Estimation and optimal design of constant stress partially accelerated life test for Gompertz distribution with type i censoring Текст научной статьи по специальности «Строительство и архитектура»

ESTIMATION AND OPTIMAL DESIGN OF CONSTANT STRESS PARTIALLY ACCELERATED LIFE TEST FOR GOMPERTZ DISTRIBUTION WITH TYPE I CENSORING

Sadia Anwar, Arif Ul Islam •

Department of Statistics & Operations Research (Aligarh Muslim University, Aligarh-202002, India)

e-mail: [email protected]

ABSTRACT

This study deals with simple Constant Stress Partially Accelerated life test (CSPALT) using type-I censoring. The lifetime distribution of the test item is assumed to follow Gompertz distribution. The Maximum Likelihood (ML) Estimation is used to estimate the distribution parameters and acceleration factor. Asymptotic confidence interval estimates of the model parameters are also evaluated by using Fisher information matrix. Statistically optimal PALT plans are developed such that the Generalized Asymptotic Variance (GAV) of the Maximum Likelihood Estimators (MLEs) of the model parameters at design stress is minimized. In the last, to illustrate the statistical properties of the parameters, a simulation study is performed

KEYWORDS: Reliability; Partially Accelerated Life Tests; Acceleration factor; constant stress; maximum likelihood estimation; Fisher information matrix; generalized asymptotic variance; optimum test plans; time censoring

1 INTRODUCTION

The life test of the Products having high reliability under normal use conditions often requires a long period of time. In such problems, accelerated life tests (ALTs) are often used to quickly obtain information on the life time distribution of products by testing them at accelerated conditions than normal use conditions to induce early failures.

In ALT , the mathematical model relating to the lifetime of an item and stress is known or can be assumed.But,in some cases these relationships are not known and can not be assumed, i.e the data obtained from ALT can not be extrapolated to use condition.So, partially accelerated life test can be used in such cases in which the test items are run at both normal and higher than normal stress conditions. PALT can be carried out using constant-stress, step-stress, or Progressive-stress (linearly increasing stress). In constant stress PALT products are tested at either usual or higher than usual condition only until the test is terminated. The approach to accelerate failures is the step stress which increases the load applied to the products in a specified discrete sequence. A sample of test items is first run at use condition and, if it does not fail for a specified time, then it is run at accelerated condition until a pre specified numbers of failures are obtained or a pre specified time has reached.

There is an amount of literature on PALT which has been studied by many authors. Bai and Chung (1) discussed the optimall designing constant strss PALT or the test item having exponential distribution under Type I censoring.Bai et.al (2) discussed the PALT plan for lognormal distribution under time censored data.after that Bai et al (3) also considered the problemof failure-censored accelerated life-test sampling plans for lognormal and Weibull distributions. Abdel-Ghani (4) investigated some lifetime models under partially accelerated life tests. Ghaly et al. (5) discussed the PALT problem of parameter estimation for Pareto using Type I censoring and after that Ghaly et

al. (6) considered the same problem under Type II censoring. Ismail (7) used the maximum likelihood method to estimate the acceleration factor and parameters of the Pareto distribution under PALT. Ismail (8) discuss the constant stress PALT for the Weibull failure distribution under failure censored case. Ismail (9) considered the problem of optimally designing a simple time-step-stress PALT which terminates after a pre-specified number of failures and developed optimum test plans for products having a two-parameter Gompertz lifetime distribution. Zarrin et al. (10) considered constant stress PALT with type-I censoring. Assuming Rayleigh distribution as the underlying lifetime distribution, the MLEs of the distribution parameter and acceleration factor were obtained. More recent Saxena et al (11) consider the PALT design for extreme value distribution using type I censoring and Kamal et al. (12) discuss the same problem for Inverted Weibull distribution.

This work was conducted for constant-stress PALT under type II censored sample. the problems of estimation in constant stress PALT are considered under Rayleigh distribution. Maximum likelihood estimates and confidence intervals for parameters and acceleration factor are obtained.

