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Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing
ARTICLE IN PRESS Journal of Statistical Planning and Inference 140 (2010) 2706–2718
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Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing Bing Xing Wang Zhejiang Gongshang University, Hangzhou, PR China
a r t i c l e i n f o
abstract
Article history: Received 3 December 2008 Received in revised form 12 October 2009 Accepted 16 March 2010 Available online 21 March 2010
This paper derives the exact confidence intervals for the exponential step-stress accelerated life-testing model as well as the approximate confidence intervals for the k-step exponential step-stress accelerated life-testing model under progressive Type-II censoring. A Monte Carlo simulation study is carried out to examine the performance of these confidence intervals. Finally, an example is given to illustrate the proposed procedures. & 2010 Elsevier B.V. All rights reserved.
Keywords: Accelerated life test Step-stress Exponential distribution Progressive Type-II censoring Interval estimation Cumulant
1. Introduction In many situations, it may be difficult to collect data on life-time of a product under normal operating conditions as the product may have a high reliability under normal conditions. For this reason, accelerated life-testing (ALT) experiments can be used to force these products to fail more quickly than under normal operating condition. The results obtained at the accelerated conditions are analyzed in terms of a model to relate life length to stress; they are extrapolated to the design stress to estimate the life distribution. The step-stress ALT (SSALT) was developed to reduce operation time and costs further. One way of applying stress to the test units is the step-stress scheme which allows the stress setting of a unit to be changed at pre-specified times or upon the occurrence of a fixed number of failures. The former is called SSALT with Type-I censoring and the latter is called SSALT with Type-II censoring. The problem of modeling data from SSALT and making inferences from such data have been studied by many authors. DeGroot and Goel (1979) proposed the tampered random variable model. Nelson (1980) proposed the cumulative exposure model. Bhattacharyya and Soejoeti (1989) proposed the tampered failure rate model. Wang and Fei (2004) discussed conditions for the coincidence of these models. Khamis and Higgins (1998), and Bagdonavicˇius et al. (2002) proposed some new models for SSALT. Tang et al. (1996) obtained the maximum likelihood estimations (MLE) for parameters in a multi-censored ALT. Teng and Yeo (2002) used the method of least-squares to estimate the life-stress relationship in SSALT. Dorp et al. (1996) gave a Bayes approach to SSALT. Dorp and Mazzuchi (2004) developed a general Bayes exponential inference model for ALT. Wang (2006) obtained unbiased estimations for the exponential distribution based on SSALT data. Balakrishnan et al. (2007) obtained point and interval estimation for the exponential simple
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ARTICLE IN PRESS B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
2707
step-stress model. Bai et al. (1989) obtained the optimum simple SSALT plans. Khamis and Higgins (1996) obtained the optimum 3-step SSALT plans. Cheng (1994) extended the results of Bai et al. (1989) to case of the k-step SSALT. Tang et al. (1999) discussed optimum plan for two-parameter exponential distribution. Many examples of ALT, as well as an excellent introduction to the methodology, are given in Nelson (1990) and Mao and Wang (1997). One popular SSALT model is the SSALT model with Type-I censoring, where stress levels change at pre-fixed time points. This model, although simple and intuitive, has the following problems: (1) when there is no observation under some particular stress-levels, the MLEs for the parameters corresponding to these stress levels do not exist; (2) there are quite complicated expressions for the distributions and the moments of the MLEs; see, for example, Balakrishnan et al. (2007). To alleviate these problems as well as for some other reasons associated with the performance of a life-test, experimenters may consider increasing a stress level only after having observed a pre-specified number of observations at the current level of stress. Under this proposed alternative model, the step-stress model is the SSALT with Type-II censoring. Compared to the SSALT model with Type-I censoring, this approach ensures the existence of the MLEs of all the parameters in the model. Progressive censoring is a generalized form of censoring which includes the conventional right censoring as a special case. Compared to the conventional censoring, however, it provides higher flexibility to the experimenter in the design stage by allowing the removal of test units at non-terminal time points and thus, it proves to be highly efficient and effective in utilizing the available resources. For this reason, we consider a more general censoring scheme called progressive Type-II censoring. Progressive Type-II censoring is a method which enables an efficient exploitation of the available resources by continual removal of a pre-specified number of surviving test units at each failure time. Another advantage of progressive censoring is that the degeneration information of the test units is obtained from those removed units. Gouno et al. (2004) and Balakrishnan and Han (2009) discussed the optimal SSALT plans under progressive Type-I censoring. Wang and Yu (2009) discussed the optimal SSALT plans under progressive Type-II censoring. A book dedicated completely to progressive censoring was published by Balakrishnan and Aggarwala (2000). The main objective of this paper is to investigate the interval estimations for the exponential SSALT model under progressive Type-II censoring. The paper is organized as follows. Section 2 gives some basic assumptions in the exponential SSALT. Section 3 derives the exact confidence intervals for the exponential SSALT model and the approximate confidence intervals for the k-step exponential SSALT model under progressive Type-II censoring. Section 4 gives a Monte Carlo simulation to study the coverage percentages and the average confidence lengths of the proposed confidence intervals. Section 5 gives an example to illustrate the proposed procedures. 2. Basic assumptions Statistical inference for the exponential SSALT usually depends on the following assumptions: 1. For any stress level xi, the lifetime distribution of a test unit is exponential with the cumulative distribution function (cdf): Fi ðtÞ ¼ 1expðt=yi Þ,
t 4 0,
ð1Þ
where yi 4 0 is the mean life of test unit at stress level xi. 2. The log mean life of a test unit logðyi Þ is a linear function of stress level xi: logðyi Þ ¼ a þ bxi ,
ð2Þ
where a and b are unknown parameters. 3. A cumulative exposure model holds: the remaining life depends only on the current cumulative failure probability and current stress level regardless of how the probability is accumulated. (Nelson, 1980). The cumulative exposure model may be replaced by the tampered random variable model or the tampered failure rate model (see DeGroot and Goel, 1979; Bhattacharyya and Soejoeti, 1989). 3. Interval estimation SSALT with progressive Type-II censoring is as follows. Suppose that all n test units are initially placed at lowest stress x1. At the first failure time t1,1, R1,1 units are randomly removed from the remaining n 1 surviving units. At the second failure time t1,2, R1,2 units are randomly removed from the remaining n 2 R1,1. The test continues until the r1th failure P time t1,r1 . At failure time t1,r1 , R1,r1 units are randomly removed from the remaining nr1 jr1¼11 R1,j surviving units. Then the stress is changed to x2. Similar to stress x1, the test is continued until r2 units have failed. At the stress x2, r2 failure times t2,j (j= 1,2,y,r2) of test units are observed. The stress is changed again, and the test continues. Suppose that the stress is finally changed to xk, and test does not stop until rk units have failed under xk. We write the failure times under the stress xi (i= 1, 2, y, k) as ti,1 r ti,2 r r ti,ri :
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At failure time ti,j, Ri,j units are randomly removed. Further, suppose that the lifetime at stress level xi has the exponential distribution (1). For i=1, 2, y, k, let 2 0 13 rl ri i X X X @rl þ ð1 þRi,j Þðti,j ti1,ri1 Þ þ 4n Rl,j A5ðti,ri ti1,ri1 Þ ð3Þ Ti ¼ j¼1
l¼1
j¼1
be the total time on test at stress xi, where t0,r0 ¼ 0. In order to obtain the confidence intervals of the exponential SSALT model, the following lemmas are necessary. Lemma 1. Suppose that T1,T2,y,Tk are defined as (3), then, under the assumptions 1–3, T1,T2,y,Tk are independent, and the distribution of the 2Ti =yi is w2 ð2ri Þ ði ¼ 1,2, . . . ,kÞ. (see Wang and Yu, 2009) Lemma 2. Let F(y) is the cdf of w2 ð2rÞ, then FðyÞ ¼ 1ey=2
r 1 X ðy=2Þj j¼0
j!
,
y 40:
It is a relationship between the gamma and Poisson distributions. See Casella and Berger (2001, P100). Lemma 3. Let X follows the log-gamma distribution with the probability function (pdf): f ðxÞ ¼
1
GðrÞ
expðrxex Þ,
then the ith cumulant of X is
ki ¼
di1 cðrÞ dr
i1
,
where cðrÞ ¼ dlogðGðrÞÞ=dr is the digamma function. Proof. We have from Lawless (1982, P22) that the moment generating function for X is MðyÞ ¼ Gðy þrÞ=GðrÞ. Therefore the i th cumulants of X is given by di logðMðyÞÞ di1 cðrÞ ki ¼ ¼ : i i1 dy dr y¼0 The proof is completed.
&
3.1. k= 2 case In this subsection, we shall derive the exact confidence intervals for simple exponential SSALT model based on progressively Type-II censored sample. Under the Assumptions 1–3, we have from Wang and Yu (2009) that the MLEs of the parameters a and b are given by
a^ ¼
x1 logðT2 =r2 Þx2 logðT1 =r1 Þ , x1 x2
ð4Þ
b^ ¼
logðT1 =r1 ÞlogðT2 =r2 Þ , x1 x2
ð5Þ
and
respectively. Therefore, the MLE of the mean life y0 at designed stress level x0 is
y^ 0 ¼ expða^ þ b^ x0 Þ ¼ r1k0 1 r2k0 T1k0 þ 1 T2k0 , where k0 ¼ ðx0 x1 Þ=ðx1 x2 Þ. Let W1 ¼ 2exp½a^ a þlogðr2 Þ þ k1 logðr2 =r1 Þ,
ð6Þ
W2 ¼ exp½ðx1 x2 Þðb^ bÞ,
ð7Þ
W3 ¼ 2r1k0 þ 1 r2k0 y^ 0 =y0 ,
ð8Þ
where k1 = x2/(x1 x2). Then W1 ¼ ð2T2 =y2 Þ1 þ k1 ð2T1 =y1 Þk1 ,
ARTICLE IN PRESS B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
W2 ¼
2709
2T1 =y1 r2 , 2T2 =y2 r1
W3 ¼ ð2T1 =y1 Þ1 þ k0 ð2T2 =y2 Þk0 : Hence it follows from Lemma 1 that W1,W2,W3 are the pivotal quantities. The following theorem gives the cumulative distribution functions of the pivotal quantities W1,W2,W3. Theorem 1. Suppose the pivotal quantities W1,W2,W3 are defined as Eqs. (6)–(8), then (1) the cdf of W1 is given by Z 1 rX 2 1 ðy=2Þj FW1 ðwÞ ¼ 1 dt, g1 ðtÞey=2 j! 0 j¼0
ð9Þ
where y ¼ ðwt k1 Þ1=ðk1 þ 1Þ , g1(t) is the pdf of w2 ð2r1 Þ; (2) W2 Fð2r1 ,2r2 Þ; (3) the cdf of W3 is given by Z 1 rX 1 1 ðy=2Þj FW3 ðwÞ ¼ 1 dt, g2 ðtÞey=2 j! 0 j¼0
ð10Þ
where y ¼ ðwt k0 Þ1=ðk0 þ 1Þ , g2(t) is the pdf of w2 ð2r2 Þ. Proof. (1) Let U1 ¼ 2 T1 =y1 , U2 ¼ 2 T2 =y2 . From Lemmas 1 and 2, we have FW1 ðwÞ ¼ PðW1 rwÞ ¼ PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ Þ ¼ E½PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ jU1 Þ Z 1 Z Z 1 PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ jU1 ¼ tÞg1 ðtÞ dt ¼ FðyÞg1 ðtÞ dt ¼ 1 ¼ 0
0
1
g1 ðtÞey=2
0
rX 2 1
ðy=2Þj dt: j! j¼0
2
where F(y) is the cdf of w ð2r2 Þ. (2) Since W2 ¼
2 T1 =y1 2r2 , 2 T2 =y2 2r1
it follows from Lemma 1 that W2 Fð2r1 ,2r2 Þ. (3) Similar to (1), we can obtain the cdf of W3. The proof is completed. & In computing the improper integral in (9) or (10), one must truncate the integral at some finite value. The numerical error caused by this truncation can be assessed by the following error bound formula. For example, in evaluating (9), if the integration range is from 0 to M, then the error is
eM ¼
Z
1
g1 ðtÞey=2 M
rX 2 1
ðy=2Þj dt r j! j¼0
Z
1
g1 ðtÞ dt ¼ eM=2 M
rX 1 1
ðM=2Þj : j! j¼0
For any specified e 4 0, one can choose a large M so that eM o e. Thus, we can evaluate the improper integrals as accurately as desired. Based on Theorem 1, it is easy to obtain the confidence intervals of the parameters a, b and y0 . Theorem 2. Suppose that a^ , b^ , y^ 0 are MLEs of the parameters a, b, y0 . Then, for any 0 o g o1, (1) a 100ð1gÞ% exact confidence interval for the parameter a is given by a^ þ logðr2 Þ þ k1 logðr2 =r1 ÞlogðW1,1g=2 =2Þ, a^ þlogðr2 Þ þ k1 logðr2 =r1 ÞlogðW1, g=2 =2Þ , where W1, g is the g percentile of W1. (2) a 100ð1gÞ% exact confidence interval for the parameter b is given by logF1g=2 ð2r1 ,2r2 Þ ^ logFg=2 ð2r1 ,2r2 Þ , b^ ; b x1 x2 x1 x2 where Fg ð2r1 ,2r2 Þ is the g percentile of F(2r1,2r2).
