Analysis method of pooled data for accelerated life testing

PERGAMON

Microelectronics Reliability 38 (1998) 1931±1934

Research Note

Analysis method of pooled data for accelerated life testing T. Seki a, *, S. Yokoyama b a

Department of Project Management, Chiba Institute of Technology 2-17-1, Tsudanuma, Narashino-shi, Chiba 275-0016, Japan Department of Industrial Engineering, Musashi Institute of Technology 1-28-1, Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan

b

Received 11 March 1997; in revised form 17 August 1998

Abstract This paper deals with the analysis method of pooled data for accelerated life testing. The moment estimators of the characteristics of life which take weights of sample size of each stress levels into consideration is proposed. The validity of this method is examined graphically through the example of the actual accelerated life test data. This consideration makes it clear that the proposed estimators ®t in better with the data than the estimators by the least squares method. # 1998 Elsevier Science Ltd. All rights reserved.

Nomenclature Si T, X tij fW(t; y0, Zi) fE(x; u, b) ta(i) r(Si) x0, x1 mXn, mXn mXn0, mXn0 E

ith stress level random variables of life-time and X = lnT jth observation under Si p.d.f. of 2-parameter Weibull distribution with common shape parameter y0 and scale parameter Zi under Si p.d.f. of type I smallest extreme value distribution 100ath of percentile of fW(t; y0, Zi) general expression of the function of Si parameters of the function of ta(i) and r(Si) nth moment and its sample moment nth central moment and its sample moment relative mean squares error

ducts for saving testing time and cost. In many cases, the interest of the reliability analysis is in estimating distribution parameters under the standard conditions. This testing gives the information to predict the life time distribution under the standard condition in a short time. Some approaches are generally used with the life-stress relationships; e.g. the graphical method and the least squares method are very useful and popular. However, the graphical method requires knowledge of the characteristics of the testing method and materials, and the least squares method does not always consider the weights of sample size of each stress level. In this paper, the moment estimators of the characteristics of life which take weights of sample size of each stress level into consideration is proposed. To check the validity of this method, the accelerated life test data on insulating ¯uids [2] are analyzed by the proposed method, and the proposed method is compared with the least squares method [3].

1. Introduction

2. Models and distribution

Accelerated life testing (ALT), which is a method of testing products and materials under severe conditions, is employed at the early stage of development of pro-

2.1. Models

* Corresponding author. E-mail: [email protected].

To get a general consideration of the analysis method of accelerated life testing, the assumptions are de®ned in the models that follow. In this consideration, the assumptions are the existence of a common

0026-2714/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 9 8 ) 0 0 1 8 8 - 7

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T. Seki, S. Yokoyama / Microelectronics Reliability 38 (1998) 1931±1934

shape parameter y0 and the existence of the general relationship between the life and the stress without a lack of generality. Suppose the variables ti1, ti2, . . . , tiki are life-times of sample size ki under the condition of higher levels of stress Si(i = 1, 2, . . . , n). Assume that the variables under the stress Si depend on the Weibull distribution fW(t; y0, Zi) as follows,   y0    y0 t y0 ÿ1 t fW …t; y0 ; Zi † ˆ exp ÿ ; …1† Zi Zi Zi where Zi is a scale parameter of Si and y0 has a shape parameter which is common value for all test stress levels, i.e. Zi is dependent on the stress level Si but y0 is not. The assumption of y0 is not against the usual ALT model. Because if the failure mode does not change according to the change of stress level, the shape parameter under each stress level must take the same value. Let ta(i) be the 100ath of percentile of fW(t; y0, Zi) under the stress Si. We can assume that the logarithm life, ln{ta(i)}, are expressed by an appropriate life-stress relationship [4], i.e. expressed by the function of stress as follows, lnfta…i† g ˆ x0 ‡ x1 r…Si †

…2†

where x0 and x1 are parameters of characteristics of the material and the testing method, and r(Si) is a simple conversion of a stress Si. If the properties of life are expressed by the inverse power model ta(i) = A/S Bi , then r(Si) is equal to ÿln(Si). If the Arrhenius model ta(i) = A exp{E/(kSi)} is assumed, then r(Si) is equal to S iÿ 1. In the above models, A, B, E and k are parameter characteristics of the material and test method. From these assumptions and the assumption that the Weibull scale parameter is equivalent to 63.2%, each Zi is expressed by the following equation: ln…Zi † ˆ x0 ‡ x1 r…Si †:

…3†

ui, bi), then the pooled variable X0{X1, X2, . . . , Xn} is distributed with the following mixed distribution [1], fE…0† …x; u1 ; u2 ; . . . ; un ; b1 ; b2 ; . . . ; bn † ˆ

n X

pi fE…i† …x; ui ; bi †;

…5†

iˆ1

where pi = ki/N and N denotes S ni= 1ki. From the assumptions in Section 2.1, Eq. (5) is rewritten as follows: fE…0† …x; x0 ; x1 ; b0 ; p1 ; p2 ; . . . ; pn ; r1 ; r2 ; . . . ; rn † ˆ

n X iˆ1

pi fE…i† …x; x0 ; x1 ; b0 ; ri †;

…6†

where b0 = y 0ÿ 1. In Eq. (6), pi and ri are also de®ned parameters from a testing plan; therefore we only estimate the three parameters i.e. b0, x0 and x1 regardless of the number of stresses. In the following section, the moment estimation method of these three parameters by the pooled data is discussed.