2 THE MODEL AND TEST METHOD

2.1 The Gompertz Distribution

The lifetimes of the test items are assumed to follow a Gompertz distribution. The probability density function (pdf) of the Gompertz distribution is given by

f(t) = 0 eat exp ^ ^ ^ t > 0,6> 0, a> 0. (1)

where 0 is the scale parameter and a is the shape parameter of the distribution. And the cumulative distribution function is given by

F(t) = 1 - exp ^ 0a t > 0,0 > 0, a > 0. (2)

The reliability function of the Gompertz distribution is given by

R(t) = exp ^ ^ t > 0,0 > 0, a > 0. (3)

And the corresponding hazard rate is given by h(t ) = 0 e at

When a ^ 0, the Gompertz distribution will tend to an exponential distribution, see Wu et al. (13).

The two-parameter Gompertz model is a commonly used survival time distribution in actuarial science, reliability and life testing. There are several forms for the Gompertz distribution given in the literature. Some of these are given in Johnson et al. (14, 15).The pdf formula given in Equation (1) is the commonly used form and it is unimodal. It has positive skewness and an increasing hazard rate function.

2.2 Constant stress PALT procedure

i. Total n items are divided randomly into two samples of sizes n(1 -5)and ns respectively where 5 is sample proportion. First sample is allocated to normal use condition and other is assigned to accelerated conditions.

ii. Each test item of every sample is run until the censoring time x and the test condition is not changed.

2.3 Assumptions

i. The life time of the test product at use condition follows the Gompertz distribution given in (1).

ii. The life time of the test product at accelerated condition is obtained by using the relation X = P lT, where p > lis an acceleration factor. Therefore, the pdf at accelerated condition is given by equation (4) as follows

f (x) = 6p eaPx exp ^ ^^ x > 0,6 > 0, a > 0. (4)

iii. The lifetimes T , i = 1,2,............n(l-s)of items allocated to normal use condition, are i.i.d.

random variables.

iv. The lifetime Xj , j = 1,2,............ns of items allocated to accelerated condition, are i.i.d

random variables.

v. The lifetimes Ti and X j are mutually statistically-independent.

3 MAXIMUM LIKELIHOOD ESTIMATION

The maximum likelihood parameter estimation is used to determine the estimates of the parameter that maximizes the likelihood of the sample data. Also the MLEs have the desirable properties of being consistent and asymptotically normal for large samples

Since, the type-I censoring test terminates after a pre specified time is reached, so, the observed lifetimes t^ <........<t( )<xand t^ <........<)<xare ordered failure times at normal use and

accelerated conditions respectively, where x is the pre specified time at which the test is terminated, nu and na are the numbers of items failed at normal use and accelerated use conditions, respectively which are given by

n(l- s) ns

nu = Z5 u, and na =Z5 aj

i=1 j=1

and n + n = r, r is the total number of the failed items.

Let 5,„ and be the indicator functions such that

5,„ =

1

t, <x otherwise

i = 1,2,

(1-s)

And

5 aj=-

x. < x otherwise

j = 1,2,.

.ns

0

1

0

Then the likelihood function for (ti, 8ui), the likelihood function for X, 8aj)and the total

likelih°°d functi°n for 8u!,........, tn[l_s); 8un[i—s) , xi; 8 a1>.........., Xns i 8 ans )are respectively given by

, , n(i-s)

Lul(tl,8ui|a,0)= n ai expi-| - |[e" - 1Jf exp<

H «-iJl

(5)

, . ns Laj(Xj , 8 aj K 0)=n

p0e apj

exp ru,

ea^J -1

exp <

.!>1

(6)

, , n[i-s)

L(t, X a, p, 0)= n

i=1

0eat' expi-i-l[eat - 1J

exp <

—

a

l[e «- 1JI

11

p0eapj expJ-f0

eap^J -1

exp <

0L-1

where 8u, =1 -8u, and 8aj =1 -8 a Taking log of above equation

(7)

n[1-s)

l = ln L = T8

^^ u

ln9 +at, -gV - 1]j-Q|?1lz(1 -8„)

+ £8 a

i=1

a

i=1

ln0 + lnp + apxj-fi]eapXj -1] -fi][eapx -^(1 -8 J

j=1

dl dl dl MLEs of a, p and 0 are obtained by solving the equations — = 0,— = 0 and— = 0.