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(3) a 100ð1gÞ% exact confidence interval for the parameter y0 is given by " # 2r11 þ k0 r2k0 y^ 0 2r11 þ k0 r2k0 y^ 0 , , W3,1g=2 W3, g=2 where W3, g is the g percentile of W3. 3.2. k Z2 case In this subsection, we shall derive the exact and the approximate confidence intervals for the kð Z2Þstep exponential SSALT model based on progressively Type-II censored sample. Under the assumptions 1–3, we have from Wang and Yu (2009) that the unbiased estimators for a, b,logðy0 Þ are given by
a~ ¼
GHIM , EGI2
b~ ¼
EMIH , EGI2
logðy~ 0 Þ ¼ a~ þ b~ x0 ,
respectively, where Ui ¼ logðTi Þcðri Þ, E ¼ P 0 M ¼ ki¼ 1 ½c ðri Þ1 xi Ui . Further Varða~ Þ ¼
G , EGI2
E , EGI2
Varðb~ Þ ¼
Pk
0 1 , i ¼ 1 ½c ðri Þ
Varðlogðy~ 0 ÞÞ ¼
ð11Þ I¼
Pk
0 1 i ¼ 1 ½c ðri Þ
xi , G ¼
Pk
0 1 i ¼ 1 ½c ðri Þ
x2i , H ¼
Pk
0 1 i ¼ 1 ½c ðri Þ
Ui ,
Gþ x20 E2x0 I : EGI2
Let
a~ a W4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varða~ Þ
b~ b W5 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðb~ Þ
logðy~ 0 Þlogðy0 Þ W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Varðlogðy~ 0 ÞÞ
Write Yi ¼ logðTi =yi Þcðri Þ, from H¼
k X
0
½c ðri Þ1 Ui ¼
i¼1
M¼
k X
k X
0
½c ðri Þ1 Yi þ aEþ bI :¼ H1 þ aE þ bI,
i¼1
0
½c ðri Þ1 xi Ui ¼
i¼1
k X
0
½c ðri Þ1 xi Yi þ aI þ bG :¼ M1 þ aI þ bG,
i¼1
we have
a~ ¼
GH1 IM 1 þ a, EGI2
b~ ¼
EM 1 IH1 þ b, EGI2
so GH1 IM1 W4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEGI2 Þ, Varða~ Þ
ð12Þ
EM 1 IH1 W5 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEGI2 Þ, Varðb~ Þ
ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a~ þ b~ x0 abx0 1 W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Varða~ ÞW4 þ Varðb~ Þx0 W5 Þ: Varðlogðy~ 0 ÞÞ Varðlogðy~ 0 ÞÞ
ð14Þ
Then it follows from Lemma 1 that W4,W5,W6 are the pivotal quantities. Because the distributions of the pivotal quantities W4,W5,W6 depend only on the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j, i= 1,2,y,k, j = 1,2,y,ri at the failure times ti.j, i= 1,2,y,k, j = 1,2,y,ri, without loss of generality, we choose Ri,j = 0, i= 1,2,y,k, j = 1,2,y,ri. Hence, in order to obtain the exact percentiles of W4, W5, W6, we need only to consider various combinations (r1,r2,y,rk). The exact confidence intervals of a, b, y0 based on W4, W5, W6 can be obtained by the following steps. Step 1: Given r1,y,rk, generate a random sample of 2Ti =yi ,i ¼ 1, . . . ,k, where 2Ti =yi ,i ¼ 1, . . . ,k are independently the
w2 ð2ri Þ distributions. Step 2: Given stress levels x0,x1,y,xk, calculate W4, W5, W6 from Eqs. (12)–(14). Step 3: Repeat step 1–2 N times. Step 4: Arrange all W4, W5, W6 values in ascending order: Wi½1 o Wi½2 o o Wi½N ,
i ¼ 4,5,6:
ARTICLE IN PRESS B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718 ½gN
½gN
½gN
Step 5: The exact g percentiles of W4,W5,W6 are W4 ,W5 ,W6 , respectively. Step 6: The exact 100ð1gÞ% confidence intervals of a, b, y0 based on W4, W5, W6 are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1g=2ÞN ½gN=2 ½a~ W4 Varða~ Þ, a~ W4 Varða~ Þ, ½ð1g=2ÞN
½b~ W5
½ð1g=2ÞN
½y~ 0 eW6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½gN=2 Varðb~ Þ, b~ W5 Varðb~ Þ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðlogðy~ 0 ÞÞ
½gN=2 , y~ 0 eW6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðlogðy~ 0 ÞÞ
2711
ð15Þ ð16Þ
,
ð17Þ
respectively. From (15)–(17), we obtain that the lengths of the exact confidence intervals for the parameters a, b,logðy0 Þ are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1g=2ÞN ½gN=2 ½ð1g=2ÞN ½gN=2 ½ð1g=2ÞN ½gN=2 W4 Þ, Varðb~ ÞðW5 W5 Þ and Varðlogðy~ 0 ÞÞðW6 W6 Þ, respectively. Varða~ ÞðW4 In particular, when k= 2, from Eqs. (4), (5), (11), we have
a~ ¼ a^
x1 ðcðr2 Þlogðr2 ÞÞx2 ðcðr1 Þlogðr1 ÞÞ , x1 x2
b~ ¼ b^
ðcðr1 Þlogðr1 ÞÞðcðr2 Þlogðr2 ÞÞ : x1 x2
and
Hence, we obtain W4 ¼
logðW1 Þc1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varða~ Þ
W5 ¼
logðW2 Þc2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðx1 x2 Þ Varðb~ Þ
logðW3 Þc3 W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðlogðy~ 0 ÞÞ
where c1 ¼
x1 cðr2 Þx2 cðr1 Þ þ logð2Þ, x1 x2
c2 ¼ ðcðr1 Þlogðr1 ÞÞðcðr2 Þlogðr2 ÞÞ, c3 ¼ logð2Þ þðk0 þ 1Þcðr1 Þk0 cðr2 Þ: Since it is difficult to obtain the exact percentiles of W4, W5, W6, we try to find approximate percentiles of W4, W5, W6 by the Cornish–Fisher expansion. Note from (12)–(14) that W4,W5,W6 are linear combinations of Y1,Y2,y,Yk, after some calculations, we have from Lemmas 1 and 3 that the first five cumulants of W4,W5,W6 are given by Pk 0 ðj1Þ ðGxi IÞj ½c ðri Þj c ðri Þ kW4 ,1 ¼ 0, kW4 ,2 ¼ 1, kW4 ,j ¼ i ¼ 1 , j ¼ 3,4,5, ð18Þ j j=2 ðEGI2 Þ ½Varða~ Þ
kW5 ,1 ¼ 0, kW5 ,2 ¼ 1, kW5 ,j ¼
Pk
0 j j ðj1Þ ðri Þ i ¼ 1 ðxi EIÞ ½c ðri Þ c , j ~ 2 ðEGI Þ ½Varðb Þj=2
Pk
kW6 ,1 ¼ 0, kW6 ,2 ¼ 1, kW6 ,j ¼
Dji ½c ðri Þj c ðri Þ , j ~ 2 ðEGI Þ ½Varðlogðy 0 ÞÞj=2 i¼1
0
j ¼ 3,4,5,
ð19Þ
ðj1Þ
j ¼ 3,4,5,
ð20Þ
respectively, where Di = G+x0 xi E (x0 + xi)I. Cornish and Fisher (1937) provided an expansion for approximating the g percentile, xg , of X based on its cumulants. Using the first five cumulants, the expansion is 1 1 1 1 1 k4 ðz3g 3zg Þ36 k23 ð2z3g 5zg Þ þ 120 k5 ðz4g 6z2g þ 3Þ24 k3 k4 ðz4g 5z2g þ 2Þ þ 324 k33 ð12z4g 53z2g þ 17Þ, xg ¼ zg þ 16 k3 ðz2g 1Þ þ 24
ð21Þ where zg is the g percentile of the standard normal distribution N(0,1). Using Eqs. (18)–(21), it is easy to obtain the approximate g percentiles W4, g ,W5, g ,W6, g of W4,W5,W6. Hence the 100ð1gÞ% approximate confidence intervals for a, b, y0 are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ ½a~ W4,1g=2 Varða~ Þ, a~ W4, g=2 Varða~ Þ, ½b~ W5,1g=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðb~ Þ, b~ W5, g=2 Varðb~ Þ,
ð23Þ
2712
Table 1 Experiment schemes and the exact and approximate percentiles of W4,W5,W6. 0.025
0.05
0.1
0.9
0.95
0.975
(9,6)
W4
2.0013 ( 1.9912) 2.0143 ( 2.0219) 2.0454 ( 2.0353) 1.9953 ( 1.9864) 2.0166 ( 2.0119) 2.0344 ( 2.0247) 1.9908 ( 1.9832) 2.0084 ( 2.0054) 2.0273 ( 2.0176) 1.9807 ( 1.9756) 1.9961 ( 1.9905) 2.0037 ( 2.0002)
1.6599 ( 1.6554) 1.6550 ( 1.6722) 1.6882 ( 1.6825) 1.6579 ( 1.6546) 1.6608 ( 1.6688) 1.6789 ( 1.6782) 1.6530 ( 1.6539) 1.6639 ( 1.6665) 1.6701 ( 1.6750) 1.6472 ( 1.6518) 1.6545 ( 1.6604) 1.6641 ( 1.6668)
1.2752 ( 1.2767) 1.2763 ( 1.2816) 1.2822 ( 1.2876) 1.2790 ( 1.2787) 1.2793 ( 1.2830) 1.2824 ( 1.2881) 1.2747 ( 1.2798) 1.2826 ( 1.2837) 1.2791 ( 1.2882) 1.2779 ( 1.2816) 1.2861 ( 1.2845) 1.2804 ( 1.2875)
1.2657 (1.2661) 1.2596 (1.2587) 1.2541 (1.2564) 1.2732 (1.2694) 1.2674 (1.2633) 1.2666 (1.2609) 1.2752 (1.2714) 1.2641 (1.2661) 1.2694 (1.2638) 1.2766 (1.2757) 1.2681 (1.2721) 1.2740 (1.2701)
1.6157 (1.6285) 1.6081 (1.6102) 1.5917 (1.6003) 1.6307 (1.6308) 1.6163 (1.6155) 1.6101 (1.6064) 1.6351 (1.6324) 1.6179 (1.6190) 1.6178 (1.6106) 1.6385 (1.6362) 1.6264 (1.6272) 1.6252 (1.6209)
1.9381 (1.9476) 1.9239 (1.9173) 1.8897 (1.8989) 1.9538 (1.9475) 1.9226 (1.9222) 1.9084 (1.9057) 1.9554 (1.9478) 1.9249 (1.9258) 1.9152 (1.9106) 1.9586 (1.9497) 1.9316 (1.9350) 1.9304 (1.9238)
1.9953 ( 2.0031) 2.0062 ( 2.0102) 2.0342 ( 2.0351) 1.9905 ( 2.0004) 1.9920 ( 2.0005) 2.0211 ( 2.0284) 1.9936 ( 1.9859) 1.9778 ( 1.9893) 2.0102 ( 2.0066)
1.6582 ( 1.6644) 1.6733 ( 1.6665) 1.6772 ( 1.6843) 1.6588 ( 1.6642) 1.6594 ( 1.6628) 1.6755 ( 1.6815) 1.6578 ( 1.6579) 1.6581 ( 1.6593) 1.6735 ( 1.6706)
1.2799 ( 1.2820) 1.2867 ( 1.2809) 1.2858 ( 1.2903) 1.2808 ( 1.2835) 1.2810 ( 1.2817) 1.2885 ( 1.2906) 1.2921 ( 1.2838) 1.2789 ( 1.2836) 1.2944 ( 1.2889)
1.2732 (1.2649) 1.2677 (1.2631) 1.2638 (1.2581) 1.2714 (1.2672) 1.2681 (1.2669) 1.2677 (1.2611) 1.2739 (1.2731) 1.2816 (1.2721) 1.2721 (1.2686)
1.6230 (1.6204) 1.6125 (1.6172) 1.6037 (1.5997) 1.6239 (1.6217) 1.6159 (1.6225) 1.6084 (1.6036) 1.6339 (1.6299) 1.6335 (1.6282) 1.6210 (1.6169)
1.9349 (1.9312) 1.9180 (1.9264) 1.8998 (1.8947) 1.9294 (1.9304) 1.9312 (1.9323) 1.8979 (1.8990) 1.9337 (1.9395) 1.9436 (1.9371) 1.9140 (1.9172)
W5 W6 (12,8)
W4 W5 W6
(15,10)
W4 W5 W6
(30,20)
W4 W5 W6
(12,8,6)
W4 W5 W6
(15,10,8)
W4 W5 W6
(30,20,15)
W4 W5 W6
ARTICLE IN PRESS
Pivotal quantities
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(r1,y,rk)
(60,40,25)
W4 W5 W6
W4 W5 W6
(15,10,8,6)
W4
W6 (30,20,15,12)
W4 W5 W6
(60,40,25,18)
W4 W5 W6
(90,60,40,25)
W4 W5 W6
1.2905 ( 1.2828) 1.2802 ( 1.2846) 1.2884 ( 1.2867)
1.2708 (1.2769) 1.2852 (1.2745) 1.2693 (1.2734)
1.6331 (1.6365) 1.6399 (1.6307) 1.6197 (1.6267)
1.9417 (1.9485) 1.9494 (1.9391) 1.9326 (1.9318)
2.0009 ( 2.0052) 2.0130 ( 2.0159) 2.0249 ( 2.0335) 1.9903 ( 2.0043) 1.9994 ( 2.0018) 2.0069 ( 2.0279) 1.9783 ( 1.9897) 1.9911 ( 1.9896) 1.9870 ( 2.0070) 1.9647 ( 1.9760) 1.9905 ( 1.9873) 1.9659 ( 1.9902) 1.9858 ( 1.9729) 1.9597 ( 1.9816) 1.9943 ( 1.9845)