3. Method 0

From Eq. (6), the nth moment of X0, m Xn(0), is denoted as, …1 m0Xn…0† ˆ X n0 fE…0† …x; x0 ; x1 ; b0 ; p1 ; p2 ; . . . ; ÿ1

 pn ; r1 ; r2 ; . . . ; rn † dx …1 n X ˆ pi X n0 fE…i† …x; x0 ; x1 ; b0 ; ri † dx iˆ1

ˆ

n X iˆ1

ÿ1

pi m0Xn…i† :

…7† 0

In the following discussion, ri implies r(Si).

Therefore, the expectation of the ®rst moment m X1(0), second and third order central moments mX2(0), mX3(0) are,

2.2. Mixed Weibull distribution and ALT

m0X1…0† ˆ x0 ‡ P 1 x1 ÿ gb0 ;

The relationship (3) in 2.1 show the relationship between the logarithm of the life T and the stress level Si. The logarithm of random variables T, i.e. X = ln(T) has p.d.f. as follows,      1 xÿu xÿu fE …x; u; b† ˆ exp exp ÿ exp ; …4† b b b

mX2…0† ˆ m0X2…0† ÿ m0X1…0† 2

where u = ln(Z), b = y ÿ 1 denote the location and scale parameter, respectively. This distribution is well known as a type I smallest extreme value distribution. If the random variables Xij, j = 1, 2, . . . , ki under the stress Si are independently distributed with fE(i)(x;

ˆ …G000 …1† ‡ g 3 ‡

2

p ˆ …P 2 ÿ P 21 †x 21 ‡ b 20 ; 6

…8†

…9†

mX3…0† ˆ m0X3…0† ÿ 3m0X1…0† mX2…0† ÿ m03 X1…0† p2 g†b 30 ‡ …P 3 ‡ 2P 31 ÿ 3P 1 P 2 †x 31 2 ˆ c00 …1†b 30 ‡ …P 3 ‡ 2P 31 ÿ 3P 1 P 2 †x 31 ; …10† where Pq = S ni= 1pir qi, i.e. it is like a weighted average of r qi. In these equations, g is the Euler's constant i.e.

T. Seki, S. Yokoyama / Microelectronics Reliability 38 (1998) 1931±1934

g 2 0.5772, G1(
) denotes the value of the third order derivative of the gamma function and c0(
) is a tetragamma function i.e. c0(1) 2ÿ2.404. From the above equations, the function of b0, H(b0), is constructed as follows, H…b0 † ˆc00 …1†b 30 ‡ …P 3 ‡ 2P 31 ÿ 3P 1 P 2 †   3=2  p2 …P 2 ÿ P 21 †ÿ1 mX2…0† ÿ b 20 ÿ mX3…0† : …11† 6 We want to estimate the parameter b0 value which satis®es the equation H(b0) = 0 in Eq. (11). Now replace mXn(0) by the sample moment mXn(0) and rewrite H(b0) as h(bÃ0), then the estimator of b0, i.e. bÃ0, is estimated as the solution of the equation h(bÃ0) = 0. On the other hand, considering the pooled data on each of the stresses; the nth moment mXn(0) is derived as follows, mXn…0† ˆ ˆ

n X

pi

iˆ1

ki 1X …xij ÿ m0X1…0† †n ki jˆ1

ki n X 1X …xij ÿ m0X1…0† †n ; N iˆ1 jˆ1

…12†

where, m0X1…0† ˆ

n X iˆ1

ki ki n X 1X 1X pi xij ˆ xij : ki jˆ1 N iˆ1 jˆ1

…13†

Furthermore, if getting the unbiased estimators of Eqs. (9), mÄX2(0), and (10), mX3(0), are needed in this method, the following modi®cations of Eq. (12) are required,  n  ki 1 X ki X m~ X2…0† ˆ mX2…0† ‡ 2 …xij ÿ m0X1…i† †2 ; …14† N k ÿ 1 jˆ1 iˆ1 i m~ X3…0† ˆ mX3…0† ‡

