -.011485 -.007666 -.008216

15,15 -.005138 -.0022865 -.001908 -.003719 -.008930 -.005695

20,20 0.000287 0.0005562 -.000367 -.006453 -005077 -.004446

25,25 -.000917 -.0003678 -.001444 -.001709 -.004725 -.005074

30,30 -.000523 -.0020488 -.003957 -.002097 -.004196 -.003842

Results of Table 1 and Table 2 depicts that average bias and MSE decrease as sample size increases for both the cases of estimation of reliability. Bias is negative in all the combinations of parameters in both situations of (s, k). This shows the consistency of MSE. Also, absolute bias increases as ^increases for a fixed value of a2.While MSE decreases as a1 increases for a fixed value of a2 for both the cases of (s, k). Also, for fixed a1 and increasing a2 MSE increases for same sample.

Table 2: Average MSE of the simulated estimates of Rsk(a1, a2)

s, k n, m 2.0,2.5 2.0,3.0 2.0,3.5 3.0,2.0 3.0,2.5 3.0,3.0

1,3 10,10 .008420 .008347 .0105324 .005115 .005915 .010999

15,15 .005872 .006666 .0071572 .002870 .004008 .008089

20,20. .0049068 .004907 .0054370 .002320 .003011 .006478

25,25 .0036479 .004291 .0045213 .001871 .002486 .005984

30,30 .002805 .003195 .0037037 .001478 .002033 .005489

2,4 10,10 .015654 .0154285 .0165471 .010602 .012716 .014210

15,15 .010693 .010985 .0111963 .004762 .008500 .009423

20,20. .0075629 .008428 .008489 .004305 .006470 .007210

25,25 .006857 .006696 .0069927 .004016 .004969 .005663

30,30 .005238 .005696 .0052105 .003354 .004067 .005053

Table 3: Average Length of the simulated 95% confidence intervals of Rsk(a1, a2)

s, k n, m 2.0,2.5 2.0,3.0 2.0,3.5 3.0,2.0 3.0,2.5 3.0,3.0

1,3 10,10 .350894 .378572 .390008 .263240 .299103 .322061

15,15 .290262 .311732 .323802 .214849 .242167 0.26764

20,20. .253883 .272129 .323815 .186494 .210426 .230430

25,25 .228945 .243448 .254189 .166324 .188198 .206166

30,30 .208672 .223215 .232022 .150368 .170951 .189016

2,4 10,10 .475256 .483035 .485837 .392269 .428761 .453479

15,15 .395910 .404009 .405260 .322269 .351963 .373558

20,20. .346274 .357539 .354290 .280074 .308866 .327652

25,25 .309512 .318022 .318480 .250861 .275602 .291288

30,30 .285038 .291335 .292680 .230350 .252819 .269990

Table 3 and Table 4 findings show that as the sample size increases, length of CI also decreases and coverage probability in most the cases crossing 0.95 and for few it is 0.98, which shows the performance of CI using Generalized X- Exponential Distribution GXED is excellent and it covers most of the cases. Among the parameters, it is observed that length of CI increases for fixed value of a1 for (1,3) while for fixed value of a2 length of CI decreases.

Table 4: Average Coverage Probability of simulated 95% confidence intervals of Rsk(a1, a2)

s, k n, m 2.0,2.5 2.0,3.0 2.0,3.5 3.0,2.0 3.0,2.5 3.0,3.0

1,3 10,10 .891333 .968000 .936667 .910667 .969333 .987333

15,15 .972667 .944444 .880000 .950000 .925084 .905333

20,20. .905333 .980810 .914000 ..968667 .912052 .956667

25,25 .951333 .912300 .949333 .969333 .946000 .966667

30,30 .953815 .965333 .926000 .952667 .896360 .936667

2,4 10,10 .964667 .957333 .9743178 .984000 .934667 .966677

15,15 .962667 .966600 .953000 .970000 .942000 .967333

20,20. .963333 .955746 .952667 .969425 .954000 .953333

25,25 .946000 .936667 .953333 .983333 .970883 .948007

30,30 .902000 .946666 .937333 .956000 .960667 .960667

3.2 Data Analysis

In this section, we will deal with two real data sets, will show how reliability in a multicomponent stress-strength model can be applied for GXED. Both data sets were discussed by Zimmer et al. (1998) and Lio et al. (2010) for Burr-X11 reliability analysis. They showed that Burr-X11 distribution fits quite well. For both the data sets, here we are using GXED.

(X):0.19 ,0.78, 0.96, 0.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71 and 72.89

(Y):0.9, 1.5, 2.3, 3.2, 3.9, 5.0, 6.2, 7.5, 8.3, 10.4, 11.1, 12.6, 15.0, 16.3, 19.3, 22.6,

24.8, 31.5 And 53.0. Iterative procedure was used to calculate the value of A using (8) and then a1 and a2 were obtained by substituting the MLE of A in (10).

The final estimates of a1 = 0.844798, a2 = 1.551717 and A= 0.04642891. Based on these estimates the MLE of Rlj3 turned out to be 0.620246 and 95% CI (.4704636, .770028) while for R2j4, came out to be 0.4250596; CI (.2752773,0.5748419).

4. Discussion

In this paper, we analyzed the behavior of Generalized X-Exponential Distribution (GXED) in calculating the multicomponent stress-strength reliability estimates. We also calculated 95% CI & coverage probability for reliability estimates and results were excellent. Coverage probability touched up to 0.98, which shows GXED estimates, very accurately.

The simulation results indicated that average bias and MSE decreased as the sample size increased for both the cases of Rsk .The real data sets also revealed GXED fits very well and provides quite close results. Hence, GXED can be used readily to calculate the reliability in a multicomponent stress- strength environment.

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