1.6622 ( 1.6661) 1.6631 ( 1.6698) 1.6812 ( 1.6839) 1.6505 ( 1.6670) 1.6634 ( 1.6638) 1.6666 ( 1.6817) 1.6499 ( 1.6605) 1.6621 ( 1.6596) 1.6615 ( 1.6711) 1.6442 ( 1.6532) 1.6637 ( 1.6595) 1.6554 ( 1.6620) 1.6565 ( 1.6518) 1.6453 ( 1.6567) 1.6647 ( 1.6589)
1.2872 ( 1.2831) 1.2794 ( 1.2819) 1.2956 ( 1.2906) 1.2769 ( 1.2851) 1.2791 ( 1.2823) 1.2811 ( 1.2912) 1.2850 ( 1.2850) 1.2832 ( 1.2840) 1.2890 ( 1.2894) 1.2813 ( 1.2833) 1.2817 ( 1.2852) 1.2888 ( 1.2868) 1.2879 ( 1.2832) 1.2767 ( 1.2848) 1.2899 ( 1.2861)
1.2706 (1.2646) 1.2629 (1.2618) 1.2695 (1.2589) 1.2701 (1.2667) 1.2614 (1.2669) 1.2664 (1.2617) 1.2757 (1.2725) 1.2655 (1.2722) 1.2694 (1.2687) 1.2777 (1.2766) 1.2683 (1.2738) 1.2767 (1.2735) 1.2764 (1.2777) 1.2711 (1.2756) 1.2733 (1.2752)
1.6268 (1.6189) 1.6146 (1.6137) 1.6098 (1.6005) 1.6197 (1.6190) 1.6140 (1.6216) 1.6054 (1.6038) 1.6289 (1.6274) 1.6230 (1.6279) 1.6241 (1.6165) 1.6387 (1.6355) 1.6217 (1.6288) 1.6315 (1.6267) 1.6296 (1.6373) 1.6377 (1.6321) 1.6246 (1.6301)
1.9381 (1.9281) 1.9138 (1.9204) 1.9086 (1.8954) 1.9250 (1.9251) 1.9260 (1.9302) 1.8996 (1.8983) 1.9414 (1.9349) 1.9419 (1.9363) 1.9193 (1.9161) 1.9538 (1.9467) 1.9314 (1.9359) 1.9410 (1.9316) 1.9406 (1.9489) 1.9524 (1.9406) 1.9351 (1.9367)
ARTICLE IN PRESS
W5
1.6569 ( 1.6522) 1.6511 ( 1.6577) 1.6647 ( 1.6619)
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(12,8,6,4)
1.9767 ( 1.9745) 1.9629 ( 1.9843) 1.9930 ( 1.9903)
2713
ARTICLE IN PRESS 2714
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
½y~ 0 eW6,1g=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Varðlogðy~ 0 ÞÞ
, y~ 0 eW6, g=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Varðlogðy~ 0 ÞÞ
,
ð24Þ
respectively. From (22)–(24), we obtain that the lengths of the approximate confidence intervals for the parameters a, b,logðy0 Þ are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varða~ ÞðW4,1g=2 W4, g=2 Þ, Varðb~ ÞðW5,1g=2 W5, g=2 Þ and Varðlogðy~ 0 ÞÞðW6,1g=2 W6, g=2 Þ, respectively. Table 1 gives the exact and approximate percentiles of W4, W5, W6 for some different combinations r1,r2,y,rk. These exact percentiles are obtained based on 100,000 simulations. The approximate percentiles of W4,W5,W6 are given in parentheses. It is observed from Table 1 that the approximate percentiles of W4, W5, W6 are close to the exact percentiles of W4, W5, W6 even if the failure numbers r1,r2,y,rk are small. Hence the lengths of the approximate confidence intervals for a, b,logðy0 Þ are also close to the lengths of the exact confidence intervals. 3.3. Bootstrap confidence intervals In this subsection, for comparison purpose, two confidence intervals based on the parametric bootstrap methods are proposed: (i) percentile bootstrap method (Efron, 1982) and (ii) studentized-t bootstrap method (Hall, 1988). From (11), we find that the estimators a~ , b~ ,logðy~ 0 Þ depend only on the total test times T1, T2,y,Tk and the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j, i= 1,2,y,k, j = 1,2,y,ri at the failure times ti.j, i= 1,2,y,k, j =1,2,y,ri. Further, note that 2Ti =yi w2 ð2ri Þ, without loss of generality, we choose Ri,j = 0, i= 1,2,y,k, j= 1,2,y,ri. Hence, in order to obtain the bootstrap sample of a~ , b~ ,logðy~ 0 Þ, we need only to consider various combinations (r1,r2,y,rk). The bootstrap sample of a~ , b~ ,logðy~ 0 Þ can be obtained by the following steps. Step 1: From SSALT sample ti,j,i=1,2,y,k,j = 1,2, y,ri, compute a~ , b~ ,logðy~ 0 Þ. Step 2: Based on a~ , b~ and r1,r2,y,rk, generate a random sample (T*1,T*2,y,T*k) of (T1,T2,y,Tk), where T1,T2,y,Tk are independent, and 2Ti =y~ i w2 ð2ri Þ. Here y~ i ¼ expða~ þ b~ xi Þ. Step 3: Based on (T*1,T*2,y,T*k) compute the bootstrap sample estimates a~ , b~ ,logðy~ 0 Þ of a, b,logðy0 Þ using (11). ~ ~ Step 4: Repeat Steps 2–3 B times. Then arrange all the values of a~ , b ,logðy 0 Þ in an ascending order to obtain the bootstrap samples:
a~ ,½1 r a~ ,½2 r r a~ ,½B , b~
,½1
r b~
,½2
,½B r r b~ ,
and ,½1
logðy~ 0
,½2
Þ r logðy~ 0
,½B
Þ r r logðy~ 0
Þ,
respectively. With the bootstrap samples generated as above, we now obtain the 100ð1gÞ% percentile bootstrap confidence intervals for a, b, y0 as ða~ ,½gB=2 ,
a~ ,½ð1g=2ÞB Þ,ðb~
,½gB=2
, b~
,½ð1g=2ÞB
Þ
and
,½gB=2
ðy~ 0
,½ð1g=2ÞB
, y~ 0
Þ
respectively. The 100ð1gÞ% Studentized-t bootstrap confidence intervals for a, b, y0 are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,½ð1g=2ÞB ,½gB=2 ða~ S1 Þ Varða~ ÞÞ, Varða~ Þ, a~ S1 ,½ð1g=2ÞB
ðb~ S2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,½gB=2 Varðb~ Þ, b~ S2 Varðb~ ÞÞ
and ,½ð1g=2ÞB
ðy~ 0 eS3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Varðlogðy~ 0 ÞÞ
,½gB=2
,ðy~ 0 eS3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Varðlogðy~ 0 ÞÞ
Þ
respectively, where
a~ ,½i a~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S,½i , 1 ¼ Varða~ Þ
,½i b~ b~ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , S,½i ¼ 2 Varðb~ Þ
,½i
S,½i 3 ¼
logðy~ 0 Þlogðy~ 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðlogðy~ 0 ÞÞ
Here ðVarða~ Þ,Varðb~ Þ,Varðlogðy~ 0 ÞÞÞ can be estimated by ðVarða~ Þ,Varðb~ Þ,Varðlogðy~ 0 ÞÞ.
4. Simulation study In order to assess the performance of all methods of constructing confidence intervals discussed in Section 3, a Monte Carlo simulation study was conducted to determine the coverage percentages and the average interval lengths of the proposed confidence intervals. The values of the parameters and stress levels are chosen to be a ¼ 4, b ¼ 1 and x0 = 0.25, x1 =0.5, x2 = 0.75, x3 = 1, x4 =1.25.
Table 2 Experiment schemes, coverage percentages and average interval lengthes in simulation study.
(9,6)
Parameters
a
logðy0 Þ (12,8)
a b logðy0 Þ
(15,10)
a b logðy0 Þ
(30,20)
a b logðy0 Þ
(12,8,6)
a b logðy0 Þ
(15,10,8)
a b
Exact
Approx.
Boot-p
Boot-t
Exact
Approx.
Boot-p
Boot-t
0.9502 (5.2601) 0.9495 (8.6114) 0.9492 (3.1759) 0.9507 (4.5280) 0.9493 (7.3912) 0.9506 (2.7338) 0.9503 (4.0265) 0.9499 (6.5648) 0.9516 (2.4333) 0.9522 (2.8136) 0.9504 (4.5849) 0.9513 (1.7006) 0.9497 (2.8581) 0.9478 (3.9795) 0.9499 (1.9228) 0.9460 (2.5064) 0.9467 (3.4689) 0.9482 (1.6912)
0.9499 (5.2593) 0.9498 (8.6135) 0.9491 (3.1752) 0.9501 (4.5105) 0.9491 (7.3818) 0.9497 (2.7252) 0.9499 (4.0109) 0.9498 (6.5613) 0.9503 (2.4244) 0.9517 (2.8036) 0.9507 (4.5823) 0.9505 (1.6962) 0.9497 (2.8611) 0.9484 (3.9920) 0.9497 (1.9208) 0.9468 (2.5134) 0.9472 (3.4774) 0.9489 (1.6947)
0.9500 (5.2545) 0.9492 (8.6046) 0.9481 (3.1734) 0.9476 (4.5061) 0.9476 (7.3767) 0.9487 (2.7236) 0.9500 (4.0086) 0.9497 (6.5582) 0.9498 (2.4238) 0.9498 (2.8026) 0.9498 (4.5813) 0.9492 (1.6960) 0.9469 (2.8596) 0.9471 (3.9900) 0.9469 (1.9206) 0.9458 (2.5116) 0.9463 (3.4744) 0.9459 (1.6939)
0.9498 (5.2545) 0.9492 (8.6046) 0.9487 (3.1734) 0.9487 (4.5061) 0.9477 (7.3767) 0.9499 (2.7236) 0.9492 (4.0086) 0.9482 (6.5582) 0.9506 (2.4238) 0.9511 (2.8026) 0.9504 (4.5813) 0.9501 (1.6960) 0.9483 (2.8596) 0.9468 (3.9900) 0.9479 (1.9206) 0.9471 (2.5116) 0.9464 (3.4744) 0.9484 (1.6939)
0.8984 (4.3738) 0.8956 (7.1352) 0.8987 (2.6471) 0.9008 (3.7707) 0.8968 (6.1489) 0.8990 (2.2805) 0.8991 (3.3550) 0.8988 (5.4774) 0.8978 (2.0292) 0.8975 (2.3468) 0.9004 (3.8299) 0.8997 (1.4218) 0.8961 (2.3862) 0.8970 (3.3321) 0.9008 (1.6036) 0.8932 (2.0990) 0.8948 (2.8961) 0.8946 (1.4171)
0.8997 (4.3848) 0.8981 (7.1774) 0.8989 (2.6495) 0.9003 (3.7670) 0.8973 (6.1625) 0.8988 (2.2774) 0.8989 (3.3532) 0.8992 (5.4836) 0.8972 (2.0279) 0.8973 (2.3484) 0.9005 (3.8378) 0.8993 (1.4212) 0.8969 (2.3888) 0.8964 (3.3300) 0.9011 (1.6051) 0.8935 (2.1010) 0.8960 (2.9049) 0.8944 (1.4176)
0.9005 (4.3824) 0.8960 (7.1757) 0.8978 (2.6485) 0.8969 (3.7646) 0.8968 (6.1598) 0.8980 (2.2763) 0.8985 (3.3518) 0.8986 (5.4818) 0.8974 (2.0275) 0.8965 (2.3475) 0.8994 (3.8366) 0.8983 (1.4204) 0.8977 (2.3879) 0.8971 (3.3294) 0.9001 (1.6050) 0.8943 (2.1006) 0.8954 (2.9033) 0.8950 (1.4178)
0.8967 (4.3824) 0.8968 (7.1757) 0.8971 (2.6485) 0.8988 (3.7646) 0.8974 (6.1598) 0.8976 (2.2763) 0.8979 (3.3518) 0.8980 (5.4818) 0.8977 (2.0275) 0.8952 (2.3475) 0.9005 (3.8366) 0.8970 (1.4204) 0.8969 (2.3879) 0.8973 (3.3294) 0.9001 (1.6050) 0.8926 (2.1006) 0.8950 (2.9033) 0.8948 (1.4178)
2715
logðy0 Þ
g ¼ 0:1
ARTICLE IN PRESS
b
g ¼ 0:05
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(r1,y,rk)
2716
Table 2 (continued ) (r1,y,rk)
(30,20,15)
Parameters
a b logðy0 Þ
a b logðy0 Þ
(12,8,6,4)
a
logðy0 Þ (15,10,8,6)
a b logðy0 Þ
(30,20,15,12)
a b logðy0 Þ
(60,40,25,18)
a b logðy0 Þ
(90,60,40,25)
a b logðy0 Þ
Exact
Approx.