5a 5p 30

<L-L-1

50 = 0 a

(1-s)

n(1-s

- n + £ 8u,e ati + £8j apj + e <"{«(1 - s)-nu } + e ap^(ns - na ) i=1 j=1

(8)

(9)

£ = £ ufi +P^8

n(1-s)

5a

i=1

j=1

ajXj

a

a

n(1-s) ns

£ 8 ufie ati +p^8 jxe ap^j +xe <"{n(1 - s)-nu }+pxe ap^(ns - na ) i=1 j=1 _

n(1-s) ns

-n + £ 8uieati + £8 jjeapj + e<"{n(1 -s)-nu }+ eap^(ns -na) i=1 j=1

(10)

pi] yy ns ns

— = ^ + aT8ai.x, -eT8

dp p aJ j

ajXje

apj -0xeapx(ns -na )

j=1

j=1

(11)

It is difficult obtain a closed form solution to nonlinear equations to (9), (10) and (11). Newton-Raphson method is used to solve these equations simultaneously to obtain a, p and 9 . The asymptotic variance-covariance matrix of a, p and0 is obtained by numerically inverting the Fisher-information matrix composed of the negative second derivatives of the natural logarithm of the likelihood function evaluated at the ML estimates. The asymptotic Fisher-information matrix can be written as:

i=1

8

8

aj

aj

0

i=1

8

8

0

8

8

aj

aj

i=1

i=1

ns

0

+

5 2l r

56

2

6

2

521 6 a

5a

2

n(1-s)

Z 5 ut2 e ati +p 2 Z5 ajj apj +x 2 e ax{n(1-s)-nu }+P 2 x 2 e apx(ns-na )

i=1

26

29

07

n(1-s)

j=1

ns

Z 5 Ultle ati +PZ5 ajxje apj +xe ax{n(1-s)-nu }+Pxe a^{ns-na )

j =1

n(1 s)

Z 5uieati +Z5jafx + eax{n(1-s)-nu }+ eap^{ns-na)

j=1

2 apj

Cs 2 i ns

£1 = -naL-a6Z5a,xjeapj -a6x2eaPx(ns-na)

5p2 p2 Z j ( a)

521 1

565a

a

n(1-s)

-n + Z 5uieati +Z5japj + eax{n(1-s)-nu }+ eap^{ns-na)

j =1

n(1-s)

Z 5U1tteati +pZ5ajx]eapj +xeax{n(1-s)-nu }+pxeap^(ns-na )

i=1

j =1

5 2l

sesp

= -Z5 jjxje ^ +xe apx(ns-na )

j=1

5 21 ns

5a5p ^ aa

= Z5ajxj -6pZ5a^eaf&j -6px2eapx(ns-na)

j=1

The variance covariance and covariance matrix of the parameter can be written as

Z =

521 521 521

5a 2 5a5p 5a56

5 21 521 521

5p5a 5p2 5p56

521 521 521

565a 565p 562

AVar(a) ACov(ap) ACov(a6) ACov(p a ) AVar (p ) ACov(p 6) ACov(6(x) ACov(6p) AVar(6)

4 INTERVAL ESTIMATES FOR MODEL PARAMETER

To construct a confidence interval for a population parameter a, assume that

La = La(y1...........yn) and Ua = U afo..............yn) are functions of the sample data y................yn

then a confidence interval for a population parameter a is given by

P[La<a< U aM

(12)

ns

+

2

a

1=1

ns

i=1

ns

i=1

ns

ns

ns

-1

where the interval [La ,Ua ] is called a two sided ^100% confidence interval for a . La and Ua

are the lower and upper confidence limits for a, respectively.

For large sample size, the MLEs, under appropriate regularity conditions, are consistent and asymptotically normally distributed.