3 N2

X n

X ki n X iˆ1 jˆ1

x 3ij ÿ

n X iˆ1

ki X ki 1 X xij x 2ij ki ÿ 1 jˆ1 l6ˆj



ki X ki …xij ÿ m0X1…i† †3 …k ÿ 1†…k ÿ 2† i i iˆ1 jˆ1 X ki X ki n  X ki ‡3 xij x 2il x 2il k ÿ 1 i iˆ1 jˆ1 l6ˆj  ki X ki X ki 1 X ÿ xil xir xir † ki ÿ 2 j6ˆl6ˆr  ki ki n n X X ki X X ‡3 …xij ÿ m0X1…i† †2 xlj ; …15† k ÿ 1 jˆ1 iˆ1 i l6ˆi jˆ1

ÿ

0

2 N3

where mX1(i) = S kij = 1xij/kl. From bÃ0 and Eq. (9), the estimator of x1, i.e. x1, is given as follows,

x^ 1 ˆ

s   p2 …P 2 ÿ P 21 †ÿ1 mX2…0† ÿ b 20 : 6

1933

…16†

And also the estimator of x0, i.e. x0 is as follows from bÃ0, x1 and Eq. (8), x^ 0 ˆ m0X1…0† ÿ P 1 x^ 1 ‡ gb^0 :

…17†

From the above estimated bÃ0, x0 and x1, the life±stress relationships and the parameters of distribution under standard condition, y and Z, can be calculated. In the special case of n = 2 and k1 = k2, H(b0) is expressed as a very simple form, since the second term of Eq. (11) is equal to 0. H(b0) of this case is as follows, H…b0 † ˆ c00 …1†b 30 ÿ mX3…0† : Therefore, bÃ0 is estimated very simply,   mX3…0† 1=3 b^0 : 00 c …1†

…18†

…19†

4. Numerical example In this section, to assess the validity of the proposed method, one of the numerical examples is presented. The accelerated life test data on insulating ¯uids are analyzed by the graphical method and by the least squares method [3]. In this example, to compare with the method [2] above, the proposed method is applied to the analysis of these same data. From the procedures which are described in Section 2, the parameters, i.e. y0, x0 and x1, are estimated as 1.104, 70.916 and 19.522, respectively. For checking the validity of these estimates, the data are plotted on the Weibull probability paper and ®tted lines are drawn for each stress level, as shown in Fig. 1. The estimated line of the usual stress, which in this case is 20 kV, is also drawn on the same paper using the estimates bÃ0, x0 and x1. Further, comparing with the estimators by least squares method [3], the estimated line using this method is also drawn. Fig. 1 shows the analysis results of the typical and general authorized accelerated life testing data by Nelson ]2,3]. He showed the appropriate common slope, i.e. common shape parameter y0, in his graphical consideration [2]. Nelson also showed another estimate of y0 by the least squares method in another paper [3]. These two results show some deference, because the ®rst result is according to his engineering experience and the second result is only according to the mathematical method. The result of the proposed method shows a similar response to the ®rst result. Then, this is only one of the actual examples, judging from Fig. 1,

1934

T. Seki, S. Yokoyama / Microelectronics Reliability 38 (1998) 1931±1934

Fig. 1. Weibull plot of insulating ¯uid data with estimated line.

Table 1 Sample size for experiments and relative errors Stress level

30 kV

40 kV

50 kV

E

Case 1 Case 2 Case 3

3 3 1

6 6 1

Ð 9 1

0.616 0.796 0.3521

MSE of bÃ0 by proposed method

1

the estimated line of 20 kV, using the proposed method, provides a better ®t to the line of best ®t for accelerated stresses than the estimated line using the least squares method. 5. Conclusions In this paper, the analysis method of pooled data for accelerated life testing is proposed. This method takes weights of sample sizes of each stress level into consideration. The validity of this method is examined through a numerical example. The examination shows that the proposed method gives good estimates. Furthermore, to discuss the accuracy of the proposed estimators when the sample sizes are very small, we calculate the following relative errors of bÃ0, E by the Monte Carlo experiments, Eˆ

MSE of b^0 by the proposed method MSE of b^0 by least squares method

…20†

where MSE is a mean square error. In the experiments, x0 = x1 = y0 = 1, standard stress = 20 kV and r(Si) = ÿ ln(Si) are assumed for some cases in Table 1. In Table 1, case 3 shows the smallest sample size. Usually, in this case, the least squares method is not available. Then the MSE of bÃ0 of case 3 is presented for reference. Judging from this short examination, it seems reasonable to suppose that the proposed method also has enough validity for very small data.

Acknowledgements This research was partially supported by The Grantin-Aid for General Scienti®c Research 07680475 from The Ministry of Education, Science, Sports and Culture.

References [1] Kao JHK. A graphical estimation of mixed Weibull parameters in life-testing of electron tubes. Technometrics 1959;1:389±407. [2] Nelson W. Graphical analysis of accelerated life test data with the inverse power law model. IEEE Trans Reliab 1972;R-21:2±11. [3] Nelson W. Analysis of accelerated life test dataÐleast square methods for the inverse power law model. IEEE Trans Reliab 1975;R-24:103±7. [4] Nelson W. Accelerated life testing. New York: Wiley, 1990:71±97.