Boot-p
Boot-t
Exact
Approx.
Boot-p
Boot-t
0.9525 (1.7761) 0.9539 (2.4657) 0.9507 (1.1943) 0.9497 (1.2860) 0.9510 (1.8181) 0.9509 (0.8589) 0.9543 (2.2715) 0.9525 (2.8176) 0.9548 (1.6182) 0.9517 (1.9517) 0.9514 (2.3769) 0.9523 (1.4026) 0.9512 (1.3667) 0.9514 (1.6659) 0.9505 (0.9813) 0.9478 (1.0031) 0.9473 (1.2640) 0.9475 (0.7104) 0.9516 (0.8256) 0.9494 (1.0386) 0.9505 (0.5859)
0.9523 (1.7752) 0.9540 (2.4689) 0.9507 (1.1942) 0.9501 (1.2875) 0.9524 (1.8233) 0.9507 (0.8581) 0.9542 (2.2683) 0.9528 (2.8245) 0.9544 (1.6163) 0.9526 (1.9588) 0.9517 (2.3810) 0.9531 (1.4097) 0.9522 (1.3684) 0.9512 (1.6629) 0.9514 (0.9855) 0.9478 (1.0042) 0.9472 (1.2645) 0.9485 (0.7131) 0.9513 (0.8247) 0.9506 (1.0413) 0.9500 (0.5847)
0.9504 (1.7740) 0.9535 (2.4669) 0.9480 (1.1940) 0.9506 (1.2877) 0.9498 (1.8229) 0.9493 (0.8582) 0.9522 (2.2682) 0.9500 (2.8237) 0.9526 (1.6168) 0.9510 (1.9584) 0.9518 (2.3802) 0.9502 (1.4096) 0.9508 (1.3683) 0.9510 (1.6623) 0.9510 (0.9854) 0.9461 (1.0041) 0.9473 (1.2642) 0.9481 (0.7131) 0.9492 (0.8241) 0.9494 (1.0405) 0.9481 (0.5841)
0.9506 (1.7740) 0.9533 (2.4669) 0.9502 (1.1940) 0.9489 (1.2877) 0.9504 (1.8229) 0.9485 (0.8582) 0.9541 (2.2682) 0.9526 (2.8237) 0.9524 (1.6168) 0.9519 (1.9584) 0.9507 (2.3802) 0.9527 (1.4096) 0.9508 (1.3683) 0.9499 (1.6623) 0.9515 (0.9854) 0.9466 (1.0041) 0.9474 (1.2642) 0.9482 (0.7131) 0.9513 (0.8241) 0.9493 (1.0405) 0.9488 (0.5841)
0.9020 (1.4886) 0.9037 (2.0697) 0.9027 (1.0027) 0.9024 (1.0798) 0.9042 (1.5294) 0.9033 (0.7186) 0.9061 (1.8967) 0.9046 (2.3519) 0.9079 (1.3539) 0.9043 (1.6302) 0.9047 (1.9845) 0.9050 (1.1748) 0.9031 (1.1432) 0.8992 (1.3915) 0.9060 (0.8254) 0.8948 (0.8404) 0.8982 (1.0589) 0.8941 (0.5977) 0.9008 (0.6910) 0.9016 (0.8716) 0.8992 (0.4905)
0.9016 (1.4868) 0.9031 (2.0671) 0.9022 (1.0005) 0.9025 (1.0793) 0.9044 (1.5282) 0.9035 (0.7195) 0.9057 (1.8944) 0.9051 (2.3560) 0.9071 (1.3512) 0.9061 (1.6381) 0.9055 (1.9894) 0.9062 (1.1797) 0.9047 (1.1464) 0.8995 (1.3925) 0.9059 (0.8259) 0.8957 (0.8419) 0.8981 (1.0599) 0.8938 (0.5980) 0.9015 (0.6916) 0.9027 (0.8731) 0.8986 (0.4904)
0.9001 (1.4859) 0.9020 (2.0660) 0.8998 (0.9999) 0.9022 (1.0793) 0.9027 (1.5280) 0.9024 (0.7195) 0.9056 (1.8938) 0.9023 (2.3552) 0.9072 (1.3509) 0.9045 (1.6377) 0.9044 (1.9890) 0.9045 (1.1797) 0.9043 (1.1461) 0.9002 (1.3921) 0.9060 (0.8258) 0.8940 (0.8418) 0.8966 (1.0595) 0.8950 (0.5982) 0.8989 (0.6912) 0.9015 (0.8727) 0.8991 (0.4902)
0.9012 (1.4859) 0.9019 (2.0660) 0.9008 (0.9999) 0.8999 (1.0793) 0.9032 (1.5280) 0.9011 (0.7195) 0.9058 (1.8938) 0.9029 (2.3552) 0.9056 (1.3509) 0.9049 (1.6377) 0.9051 (1.9890) 0.9045 (1.1797) 0.9028 (1.1461) 0.9010 (1.3921) 0.9045 (0.8258) 0.8952 (0.8418) 0.8966 (1.0595) 0.8931 (0.5982) 0.9005 (0.6912) 0.9011 (0.8727) 0.8964 (0.4902)
ARTICLE IN PRESS
b
g ¼ 0:1
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(60,40,25)
g ¼ 0:05
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Because the distributions of the pivotal quantities W4,W5,W6 depend only on the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j at the failure times ti.j, i= 1,2,y,k, j= 1,2,y,ri, without loss of generality, we choose Ri,j = 0, i=1,2,y,k, j= 1,2,y,ri. We consider only various combinations (r1,r2,y,rk). The column 1 in Table 2 lists the different censoring schemes used in the simulation study. The confidence coefficients are 0.9 and 0.95. For the proposed confidence intervals, note that 2Ti =yi follows the w2 ð2ri Þ distribution, we generate k random numbers with the w2 ð2ri Þ ði ¼ 1, . . . ,kÞ distributions for each censoring scheme. We compute the proposed confidence intervals. Based on 10,000 Monte Carlo simulations with B =1000 replications, the coverage percentages and the average interval lengths of the proposed confidence intervals for the different (r1,r2,y,rk) schemes are obtained. These values are tabulated in Table 2. The average interval lengths are given in parentheses. From Table 2, we observe that all the coverage percentages of the proposed confidence intervals are close to the nominal coverage probabilities, even for small sample sizes. We also observe that the differences of the average lengths of the proposed confidence intervals are small. As the number of the accelerated stress levels or the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk increase, the average lengths of confidence intervals reduce appreciably. Considering the above simulation results and the fact that the percentage points of the pivotal quantities W4,W5,W6 are obtained by simulation method when k 42, the approximate confidence intervals are suggested unless r1,r2,y,rk are very small. 5. An illustrative example To illustrate the proposed procedures, recall the example of Section 4 of Wang and Fei (2003). This example is to obtain all the reliability indices of a kind of electronic components at the normal temperature of S0 = 25 1C, Now n = 100 units from a batch of products are randomly selected for the simple accelerated life test model logðyÞ ¼ 1=ð273:15 þSÞ. The accelerated temperature levels are S1 = 100 1C and S2 = 150 1C. At S1 = 100 1C, when 30 products have failed, the stress level rises to S2 = 150 1C and the test continues until 20 more products have failed. Their failure times are as follows. Failure times for the stress level S1: 32, 54, 59, 86, 117, 123, 213, 267, 268, 273, 299, 311, 321, 333, 339, 386, 408, 422, 435, 437, 476, 518, 570, 632, 666, 697, 796, 854, 858, 910. Failure times for the stress level S2: 16, 19, 21, 36, 37, 63, 70, 75, 83, 95, 100, 106, 110, 113, 116, 135, 136, 149, 172, 186. These times are the differences t2,j t1,r1 . This step-stress accelerated life testing is a SSALT with Type-II censoring, which is a specific case about the progressive censoring scheme with Ri,j =0, i= 1,2,j =1,2,y,ri. From Theorem 1, the 5% and 95% percentiles of the pivotal quantities W1, W2, W3 are given by W1,0.05 = 0.0349, W1,0.95 = 80.9612, F0.05(60,40)= 0.6272, F0.95(60,40)= 1.6373, W3,0.05 =41.2187, W3,0.95 =487.3390, respectively. Thus the 90% exact confidence intervals for a, b, y0 are given by ½8:6949,0:9458,
½3218:8,6246:4,
½18 504,218 780,
respectively. The 90% approximate confidence intervals for a, b, y0 are given by ½8:7009,0:9495,
½3221:3,6251:1,
½18 499,218 645,
respectively. It is observed that two methods give results that are almost in agreement.
Acknowledgements The authors thank the Associate Editor and the referees for their detailed comments and suggestions, which helped to improve the manuscript. References Bagdonavicˇius, V.B., Le´o, G., Nikulin, M., 2002. Parametric inference for step-stress models. IEEE Trans. Reliab. R-51, 27–31. Bai, D.S., Kim, M.S., Lee, S.H., 1989. Optimum simple step-stress accelerated life tests with censoring. IEEE Trans. Reliab. R-38, 529–532. ¨ Balakrishnan, N., Aggarwala, R., 2000. Progressive Censoring: Theory, Methods and Applications. Birkhauser, Boston. Balakrishnan, N., Han, D., 2009. Optimal step-stress testing for progressively Type-I censored data from exponential distribution. J. Statist. Plann. Inference 139, 1782–1798. Balakrishnan, N., Kundu, D., Ng, H.K.T., Kannan, N., 2007. Point and interval estimation for a simple step-stress model with type-II censoring. J. Quality Technol. 39, 35–47. Bhattacharyya, G.K., Soejoeti, Z., 1989. A tampered failure rate model for step-stress accelerated life test. Comm. Statist. Theory & Method 18, 1627–1643. Casella, G., Berger, R.L., 2001. Statistical Inference, second ed. Duxbury Press, Belmont. Cornish, E.A., Fisher, R.A., 1937. Moments and cumulants in the specification of distributions. Rev. Int. Statist. Inst. 5, 307–320. Cheng, Y.M., 1994. Optimum plans for step-stress accelerated life testing. Chinese J. Appl. Probab. Statist. 10, 52–61. DeGroot, M.H., Goel, P.K., 1979. Bayesian estimation and optimal designs in partially accelerated life testing. Naval Res. Logist. Quart. 26, 223–235. Dorp, J.R., Mazzuchi, T.A., 2004. A general Bayes exponential inference model for accelerated life testing. J. Statist. Plann. Inference 119, 55–74. Dorp, J.R., Mazzuchi, T.A., Fornell, G.E., Pollock, L.R., 1996. A Bayes approach to step-stress accelerated life testing. IEEE Trans. Reliab. R-45, 491–498. Efron, B., The Jackknife, the bootstrap and other re-sampling plans, in: CBMS/NSF Regional Conference Series in Applied Mathematics, vol. 38, SIAM, Philadelphia, PA, 1982.