Therefore, the two sided approximate ^100% confidence limits for the MLE a of a population parameter a can be constructed, such that

P

G -G

- Z Z

«KG )

(13)

where z is the

10011 -

standard normal percentile.

Therefore, the two sided approximate ^100% confidence limits for a, p and 0 are given respectively as follows

La = « - )

Lß=ß - z«(jß ) La = 0 - z«(0

Ua=â + z«(a) U ß=ß + z«(j )

UQ =0 + z«(0

5 OPTIMUM SIMPLEC ONSTANT-STRESS TEST PLAN

In this section the problem of optimally designing a simple constant stress PALT, which terminates after a pre specified time is discussed. Optimum test plan for the items having Gompertz distribution is developed.

Most of the test plans allocate the same number of test units at each stress i.e. they are equally spaced test stresses. Such test plans are usually inefficient for estimating the mean life at design stress, see Yang (16). To decide the optimal sample proportion allocated to each stress, statistically optimum test plans are developed. Therefore, to determine the optimal sample proportion s* allocated to accelerated condition, s is chosen such that the GAV of the ML estimators of the model parameters is minimized. The GAV of the ML estimators of the model parameters as an optimality criterion is defined as the reciprocal of the determinant of the Fisher-Information matrix F (Bai, Kim and Chun [3]). That is

GAV\

J_ F

The minimization of the GAV over s solves the following equation

3GAV _ 5s

The solution to the above equation is not in the closed form, so the Newton- Raphson method is applied to determine s* which minimize the GAV. Accordingly, the corresponding expected optimal numbers of items failed at normal use and accelerated use conditions can be respectively as follows

* I-1 *

n. = n(1 - 5

(l-5*)Pu and n* = ns*p

where,

Pu = Probability that an item tested only use condition fails by x. Pa = Probability that an item tested only accelerated condition fails by x

th

2

6 SIMULATION STUDY

In order to obtain MLEs of p, a and 9 and to study the properties of these estimates through Mean squared errors (MSEs), variance of the estimators and confidence limits for 95% and 99% asymptotic confidence interval, a simulation study is performed. Furthermore, optimum test plans are developed.

For this purpose, several data sets generated from Gompertz distribution under type-I censored data are considered with sample sizes 100, 200, 300, 400 and 500 using 500 replications for each sample size. Under Type I censoring choose a proportion of sample units allocated to accelerated condition to be 5 = 30% and censoring time of a PALT to be x = 55 .The combinations (p, a, 9) of values of the parameters are chosen to be (1.4,0.4,3) and (1.2,0.6,5).Computer programs are prepared and the Newton-Raphson method is used for the practical application of the ML estimators of a, p and 9 . Table (1) and Table (3) give the MSE, variance of the estimators and the two sided approximate confidence limits at 95% and 99% level of significance. Tables (2) and (4) represent the results of the test design in which, the optimal sample-proportion 5 * allocated to accelerated use condition, the expected fraction failing at each stress, represented by n* and n* and

the optimal GAV of the MLEs of the model parameters are obtained numerically for each sample size.

Table 1: Simulation result for the parameters (p, a, 9) set as (1.4, 0.4, 3) respectively, given as 5=0.30 and x = 55 for different sized samples under type-I censoring in constant-stress PALT

n Para meters [p i a ,9 J MSE Variance 95% LCL UCL 99% LCL UCL

100 1.4921 0.5193 3.8653 0.0248 0.0433 0.0297 1.9972 0.9341 2.6411 1.1429 1.9733 0.1933 0.8153 2.4549 3.7177 1.1356 1.9334 0.1842 0.8615 2.4426 3.6961

200 1.4755 0.5014 3.6411 0.0212 0.0398 0.0172 1.6546 0.7857 2.1981 1.2684 1.9454 0.2408 0.6842 2.5113 3.6883 1.2569 1.9831 0.2371 0.6648 2.5049 3.6749

300 1.4592 0.4822 3.3983 0.0170 0.1669 0.0241 1.3427 0.3942 1.9428 1.3102 1.8931 0.3889 0.5978 2.6906 3.5917 1.2994 1.8912 0.3313 0.5410 2.6769 3.5236