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Gouno, E., Sen, A., Balakrishnan, N., 2004. Optimal step-stress test under progressive type-I censoring. IEEE Trans. Reliab. R-53, 388–393. Hall, P., 1988. Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16, 927–953. Khamis, I., Higgins, J.J., 1996. Optimum 3-step step-stress tests. IEEE Trans. Reliab. R-45, 341–345. Khamis, I., Higgins, J.J., 1998. A new model for step-stress testing. IEEE Trans. Reliab. R-47, 131–134. Lawless, J.F., 1982. Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York. Mao, S.S., Wang, L.L., 1997. Accelerated Life Testing. Science Press, Beijing. Nelson, W., 1980. Accelerated life testing–step-stress models and data analyses. IEEE Trans. Reliab. R-29, 103–108. Nelson, W., 1990. Accelerated Life Testing: Statistical Models, Test Plans, and Data Analyses. John Wiley & Sons, New York. Tang, L.C., Goh, T.N., Sun, Y.S., Ong, H.L., 1999. Planning accelerated life tests for censored two-parameter exponential distributions. Naval Res. Logist. 46, 169–186. Tang, L.C., Sun, Y.S., Goh, T.N., Ong, H.L., 1996. Analysis of step-stress accelerated-life-test data: a new approach. IEEE Trans. Reliab. R-45, 69–74. Teng, S.L., Yeo, K.P., 2002. A least-squares approach to analyzing life-stress relationship in step-stress accelerated life tests. IEEE Trans. Reliab. R-51, 177–182. Wang, B.X., 2006. Unbiased estimations for the exponential distribution based on step-stress accelerated life-testing data. Appl. Math. Comput. 173, 1227–1237. Wang, B.X., Yu, K., 2009. Optimum plan for step-stress model with progressive type-II censoring. Test 18, 115–135. Wang, R.H., Fei, H.L., 2004. Conditions for the coincidence of the TFR,TRV and CE models. Statist. Papers 45, 393–412. Wang, R.H., Fei, H.L., 2003. Uniqueness of the maximum likelihood estimate of the Weibull distribution tampered failure rate model. Comm. Statist. Theory and Methods 32, 2321–2338.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing Bing Xing Wang Zhejiang Gongshang University, Hangzhou, PR China
a r t i c l e i n f o
abstract
Article history: Received 3 December 2008 Received in revised form 12 October 2009 Accepted 16 March 2010 Available online 21 March 2010
This paper derives the exact confidence intervals for the exponential step-stress accelerated life-testing model as well as the approximate confidence intervals for the k-step exponential step-stress accelerated life-testing model under progressive Type-II censoring. A Monte Carlo simulation study is carried out to examine the performance of these confidence intervals. Finally, an example is given to illustrate the proposed procedures. & 2010 Elsevier B.V. All rights reserved.
Keywords: Accelerated life test Step-stress Exponential distribution Progressive Type-II censoring Interval estimation Cumulant
1. Introduction In many situations, it may be difficult to collect data on life-time of a product under normal operating conditions as the product may have a high reliability under normal conditions. For this reason, accelerated life-testing (ALT) experiments can be used to force these products to fail more quickly than under normal operating condition. The results obtained at the accelerated conditions are analyzed in terms of a model to relate life length to stress; they are extrapolated to the design stress to estimate the life distribution. The step-stress ALT (SSALT) was developed to reduce operation time and costs further. One way of applying stress to the test units is the step-stress scheme which allows the stress setting of a unit to be changed at pre-specified times or upon the occurrence of a fixed number of failures. The former is called SSALT with Type-I censoring and the latter is called SSALT with Type-II censoring. The problem of modeling data from SSALT and making inferences from such data have been studied by many authors. DeGroot and Goel (1979) proposed the tampered random variable model. Nelson (1980) proposed the cumulative exposure model. Bhattacharyya and Soejoeti (1989) proposed the tampered failure rate model. Wang and Fei (2004) discussed conditions for the coincidence of these models. Khamis and Higgins (1998), and Bagdonavicˇius et al. (2002) proposed some new models for SSALT. Tang et al. (1996) obtained the maximum likelihood estimations (MLE) for parameters in a multi-censored ALT. Teng and Yeo (2002) used the method of least-squares to estimate the life-stress relationship in SSALT. Dorp et al. (1996) gave a Bayes approach to SSALT. Dorp and Mazzuchi (2004) developed a general Bayes exponential inference model for ALT. Wang (2006) obtained unbiased estimations for the exponential distribution based on SSALT data. Balakrishnan et al. (2007) obtained point and interval estimation for the exponential simple
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.03.033
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step-stress model. Bai et al. (1989) obtained the optimum simple SSALT plans. Khamis and Higgins (1996) obtained the optimum 3-step SSALT plans. Cheng (1994) extended the results of Bai et al. (1989) to case of the k-step SSALT. Tang et al. (1999) discussed optimum plan for two-parameter exponential distribution. Many examples of ALT, as well as an excellent introduction to the methodology, are given in Nelson (1990) and Mao and Wang (1997). One popular SSALT model is the SSALT model with Type-I censoring, where stress levels change at pre-fixed time points. This model, although simple and intuitive, has the following problems: (1) when there is no observation under some particular stress-levels, the MLEs for the parameters corresponding to these stress levels do not exist; (2) there are quite complicated expressions for the distributions and the moments of the MLEs; see, for example, Balakrishnan et al. (2007). To alleviate these problems as well as for some other reasons associated with the performance of a life-test, experimenters may consider increasing a stress level only after having observed a pre-specified number of observations at the current level of stress. Under this proposed alternative model, the step-stress model is the SSALT with Type-II censoring. Compared to the SSALT model with Type-I censoring, this approach ensures the existence of the MLEs of all the parameters in the model. Progressive censoring is a generalized form of censoring which includes the conventional right censoring as a special case. Compared to the conventional censoring, however, it provides higher flexibility to the experimenter in the design stage by allowing the removal of test units at non-terminal time points and thus, it proves to be highly efficient and effective in utilizing the available resources. For this reason, we consider a more general censoring scheme called progressive Type-II censoring. Progressive Type-II censoring is a method which enables an efficient exploitation of the available resources by continual removal of a pre-specified number of surviving test units at each failure time. Another advantage of progressive censoring is that the degeneration information of the test units is obtained from those removed units. Gouno et al. (2004) and Balakrishnan and Han (2009) discussed the optimal SSALT plans under progressive Type-I censoring. Wang and Yu (2009) discussed the optimal SSALT plans under progressive Type-II censoring. A book dedicated completely to progressive censoring was published by Balakrishnan and Aggarwala (2000). The main objective of this paper is to investigate the interval estimations for the exponential SSALT model under progressive Type-II censoring. The paper is organized as follows. Section 2 gives some basic assumptions in the exponential SSALT. Section 3 derives the exact confidence intervals for the exponential SSALT model and the approximate confidence intervals for the k-step exponential SSALT model under progressive Type-II censoring. Section 4 gives a Monte Carlo simulation to study the coverage percentages and the average confidence lengths of the proposed confidence intervals. Section 5 gives an example to illustrate the proposed procedures. 2. Basic assumptions Statistical inference for the exponential SSALT usually depends on the following assumptions: 1. For any stress level xi, the lifetime distribution of a test unit is exponential with the cumulative distribution function (cdf): Fi ðtÞ ¼ 1expðt=yi Þ,
t 4 0,
ð1Þ
where yi 4 0 is the mean life of test unit at stress level xi. 2. The log mean life of a test unit logðyi Þ is a linear function of stress level xi: logðyi Þ ¼ a þ bxi ,
ð2Þ
where a and b are unknown parameters. 3. A cumulative exposure model holds: the remaining life depends only on the current cumulative failure probability and current stress level regardless of how the probability is accumulated. (Nelson, 1980). The cumulative exposure model may be replaced by the tampered random variable model or the tampered failure rate model (see DeGroot and Goel, 1979; Bhattacharyya and Soejoeti, 1989). 3. Interval estimation SSALT with progressive Type-II censoring is as follows. Suppose that all n test units are initially placed at lowest stress x1. At the first failure time t1,1, R1,1 units are randomly removed from the remaining n 1 surviving units. At the second failure time t1,2, R1,2 units are randomly removed from the remaining n 2 R1,1. The test continues until the r1th failure P time t1,r1 . At failure time t1,r1 , R1,r1 units are randomly removed from the remaining nr1 jr1¼11 R1,j surviving units. Then the stress is changed to x2. Similar to stress x1, the test is continued until r2 units have failed. At the stress x2, r2 failure times t2,j (j= 1,2,y,r2) of test units are observed. The stress is changed again, and the test continues. Suppose that the stress is finally changed to xk, and test does not stop until rk units have failed under xk. We write the failure times under the stress xi (i= 1, 2, y, k) as ti,1 r ti,2 r r ti,ri :
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At failure time ti,j, Ri,j units are randomly removed. Further, suppose that the lifetime at stress level xi has the exponential distribution (1). For i=1, 2, y, k, let 2 0 13 rl ri i X X X @rl þ ð1 þRi,j Þðti,j ti1,ri1 Þ þ 4n Rl,j A5ðti,ri ti1,ri1 Þ ð3Þ Ti ¼ j¼1
l¼1
j¼1
be the total time on test at stress xi, where t0,r0 ¼ 0. In order to obtain the confidence intervals of the exponential SSALT model, the following lemmas are necessary. Lemma 1. Suppose that T1,T2,y,Tk are defined as (3), then, under the assumptions 1–3, T1,T2,y,Tk are independent, and the distribution of the 2Ti =yi is w2 ð2ri Þ ði ¼ 1,2, . . . ,kÞ. (see Wang and Yu, 2009) Lemma 2. Let F(y) is the cdf of w2 ð2rÞ, then FðyÞ ¼ 1ey=2
r 1 X ðy=2Þj j¼0
j!
,
y 40:
It is a relationship between the gamma and Poisson distributions. See Casella and Berger (2001, P100). Lemma 3. Let X follows the log-gamma distribution with the probability function (pdf): f ðxÞ ¼
1
GðrÞ
expðrxex Þ,
then the ith cumulant of X is
ki ¼
di1 cðrÞ dr
i1
,
where cðrÞ ¼ dlogðGðrÞÞ=dr is the digamma function. Proof. We have from Lawless (1982, P22) that the moment generating function for X is MðyÞ ¼ Gðy þrÞ=GðrÞ. Therefore the i th cumulants of X is given by di logðMðyÞÞ di1 cðrÞ ki ¼ ¼ : i i1 dy dr y¼0 The proof is completed.
&
3.1. k= 2 case In this subsection, we shall derive the exact confidence intervals for simple exponential SSALT model based on progressively Type-II censored sample. Under the Assumptions 1–3, we have from Wang and Yu (2009) that the MLEs of the parameters a and b are given by
a^ ¼
x1 logðT2 =r2 Þx2 logðT1 =r1 Þ , x1 x2
ð4Þ
b^ ¼
logðT1 =r1 ÞlogðT2 =r2 Þ , x1 x2
ð5Þ
and
respectively. Therefore, the MLE of the mean life y0 at designed stress level x0 is
y^ 0 ¼ expða^ þ b^ x0 Þ ¼ r1k0 1 r2k0 T1k0 þ 1 T2k0 , where k0 ¼ ðx0 x1 Þ=ðx1 x2 Þ. Let W1 ¼ 2exp½a^ a þlogðr2 Þ þ k1 logðr2 =r1 Þ,
ð6Þ
W2 ¼ exp½ðx1 x2 Þðb^ bÞ,
ð7Þ
W3 ¼ 2r1k0 þ 1 r2k0 y^ 0 =y0 ,
ð8Þ
where k1 = x2/(x1 x2). Then W1 ¼ ð2T2 =y2 Þ1 þ k1 ð2T1 =y1 Þk1 ,
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W2 ¼
2709
2T1 =y1 r2 , 2T2 =y2 r1
W3 ¼ ð2T1 =y1 Þ1 þ k0 ð2T2 =y2 Þk0 : Hence it follows from Lemma 1 that W1,W2,W3 are the pivotal quantities. The following theorem gives the cumulative distribution functions of the pivotal quantities W1,W2,W3. Theorem 1. Suppose the pivotal quantities W1,W2,W3 are defined as Eqs. (6)–(8), then (1) the cdf of W1 is given by Z 1 rX 2 1 ðy=2Þj FW1 ðwÞ ¼ 1 dt, g1 ðtÞey=2 j! 0 j¼0
ð9Þ
where y ¼ ðwt k1 Þ1=ðk1 þ 1Þ , g1(t) is the pdf of w2 ð2r1 Þ; (2) W2 Fð2r1 ,2r2 Þ; (3) the cdf of W3 is given by Z 1 rX 1 1 ðy=2Þj FW3 ðwÞ ¼ 1 dt, g2 ðtÞey=2 j! 0 j¼0
ð10Þ
where y ¼ ðwt k0 Þ1=ðk0 þ 1Þ , g2(t) is the pdf of w2 ð2r2 Þ. Proof. (1) Let U1 ¼ 2 T1 =y1 , U2 ¼ 2 T2 =y2 . From Lemmas 1 and 2, we have FW1 ðwÞ ¼ PðW1 rwÞ ¼ PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ Þ ¼ E½PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ jU1 Þ Z 1 Z Z 1 PðU2 r ðwU k11 Þ1=ð1 þ k1 Þ jU1 ¼ tÞg1 ðtÞ dt ¼ FðyÞg1 ðtÞ dt ¼ 1 ¼ 0
0
1
g1 ðtÞey=2
0
rX 2 1
ðy=2Þj dt: j! j¼0
2
where F(y) is the cdf of w ð2r2 Þ. (2) Since W2 ¼
2 T1 =y1 2r2 , 2 T2 =y2 2r1
it follows from Lemma 1 that W2 Fð2r1 ,2r2 Þ. (3) Similar to (1), we can obtain the cdf of W3. The proof is completed. & In computing the improper integral in (9) or (10), one must truncate the integral at some finite value. The numerical error caused by this truncation can be assessed by the following error bound formula. For example, in evaluating (9), if the integration range is from 0 to M, then the error is
eM ¼
Z
1
g1 ðtÞey=2 M
rX 2 1
ðy=2Þj dt r j! j¼0
Z
1
g1 ðtÞ dt ¼ eM=2 M
rX 1 1
ðM=2Þj : j! j¼0
For any specified e 4 0, one can choose a large M so that eM o e. Thus, we can evaluate the improper integrals as accurately as desired. Based on Theorem 1, it is easy to obtain the confidence intervals of the parameters a, b and y0 . Theorem 2. Suppose that a^ , b^ , y^ 0 are MLEs of the parameters a, b, y0 . Then, for any 0 o g o1, (1) a 100ð1gÞ% exact confidence interval for the parameter a is given by a^ þ logðr2 Þ þ k1 logðr2 =r1 ÞlogðW1,1g=2 =2Þ, a^ þlogðr2 Þ þ k1 logðr2 =r1 ÞlogðW1, g=2 =2Þ , where W1, g is the g percentile of W1. (2) a 100ð1gÞ% exact confidence interval for the parameter b is given by logF1g=2 ð2r1 ,2r2 Þ ^ logFg=2 ð2r1 ,2r2 Þ , b^ ; b x1 x2 x1 x2 where Fg ð2r1 ,2r2 Þ is the g percentile of F(2r1,2r2).
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(3) a 100ð1gÞ% exact confidence interval for the parameter y0 is given by " # 2r11 þ k0 r2k0 y^ 0 2r11 þ k0 r2k0 y^ 0 , , W3,1g=2 W3, g=2 where W3, g is the g percentile of W3. 3.2. k Z2 case In this subsection, we shall derive the exact and the approximate confidence intervals for the kð Z2Þstep exponential SSALT model based on progressively Type-II censored sample. Under the assumptions 1–3, we have from Wang and Yu (2009) that the unbiased estimators for a, b,logðy0 Þ are given by
a~ ¼
GHIM , EGI2
b~ ¼
EMIH , EGI2
logðy~ 0 Þ ¼ a~ þ b~ x0 ,
respectively, where Ui ¼ logðTi Þcðri Þ, E ¼ P 0 M ¼ ki¼ 1 ½c ðri Þ1 xi Ui . Further Varða~ Þ ¼
G , EGI2
E , EGI2
Varðb~ Þ ¼
Pk
0 1 , i ¼ 1 ½c ðri Þ
Varðlogðy~ 0 ÞÞ ¼
ð11Þ I¼
Pk
0 1 i ¼ 1 ½c ðri Þ
xi , G ¼
Pk
0 1 i ¼ 1 ½c ðri Þ
x2i , H ¼
Pk
0 1 i ¼ 1 ½c ðri Þ
Ui ,
Gþ x20 E2x0 I : EGI2
Let
a~ a W4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varða~ Þ
b~ b W5 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðb~ Þ
logðy~ 0 Þlogðy0 Þ W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Varðlogðy~ 0 ÞÞ
Write Yi ¼ logðTi =yi Þcðri Þ, from H¼
k X
0
½c ðri Þ1 Ui ¼
i¼1
M¼
k X
k X
0
½c ðri Þ1 Yi þ aEþ bI :¼ H1 þ aE þ bI,
i¼1
0
½c ðri Þ1 xi Ui ¼
i¼1
k X
0
½c ðri Þ1 xi Yi þ aI þ bG :¼ M1 þ aI þ bG,
i¼1
we have
a~ ¼
GH1 IM 1 þ a, EGI2
b~ ¼
EM 1 IH1 þ b, EGI2
so GH1 IM1 W4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEGI2 Þ, Varða~ Þ
ð12Þ
EM 1 IH1 W5 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEGI2 Þ, Varðb~ Þ
ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a~ þ b~ x0 abx0 1 W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Varða~ ÞW4 þ Varðb~ Þx0 W5 Þ: Varðlogðy~ 0 ÞÞ Varðlogðy~ 0 ÞÞ
ð14Þ
Then it follows from Lemma 1 that W4,W5,W6 are the pivotal quantities. Because the distributions of the pivotal quantities W4,W5,W6 depend only on the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j, i= 1,2,y,k, j = 1,2,y,ri at the failure times ti.j, i= 1,2,y,k, j = 1,2,y,ri, without loss of generality, we choose Ri,j = 0, i= 1,2,y,k, j = 1,2,y,ri. Hence, in order to obtain the exact percentiles of W4, W5, W6, we need only to consider various combinations (r1,r2,y,rk). The exact confidence intervals of a, b, y0 based on W4, W5, W6 can be obtained by the following steps. Step 1: Given r1,y,rk, generate a random sample of 2Ti =yi ,i ¼ 1, . . . ,k, where 2Ti =yi ,i ¼ 1, . . . ,k are independently the
w2 ð2ri Þ distributions. Step 2: Given stress levels x0,x1,y,xk, calculate W4, W5, W6 from Eqs. (12)–(14). Step 3: Repeat step 1–2 N times. Step 4: Arrange all W4, W5, W6 values in ascending order: Wi½1 o Wi½2 o o Wi½N ,
i ¼ 4,5,6:
ARTICLE IN PRESS B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718 ½gN
½gN
½gN
Step 5: The exact g percentiles of W4,W5,W6 are W4 ,W5 ,W6 , respectively. Step 6: The exact 100ð1gÞ% confidence intervals of a, b, y0 based on W4, W5, W6 are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1g=2ÞN ½gN=2 ½a~ W4 Varða~ Þ, a~ W4 Varða~ Þ, ½ð1g=2ÞN
½b~ W5
½ð1g=2ÞN
½y~ 0 eW6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½gN=2 Varðb~ Þ, b~ W5 Varðb~ Þ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðlogðy~ 0 ÞÞ
½gN=2 , y~ 0 eW6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðlogðy~ 0 ÞÞ
2711
ð15Þ ð16Þ
,
ð17Þ
respectively. From (15)–(17), we obtain that the lengths of the exact confidence intervals for the parameters a, b,logðy0 Þ are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1g=2ÞN ½gN=2 ½ð1g=2ÞN ½gN=2 ½ð1g=2ÞN ½gN=2 W4 Þ, Varðb~ ÞðW5 W5 Þ and Varðlogðy~ 0 ÞÞðW6 W6 Þ, respectively. Varða~ ÞðW4 In particular, when k= 2, from Eqs. (4), (5), (11), we have
a~ ¼ a^
x1 ðcðr2 Þlogðr2 ÞÞx2 ðcðr1 Þlogðr1 ÞÞ , x1 x2
b~ ¼ b^
ðcðr1 Þlogðr1 ÞÞðcðr2 Þlogðr2 ÞÞ : x1 x2
and
Hence, we obtain W4 ¼
logðW1 Þc1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varða~ Þ
W5 ¼
logðW2 Þc2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðx1 x2 Þ Varðb~ Þ
logðW3 Þc3 W6 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðlogðy~ 0 ÞÞ
where c1 ¼
x1 cðr2 Þx2 cðr1 Þ þ logð2Þ, x1 x2
c2 ¼ ðcðr1 Þlogðr1 ÞÞðcðr2 Þlogðr2 ÞÞ, c3 ¼ logð2Þ þðk0 þ 1Þcðr1 Þk0 cðr2 Þ: Since it is difficult to obtain the exact percentiles of W4, W5, W6, we try to find approximate percentiles of W4, W5, W6 by the Cornish–Fisher expansion. Note from (12)–(14) that W4,W5,W6 are linear combinations of Y1,Y2,y,Yk, after some calculations, we have from Lemmas 1 and 3 that the first five cumulants of W4,W5,W6 are given by Pk 0 ðj1Þ ðGxi IÞj ½c ðri Þj c ðri Þ kW4 ,1 ¼ 0, kW4 ,2 ¼ 1, kW4 ,j ¼ i ¼ 1 , j ¼ 3,4,5, ð18Þ j j=2 ðEGI2 Þ ½Varða~ Þ
kW5 ,1 ¼ 0, kW5 ,2 ¼ 1, kW5 ,j ¼
Pk
0 j j ðj1Þ ðri Þ i ¼ 1 ðxi EIÞ ½c ðri Þ c , j ~ 2 ðEGI Þ ½Varðb Þj=2
Pk
kW6 ,1 ¼ 0, kW6 ,2 ¼ 1, kW6 ,j ¼
Dji ½c ðri Þj c ðri Þ , j ~ 2 ðEGI Þ ½Varðlogðy 0 ÞÞj=2 i¼1
0
j ¼ 3,4,5,
ð19Þ
ðj1Þ
j ¼ 3,4,5,
ð20Þ
respectively, where Di = G+x0 xi E (x0 + xi)I. Cornish and Fisher (1937) provided an expansion for approximating the g percentile, xg , of X based on its cumulants. Using the first five cumulants, the expansion is 1 1 1 1 1 k4 ðz3g 3zg Þ36 k23 ð2z3g 5zg Þ þ 120 k5 ðz4g 6z2g þ 3Þ24 k3 k4 ðz4g 5z2g þ 2Þ þ 324 k33 ð12z4g 53z2g þ 17Þ, xg ¼ zg þ 16 k3 ðz2g 1Þ þ 24
ð21Þ where zg is the g percentile of the standard normal distribution N(0,1). Using Eqs. (18)–(21), it is easy to obtain the approximate g percentiles W4, g ,W5, g ,W6, g of W4,W5,W6. Hence the 100ð1gÞ% approximate confidence intervals for a, b, y0 are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ ½a~ W4,1g=2 Varða~ Þ, a~ W4, g=2 Varða~ Þ, ½b~ W5,1g=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðb~ Þ, b~ W5, g=2 Varðb~ Þ,
ð23Þ
2712