400 1.4204 0.4185 3.2864 0.0043 0.0136 0.0109 1.2099 0.2082 1.1839 1.3863 1.7394 0.3862 0.4604 2.7694 3.3019 1.2312 1.7661 0.3042 0.4018 2.7526 3.3924

500 1.4112 0.4023 3.1482 0.0015 0.0049 0.0068 1.1478 0.2143 1.1017 1.3900 1.6876 0.3911 0.4395 2.8114 3.1198 1.3757 1.7019 0.3817 0.4962 2.8939 3.2991

Table 2: The results of optimal design of the life test for different sized samples under type-I

censoring in constant-stress PALT

n * * n* * n* Optimal GAV

100 0.3521 38 42 1.0941

200 0.3832 45 115 0.0834

300 0.4645 56 184 0.0210

400 0.5137 73 247 0.0026

500 0.5592 88 312 0.0019

Table 3: Simulation result for the parameters (j, a, 0) set as (1.2, 0.6, 5) respectively, given as 5=0.30 and x = 55 for different sized samples under type-I censoring in constant-stress PALT

n Para meters fj i a J0 J MSE Variance 95% LCL UCL 99% LCL UCL

100 1.3162 0.8673 7.8018 0.0639 0.0911 0.0315 3.5956 1.8960 1.8555 0.7998 1.8944 0.2674 0.9032 3.7793 7.4439 0.5783 1.7603 0.2386 0.8172 3.4491 6.9927

200 1.3096 0.7791 6.5409 0.0389 0.0617 0.0276 2.8720 0.8472 1.3879 1.0260 1.7503 0.4055 0.8541 4.6841 6.8617 1.0168 1.6118 0.4302 0.8017 3.5169 6.8005

300 1.2873 0.6371 5.7365 0.0406 0.0235 0.0227 1.5612 0.7395 0.7258 1.0704 1.5633 0.4559 0.8878 4.7018 6.5609 1.1027 1.4997 0.4399 0.7548 4.5918 6.7541

400 1.2415 0.6079 5.6671 0.0209 0.0195 0.0185 1.3291 0.2959 0.3940 1.1712 1.3093 0.5874 0.7040 4.5084 6.6012 1.1626 1.2963 0.5032 0.6817 4.7203 6.6019

500 1.2055 0.5909 5.2841 0.0186 0.0122 0.0096 1.1089 0.2468 0.1874 1.1831 1.2994 0.5937 0.6991 4.9163 6.8278 1.1291 1.3022 0.5135 0.6530 4.8021 5.9879

Table 4: The results of optimal design of the life test for different sized samples under type-I

censoring in constant-stress PALT

n * * n* * n* Optimal GAV

100 0.3988 36 44 3.0849

200 0.4503 62 88 1.8692

300 0.5844 86 144 0.0672

400 0.5994 98 262 0.0151

500 0.6348 104 316 0.0039

7 Conclusions

This study deals with the problem of estimation and optimally designing simple constant stress PALT for the Gompertz distribution under type-I censored data. From the table (1) and (2), it is observed that that the ML estimates approximate the true values of the parameters as the sample size n increases. Also, we find that, for a fixed a, p and 0 the mean squared errors and asymptotic variances of the estimators are decreasing with the increasing value of n. It is also noticed that when the sample size increases, the interval of the estimators are decreases.

Tables (2) and (4) present the optimal GAV of the ML estimators of the model parameters which is obtained numerically with in place of s for different sized samples. As expected, the optimal GAV decreases as the sample size n increases. The minimization of the GAV of the MLEs of model parameters was adopted as an optimality criterion. It may be concluded that the PALT model is an appropriate plan. In practice, the optimum test plans are important for improving the level of precision in parameter estimation and thus improving the quality of the inference. So, statistically, optimum plans are needed, and the experimenters are advised to use it for estimating the life distribution at design stress because it enables us to save time and money in a limited time without necessarily using a high stress to all test units.

As a result , it is right to say that the proposed model work well which helps to save time and money considerably without using a high stress to all test units.

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