Table 1 Experiment schemes and the exact and approximate percentiles of W4,W5,W6. 0.025
0.05
0.1
0.9
0.95
0.975
(9,6)
W4
2.0013 ( 1.9912) 2.0143 ( 2.0219) 2.0454 ( 2.0353) 1.9953 ( 1.9864) 2.0166 ( 2.0119) 2.0344 ( 2.0247) 1.9908 ( 1.9832) 2.0084 ( 2.0054) 2.0273 ( 2.0176) 1.9807 ( 1.9756) 1.9961 ( 1.9905) 2.0037 ( 2.0002)
1.6599 ( 1.6554) 1.6550 ( 1.6722) 1.6882 ( 1.6825) 1.6579 ( 1.6546) 1.6608 ( 1.6688) 1.6789 ( 1.6782) 1.6530 ( 1.6539) 1.6639 ( 1.6665) 1.6701 ( 1.6750) 1.6472 ( 1.6518) 1.6545 ( 1.6604) 1.6641 ( 1.6668)
1.2752 ( 1.2767) 1.2763 ( 1.2816) 1.2822 ( 1.2876) 1.2790 ( 1.2787) 1.2793 ( 1.2830) 1.2824 ( 1.2881) 1.2747 ( 1.2798) 1.2826 ( 1.2837) 1.2791 ( 1.2882) 1.2779 ( 1.2816) 1.2861 ( 1.2845) 1.2804 ( 1.2875)
1.2657 (1.2661) 1.2596 (1.2587) 1.2541 (1.2564) 1.2732 (1.2694) 1.2674 (1.2633) 1.2666 (1.2609) 1.2752 (1.2714) 1.2641 (1.2661) 1.2694 (1.2638) 1.2766 (1.2757) 1.2681 (1.2721) 1.2740 (1.2701)
1.6157 (1.6285) 1.6081 (1.6102) 1.5917 (1.6003) 1.6307 (1.6308) 1.6163 (1.6155) 1.6101 (1.6064) 1.6351 (1.6324) 1.6179 (1.6190) 1.6178 (1.6106) 1.6385 (1.6362) 1.6264 (1.6272) 1.6252 (1.6209)
1.9381 (1.9476) 1.9239 (1.9173) 1.8897 (1.8989) 1.9538 (1.9475) 1.9226 (1.9222) 1.9084 (1.9057) 1.9554 (1.9478) 1.9249 (1.9258) 1.9152 (1.9106) 1.9586 (1.9497) 1.9316 (1.9350) 1.9304 (1.9238)
1.9953 ( 2.0031) 2.0062 ( 2.0102) 2.0342 ( 2.0351) 1.9905 ( 2.0004) 1.9920 ( 2.0005) 2.0211 ( 2.0284) 1.9936 ( 1.9859) 1.9778 ( 1.9893) 2.0102 ( 2.0066)
1.6582 ( 1.6644) 1.6733 ( 1.6665) 1.6772 ( 1.6843) 1.6588 ( 1.6642) 1.6594 ( 1.6628) 1.6755 ( 1.6815) 1.6578 ( 1.6579) 1.6581 ( 1.6593) 1.6735 ( 1.6706)
1.2799 ( 1.2820) 1.2867 ( 1.2809) 1.2858 ( 1.2903) 1.2808 ( 1.2835) 1.2810 ( 1.2817) 1.2885 ( 1.2906) 1.2921 ( 1.2838) 1.2789 ( 1.2836) 1.2944 ( 1.2889)
1.2732 (1.2649) 1.2677 (1.2631) 1.2638 (1.2581) 1.2714 (1.2672) 1.2681 (1.2669) 1.2677 (1.2611) 1.2739 (1.2731) 1.2816 (1.2721) 1.2721 (1.2686)
1.6230 (1.6204) 1.6125 (1.6172) 1.6037 (1.5997) 1.6239 (1.6217) 1.6159 (1.6225) 1.6084 (1.6036) 1.6339 (1.6299) 1.6335 (1.6282) 1.6210 (1.6169)
1.9349 (1.9312) 1.9180 (1.9264) 1.8998 (1.8947) 1.9294 (1.9304) 1.9312 (1.9323) 1.8979 (1.8990) 1.9337 (1.9395) 1.9436 (1.9371) 1.9140 (1.9172)
W5 W6 (12,8)
W4 W5 W6
(15,10)
W4 W5 W6
(30,20)
W4 W5 W6
(12,8,6)
W4 W5 W6
(15,10,8)
W4 W5 W6
(30,20,15)
W4 W5 W6
ARTICLE IN PRESS
Pivotal quantities
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(r1,y,rk)
(60,40,25)
W4 W5 W6
W4 W5 W6
(15,10,8,6)
W4
W6 (30,20,15,12)
W4 W5 W6
(60,40,25,18)
W4 W5 W6
(90,60,40,25)
W4 W5 W6
1.2905 ( 1.2828) 1.2802 ( 1.2846) 1.2884 ( 1.2867)
1.2708 (1.2769) 1.2852 (1.2745) 1.2693 (1.2734)
1.6331 (1.6365) 1.6399 (1.6307) 1.6197 (1.6267)
1.9417 (1.9485) 1.9494 (1.9391) 1.9326 (1.9318)
2.0009 ( 2.0052) 2.0130 ( 2.0159) 2.0249 ( 2.0335) 1.9903 ( 2.0043) 1.9994 ( 2.0018) 2.0069 ( 2.0279) 1.9783 ( 1.9897) 1.9911 ( 1.9896) 1.9870 ( 2.0070) 1.9647 ( 1.9760) 1.9905 ( 1.9873) 1.9659 ( 1.9902) 1.9858 ( 1.9729) 1.9597 ( 1.9816) 1.9943 ( 1.9845)
1.6622 ( 1.6661) 1.6631 ( 1.6698) 1.6812 ( 1.6839) 1.6505 ( 1.6670) 1.6634 ( 1.6638) 1.6666 ( 1.6817) 1.6499 ( 1.6605) 1.6621 ( 1.6596) 1.6615 ( 1.6711) 1.6442 ( 1.6532) 1.6637 ( 1.6595) 1.6554 ( 1.6620) 1.6565 ( 1.6518) 1.6453 ( 1.6567) 1.6647 ( 1.6589)
1.2872 ( 1.2831) 1.2794 ( 1.2819) 1.2956 ( 1.2906) 1.2769 ( 1.2851) 1.2791 ( 1.2823) 1.2811 ( 1.2912) 1.2850 ( 1.2850) 1.2832 ( 1.2840) 1.2890 ( 1.2894) 1.2813 ( 1.2833) 1.2817 ( 1.2852) 1.2888 ( 1.2868) 1.2879 ( 1.2832) 1.2767 ( 1.2848) 1.2899 ( 1.2861)
1.2706 (1.2646) 1.2629 (1.2618) 1.2695 (1.2589) 1.2701 (1.2667) 1.2614 (1.2669) 1.2664 (1.2617) 1.2757 (1.2725) 1.2655 (1.2722) 1.2694 (1.2687) 1.2777 (1.2766) 1.2683 (1.2738) 1.2767 (1.2735) 1.2764 (1.2777) 1.2711 (1.2756) 1.2733 (1.2752)
1.6268 (1.6189) 1.6146 (1.6137) 1.6098 (1.6005) 1.6197 (1.6190) 1.6140 (1.6216) 1.6054 (1.6038) 1.6289 (1.6274) 1.6230 (1.6279) 1.6241 (1.6165) 1.6387 (1.6355) 1.6217 (1.6288) 1.6315 (1.6267) 1.6296 (1.6373) 1.6377 (1.6321) 1.6246 (1.6301)
1.9381 (1.9281) 1.9138 (1.9204) 1.9086 (1.8954) 1.9250 (1.9251) 1.9260 (1.9302) 1.8996 (1.8983) 1.9414 (1.9349) 1.9419 (1.9363) 1.9193 (1.9161) 1.9538 (1.9467) 1.9314 (1.9359) 1.9410 (1.9316) 1.9406 (1.9489) 1.9524 (1.9406) 1.9351 (1.9367)
ARTICLE IN PRESS
W5
1.6569 ( 1.6522) 1.6511 ( 1.6577) 1.6647 ( 1.6619)
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(12,8,6,4)
1.9767 ( 1.9745) 1.9629 ( 1.9843) 1.9930 ( 1.9903)
2713
ARTICLE IN PRESS 2714
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
½y~ 0 eW6,1g=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Varðlogðy~ 0 ÞÞ
, y~ 0 eW6, g=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Varðlogðy~ 0 ÞÞ
,
ð24Þ
respectively. From (22)–(24), we obtain that the lengths of the approximate confidence intervals for the parameters a, b,logðy0 Þ are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varða~ ÞðW4,1g=2 W4, g=2 Þ, Varðb~ ÞðW5,1g=2 W5, g=2 Þ and Varðlogðy~ 0 ÞÞðW6,1g=2 W6, g=2 Þ, respectively. Table 1 gives the exact and approximate percentiles of W4, W5, W6 for some different combinations r1,r2,y,rk. These exact percentiles are obtained based on 100,000 simulations. The approximate percentiles of W4,W5,W6 are given in parentheses. It is observed from Table 1 that the approximate percentiles of W4, W5, W6 are close to the exact percentiles of W4, W5, W6 even if the failure numbers r1,r2,y,rk are small. Hence the lengths of the approximate confidence intervals for a, b,logðy0 Þ are also close to the lengths of the exact confidence intervals. 3.3. Bootstrap confidence intervals In this subsection, for comparison purpose, two confidence intervals based on the parametric bootstrap methods are proposed: (i) percentile bootstrap method (Efron, 1982) and (ii) studentized-t bootstrap method (Hall, 1988). From (11), we find that the estimators a~ , b~ ,logðy~ 0 Þ depend only on the total test times T1, T2,y,Tk and the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j, i= 1,2,y,k, j = 1,2,y,ri at the failure times ti.j, i= 1,2,y,k, j =1,2,y,ri. Further, note that 2Ti =yi w2 ð2ri Þ, without loss of generality, we choose Ri,j = 0, i= 1,2,y,k, j= 1,2,y,ri. Hence, in order to obtain the bootstrap sample of a~ , b~ ,logðy~ 0 Þ, we need only to consider various combinations (r1,r2,y,rk). The bootstrap sample of a~ , b~ ,logðy~ 0 Þ can be obtained by the following steps. Step 1: From SSALT sample ti,j,i=1,2,y,k,j = 1,2, y,ri, compute a~ , b~ ,logðy~ 0 Þ. Step 2: Based on a~ , b~ and r1,r2,y,rk, generate a random sample (T*1,T*2,y,T*k) of (T1,T2,y,Tk), where T1,T2,y,Tk are independent, and 2Ti =y~ i w2 ð2ri Þ. Here y~ i ¼ expða~ þ b~ xi Þ. Step 3: Based on (T*1,T*2,y,T*k) compute the bootstrap sample estimates a~ , b~ ,logðy~ 0 Þ of a, b,logðy0 Þ using (11). ~ ~ Step 4: Repeat Steps 2–3 B times. Then arrange all the values of a~ , b ,logðy 0 Þ in an ascending order to obtain the bootstrap samples:
a~ ,½1 r a~ ,½2 r r a~ ,½B , b~
,½1
r b~
,½2
,½B r r b~ ,
and ,½1
logðy~ 0
,½2
Þ r logðy~ 0
,½B
Þ r r logðy~ 0
Þ,
respectively. With the bootstrap samples generated as above, we now obtain the 100ð1gÞ% percentile bootstrap confidence intervals for a, b, y0 as ða~ ,½gB=2 ,
a~ ,½ð1g=2ÞB Þ,ðb~
,½gB=2
, b~
,½ð1g=2ÞB
Þ
and
,½gB=2
ðy~ 0
,½ð1g=2ÞB
, y~ 0
Þ
respectively. The 100ð1gÞ% Studentized-t bootstrap confidence intervals for a, b, y0 are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,½ð1g=2ÞB ,½gB=2 ða~ S1 Þ Varða~ ÞÞ, Varða~ Þ, a~ S1 ,½ð1g=2ÞB
ðb~ S2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,½gB=2 Varðb~ Þ, b~ S2 Varðb~ ÞÞ
and ,½ð1g=2ÞB
ðy~ 0 eS3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Varðlogðy~ 0 ÞÞ
,½gB=2
,ðy~ 0 eS3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Varðlogðy~ 0 ÞÞ
Þ
respectively, where
a~ ,½i a~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S,½i , 1 ¼ Varða~ Þ
,½i b~ b~ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , S,½i ¼ 2 Varðb~ Þ
,½i
S,½i 3 ¼
logðy~ 0 Þlogðy~ 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Varðlogðy~ 0 ÞÞ
Here ðVarða~ Þ,Varðb~ Þ,Varðlogðy~ 0 ÞÞÞ can be estimated by ðVarða~ Þ,Varðb~ Þ,Varðlogðy~ 0 ÞÞ.
4. Simulation study In order to assess the performance of all methods of constructing confidence intervals discussed in Section 3, a Monte Carlo simulation study was conducted to determine the coverage percentages and the average interval lengths of the proposed confidence intervals. The values of the parameters and stress levels are chosen to be a ¼ 4, b ¼ 1 and x0 = 0.25, x1 =0.5, x2 = 0.75, x3 = 1, x4 =1.25.
Table 2 Experiment schemes, coverage percentages and average interval lengthes in simulation study.
(9,6)
Parameters
a
logðy0 Þ (12,8)
a b logðy0 Þ
(15,10)
a b logðy0 Þ
(30,20)
a b logðy0 Þ
(12,8,6)
a b logðy0 Þ
(15,10,8)
a b
Exact
Approx.
Boot-p
Boot-t
Exact
Approx.
Boot-p
Boot-t
0.9502 (5.2601) 0.9495 (8.6114) 0.9492 (3.1759) 0.9507 (4.5280) 0.9493 (7.3912) 0.9506 (2.7338) 0.9503 (4.0265) 0.9499 (6.5648) 0.9516 (2.4333) 0.9522 (2.8136) 0.9504 (4.5849) 0.9513 (1.7006) 0.9497 (2.8581) 0.9478 (3.9795) 0.9499 (1.9228) 0.9460 (2.5064) 0.9467 (3.4689) 0.9482 (1.6912)
0.9499 (5.2593) 0.9498 (8.6135) 0.9491 (3.1752) 0.9501 (4.5105) 0.9491 (7.3818) 0.9497 (2.7252) 0.9499 (4.0109) 0.9498 (6.5613) 0.9503 (2.4244) 0.9517 (2.8036) 0.9507 (4.5823) 0.9505 (1.6962) 0.9497 (2.8611) 0.9484 (3.9920) 0.9497 (1.9208) 0.9468 (2.5134) 0.9472 (3.4774) 0.9489 (1.6947)
0.9500 (5.2545) 0.9492 (8.6046) 0.9481 (3.1734) 0.9476 (4.5061) 0.9476 (7.3767) 0.9487 (2.7236) 0.9500 (4.0086) 0.9497 (6.5582) 0.9498 (2.4238) 0.9498 (2.8026) 0.9498 (4.5813) 0.9492 (1.6960) 0.9469 (2.8596) 0.9471 (3.9900) 0.9469 (1.9206) 0.9458 (2.5116) 0.9463 (3.4744) 0.9459 (1.6939)
0.9498 (5.2545) 0.9492 (8.6046) 0.9487 (3.1734) 0.9487 (4.5061) 0.9477 (7.3767) 0.9499 (2.7236) 0.9492 (4.0086) 0.9482 (6.5582) 0.9506 (2.4238) 0.9511 (2.8026) 0.9504 (4.5813) 0.9501 (1.6960) 0.9483 (2.8596) 0.9468 (3.9900) 0.9479 (1.9206) 0.9471 (2.5116) 0.9464 (3.4744) 0.9484 (1.6939)
0.8984 (4.3738) 0.8956 (7.1352) 0.8987 (2.6471) 0.9008 (3.7707) 0.8968 (6.1489) 0.8990 (2.2805) 0.8991 (3.3550) 0.8988 (5.4774) 0.8978 (2.0292) 0.8975 (2.3468) 0.9004 (3.8299) 0.8997 (1.4218) 0.8961 (2.3862) 0.8970 (3.3321) 0.9008 (1.6036) 0.8932 (2.0990) 0.8948 (2.8961) 0.8946 (1.4171)
0.8997 (4.3848) 0.8981 (7.1774) 0.8989 (2.6495) 0.9003 (3.7670) 0.8973 (6.1625) 0.8988 (2.2774) 0.8989 (3.3532) 0.8992 (5.4836) 0.8972 (2.0279) 0.8973 (2.3484) 0.9005 (3.8378) 0.8993 (1.4212) 0.8969 (2.3888) 0.8964 (3.3300) 0.9011 (1.6051) 0.8935 (2.1010) 0.8960 (2.9049) 0.8944 (1.4176)
0.9005 (4.3824) 0.8960 (7.1757) 0.8978 (2.6485) 0.8969 (3.7646) 0.8968 (6.1598) 0.8980 (2.2763) 0.8985 (3.3518) 0.8986 (5.4818) 0.8974 (2.0275) 0.8965 (2.3475) 0.8994 (3.8366) 0.8983 (1.4204) 0.8977 (2.3879) 0.8971 (3.3294) 0.9001 (1.6050) 0.8943 (2.1006) 0.8954 (2.9033) 0.8950 (1.4178)
0.8967 (4.3824) 0.8968 (7.1757) 0.8971 (2.6485) 0.8988 (3.7646) 0.8974 (6.1598) 0.8976 (2.2763) 0.8979 (3.3518) 0.8980 (5.4818) 0.8977 (2.0275) 0.8952 (2.3475) 0.9005 (3.8366) 0.8970 (1.4204) 0.8969 (2.3879) 0.8973 (3.3294) 0.9001 (1.6050) 0.8926 (2.1006) 0.8950 (2.9033) 0.8948 (1.4178)
2715
logðy0 Þ
g ¼ 0:1
ARTICLE IN PRESS
b
g ¼ 0:05
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
(r1,y,rk)
2716
Table 2 (continued ) (r1,y,rk)
(30,20,15)
Parameters
a b logðy0 Þ
a b logðy0 Þ
(12,8,6,4)
a
logðy0 Þ (15,10,8,6)
a b logðy0 Þ
(30,20,15,12)
a b logðy0 Þ
(60,40,25,18)
a b logðy0 Þ
(90,60,40,25)
a b logðy0 Þ
Exact
Approx.
Boot-p
Boot-t
Exact
Approx.
Boot-p
Boot-t
0.9525 (1.7761) 0.9539 (2.4657) 0.9507 (1.1943) 0.9497 (1.2860) 0.9510 (1.8181) 0.9509 (0.8589) 0.9543 (2.2715) 0.9525 (2.8176) 0.9548 (1.6182) 0.9517 (1.9517) 0.9514 (2.3769) 0.9523 (1.4026) 0.9512 (1.3667) 0.9514 (1.6659) 0.9505 (0.9813) 0.9478 (1.0031) 0.9473 (1.2640) 0.9475 (0.7104) 0.9516 (0.8256) 0.9494 (1.0386) 0.9505 (0.5859)
0.9523 (1.7752) 0.9540 (2.4689) 0.9507 (1.1942) 0.9501 (1.2875) 0.9524 (1.8233) 0.9507 (0.8581) 0.9542 (2.2683) 0.9528 (2.8245) 0.9544 (1.6163) 0.9526 (1.9588) 0.9517 (2.3810) 0.9531 (1.4097) 0.9522 (1.3684) 0.9512 (1.6629) 0.9514 (0.9855) 0.9478 (1.0042) 0.9472 (1.2645) 0.9485 (0.7131) 0.9513 (0.8247) 0.9506 (1.0413) 0.9500 (0.5847)
0.9504 (1.7740) 0.9535 (2.4669) 0.9480 (1.1940) 0.9506 (1.2877) 0.9498 (1.8229) 0.9493 (0.8582) 0.9522 (2.2682) 0.9500 (2.8237) 0.9526 (1.6168) 0.9510 (1.9584) 0.9518 (2.3802) 0.9502 (1.4096) 0.9508 (1.3683) 0.9510 (1.6623) 0.9510 (0.9854) 0.9461 (1.0041) 0.9473 (1.2642) 0.9481 (0.7131) 0.9492 (0.8241) 0.9494 (1.0405) 0.9481 (0.5841)
0.9506 (1.7740) 0.9533 (2.4669) 0.9502 (1.1940) 0.9489 (1.2877) 0.9504 (1.8229) 0.9485 (0.8582) 0.9541 (2.2682) 0.9526 (2.8237) 0.9524 (1.6168) 0.9519 (1.9584) 0.9507 (2.3802) 0.9527 (1.4096) 0.9508 (1.3683) 0.9499 (1.6623) 0.9515 (0.9854) 0.9466 (1.0041) 0.9474 (1.2642) 0.9482 (0.7131) 0.9513 (0.8241) 0.9493 (1.0405) 0.9488 (0.5841)
0.9020 (1.4886) 0.9037 (2.0697) 0.9027 (1.0027) 0.9024 (1.0798) 0.9042 (1.5294) 0.9033 (0.7186) 0.9061 (1.8967) 0.9046 (2.3519) 0.9079 (1.3539) 0.9043 (1.6302) 0.9047 (1.9845) 0.9050 (1.1748) 0.9031 (1.1432) 0.8992 (1.3915) 0.9060 (0.8254) 0.8948 (0.8404) 0.8982 (1.0589) 0.8941 (0.5977) 0.9008 (0.6910) 0.9016 (0.8716) 0.8992 (0.4905)
0.9016 (1.4868) 0.9031 (2.0671) 0.9022 (1.0005) 0.9025 (1.0793) 0.9044 (1.5282) 0.9035 (0.7195) 0.9057 (1.8944) 0.9051 (2.3560) 0.9071 (1.3512) 0.9061 (1.6381) 0.9055 (1.9894) 0.9062 (1.1797) 0.9047 (1.1464) 0.8995 (1.3925) 0.9059 (0.8259) 0.8957 (0.8419) 0.8981 (1.0599) 0.8938 (0.5980) 0.9015 (0.6916) 0.9027 (0.8731) 0.8986 (0.4904)
0.9001 (1.4859) 0.9020 (2.0660) 0.8998 (0.9999) 0.9022 (1.0793) 0.9027 (1.5280) 0.9024 (0.7195) 0.9056 (1.8938) 0.9023 (2.3552) 0.9072 (1.3509) 0.9045 (1.6377) 0.9044 (1.9890) 0.9045 (1.1797) 0.9043 (1.1461) 0.9002 (1.3921) 0.9060 (0.8258) 0.8940 (0.8418) 0.8966 (1.0595) 0.8950 (0.5982) 0.8989 (0.6912) 0.9015 (0.8727) 0.8991 (0.4902)
0.9012 (1.4859) 0.9019 (2.0660) 0.9008 (0.9999) 0.8999 (1.0793) 0.9032 (1.5280) 0.9011 (0.7195) 0.9058 (1.8938) 0.9029 (2.3552) 0.9056 (1.3509) 0.9049 (1.6377) 0.9051 (1.9890) 0.9045 (1.1797) 0.9028 (1.1461) 0.9010 (1.3921) 0.9045 (0.8258) 0.8952 (0.8418) 0.8966 (1.0595) 0.8931 (0.5982) 0.9005 (0.6912) 0.9011 (0.8727) 0.8964 (0.4902)
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b
g ¼ 0:1
B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
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g ¼ 0:05
ARTICLE IN PRESS B.X. Wang / Journal of Statistical Planning and Inference 140 (2010) 2706–2718
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Because the distributions of the pivotal quantities W4,W5,W6 depend only on the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk, and not on the sample size n and the removed numbers Ri,j at the failure times ti.j, i= 1,2,y,k, j= 1,2,y,ri, without loss of generality, we choose Ri,j = 0, i=1,2,y,k, j= 1,2,y,ri. We consider only various combinations (r1,r2,y,rk). The column 1 in Table 2 lists the different censoring schemes used in the simulation study. The confidence coefficients are 0.9 and 0.95. For the proposed confidence intervals, note that 2Ti =yi follows the w2 ð2ri Þ distribution, we generate k random numbers with the w2 ð2ri Þ ði ¼ 1, . . . ,kÞ distributions for each censoring scheme. We compute the proposed confidence intervals. Based on 10,000 Monte Carlo simulations with B =1000 replications, the coverage percentages and the average interval lengths of the proposed confidence intervals for the different (r1,r2,y,rk) schemes are obtained. These values are tabulated in Table 2. The average interval lengths are given in parentheses. From Table 2, we observe that all the coverage percentages of the proposed confidence intervals are close to the nominal coverage probabilities, even for small sample sizes. We also observe that the differences of the average lengths of the proposed confidence intervals are small. As the number of the accelerated stress levels or the failure numbers r1,r2,y,rk at stress levels x1,x2,y,xk increase, the average lengths of confidence intervals reduce appreciably. Considering the above simulation results and the fact that the percentage points of the pivotal quantities W4,W5,W6 are obtained by simulation method when k 42, the approximate confidence intervals are suggested unless r1,r2,y,rk are very small. 5. An illustrative example To illustrate the proposed procedures, recall the example of Section 4 of Wang and Fei (2003). This example is to obtain all the reliability indices of a kind of electronic components at the normal temperature of S0 = 25 1C, Now n = 100 units from a batch of products are randomly selected for the simple accelerated life test model logðyÞ ¼ 1=ð273:15 þSÞ. The accelerated temperature levels are S1 = 100 1C and S2 = 150 1C. At S1 = 100 1C, when 30 products have failed, the stress level rises to S2 = 150 1C and the test continues until 20 more products have failed. Their failure times are as follows. Failure times for the stress level S1: 32, 54, 59, 86, 117, 123, 213, 267, 268, 273, 299, 311, 321, 333, 339, 386, 408, 422, 435, 437, 476, 518, 570, 632, 666, 697, 796, 854, 858, 910. Failure times for the stress level S2: 16, 19, 21, 36, 37, 63, 70, 75, 83, 95, 100, 106, 110, 113, 116, 135, 136, 149, 172, 186. These times are the differences t2,j t1,r1 . This step-stress accelerated life testing is a SSALT with Type-II censoring, which is a specific case about the progressive censoring scheme with Ri,j =0, i= 1,2,j =1,2,y,ri. From Theorem 1, the 5% and 95% percentiles of the pivotal quantities W1, W2, W3 are given by W1,0.05 = 0.0349, W1,0.95 = 80.9612, F0.05(60,40)= 0.6272, F0.95(60,40)= 1.6373, W3,0.05 =41.2187, W3,0.95 =487.3390, respectively. Thus the 90% exact confidence intervals for a, b, y0 are given by ½8:6949,0:9458,
½3218:8,6246:4,
½18 504,218 780,
respectively. The 90% approximate confidence intervals for a, b, y0 are given by ½8:7009,0:9495,
½3221:3,6251:1,
½18 499,218 645,
respectively. It is observed that two methods give results that are almost in agreement.
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