臺灣博碩士論文加值系統

在型 I 設限和型 I 混合設限計劃下，本論文針對一般性的對數位置尺度(韋伯和對數常態)和指數壽命分佈討論 k 階段逐步應力加速壽命試驗之不等長的應力持續時間。根據累積暴露模型，我們假設一般性的對數位置尺度壽命模式的平均壽命和應力是呈現線性關係，而指數壽命模式的平均壽命和應力是呈現對數線性關係。依據變異數最佳化準則，我們的數值結果顯示指數、韋伯和對數常態分佈的最佳化 k ( ≥ 3 ) 階段逐步應力加速壽命試驗之不等長的應力持續時間，都會縮減為二階段逐步應力加速壽命試驗。利用歸納法，我們更進一步對型 I 設限計劃下的指數壽命模式驗證了此一結果。

In this dissertation, we discuss a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps. Under the Type-I and Type-I hybrid
censoring schemes, the general log-location-scale and exponential lifetime distributions with mean lives which are a linear function of stress for the former and a
log-linear function of stress for the latter, along with a cumulative exposure model, are considered as the working models. The determination of the optimal unequal duration steps for exponential, Weibull and lognormal distributions are addressed using the variance-optimality criterion. Numerical results show that for the general log-location-scale and exponential distributions, the optimal k-level step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model when the available data is either Type-I or Type-I hybrid censored data.
Moreover, using the induction argument, we are capable to give a theoretical proof for this result based on a Type-I exponential censored data.

Contents
1 Introduction 1
2 Model Assumptions 5
2.1 Log-Location-Scale Distribution . . . . . . . . . . .6
2.2 Exponential Distribution . . . . . . . . . . . . . . 7
3 Maximum Likelihood Estimation 8
3.1 Type-I Censored Case . . . . . . . . . . . . . . . . 8
3.1.1 Log-Location-Scale Distribution . . . . . . . . . .8
3.1.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Type-I Hybrid Censored Case: Log-Location-Scale Distribution . . . . . .10
4 Optimal Test Plan 12
4.1 Type-I Censored Case . . . . . . . . . . . . . . . . 12
4.1.1 Log-Location-Scale Distribution . . . . . . . . . .12
4.1.2 Exponential Distribution . . . . . . . . . . . . . 14
4.2 Type-I Hybrid Censored Case: Log-Location-Scale Distribution . . . . . .27
5 Concluding Remarks 39
References 40
Appendix 43


List of Tables
1 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 2-level step-stress
setting based on complete data and Type-I censored data in the Weibull case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 3-level step-stress
setting based on complete data and Type-I censored data in the Weibull case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 4-level step-stress
setting based on complete data and Type-I censored data in the Weibull case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 2-level step-stress set-
ting based on complete data and Type-I censored data in the lognormal case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 3-level step-stress set-
ting based on complete data and Type-I censored data in the lognormal case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 4-level step-stress set-
ting based on complete data and Type-I censored data in the lognormal case.
The searching range for the optimal change points in the Type-I censored case
is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7 Optimal change points c and the associated asymptotic variance φ∗
k( ) (in
parentheses) according to the C-optimality under the 2-level step-stress setting
based on complete data and Type-I censored data in the exponential case. The
searching range for the optimal change points in the Type-I censored case is
(0, 10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8 Optimal change points c and the associated asymptotic variance φ∗
k( ) (in
parentheses) according to the C-optimality under the 3-level step-stress setting
based on complete data and Type-I censored data in the exponential case. The
searching range for the optimal change points in the Type-I censored case is
(0, 10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9 Optimal change points c and the associated asymptotic variance φ∗
k( ) (in
parentheses) according to the C-optimality under the 4-level step-stress setting
based on complete data and Type-I censored data in the exponential case. The
searching range for the optimal change points in the Type-I censored case is
(0, 10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10 Optimal change points d and the associated determinant (in parentheses)
according to the D-optimality under the 2-level step-stress setting based on
complete data and Type-I censored data in the exponential case. The search-
ing range for the optimal change points in the Type-I censored case is (0,
10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11 Optimal change points d and the associated determinant (in parentheses)
according to the D-optimality under the 3-level step-stress setting based on
complete data and Type-I censored data in the exponential case. The search-
ing range for the optimal change points in the Type-I censored case is (0,
10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
12 Optimal change points d and the associated determinant (in parentheses)
according to the D-optimality under the 4-level step-stress setting based on
complete data and Type-I censored data in the exponential case. The search-
ing range for the optimal change points in the Type-I censored case is (0,
10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
13 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 2-level step-stress set-
ting based on Type-I hybrid censored data in theWeibull case for n = 40. The
searching range for the optimal change points in the Type-I hybrid censored
case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
14 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 2-level step-stress
setting based on Type-I hybrid censored data in the Weibull case for n = 100.
The searching range for the optimal change points in the Type-I hybrid cen-
sored case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
15 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 3-level step-stress set-
ting based on Type-I hybrid censored data in theWeibull case for n = 40. The
searching range for the optimal change points in the Type-I hybrid censored
case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
16 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 3-level step-stress
setting based on Type-I hybrid censored data in the Weibull case for n = 100.
The searching range for the optimal change points in the Type-I hybrid cen-
sored case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
17 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 4-level step-stress set-
ting based on Type-I hybrid censored data in theWeibull case for n = 40. The
searching range for the optimal change points in the Type-I hybrid censored
case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
18 Optimal change points c and the associated asymptotic variance nAVar(yp)/σ2
(in parentheses) according to the C-optimality under the 4-level step-stress
setting based on Type-I hybrid censored data in the Weibull case for n = 100.
The searching range for the optimal change points in the Type-I hybrid cen-
sored case is (0, 20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


List of Figures
1 The value of ρ3(h∗) for different h∗ and τ1 when x0 = 0, x1 = 0.2, x2 = 0.6, x3 =
1.0, β0 = 1 and θ1 = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 The value of χ(h∗) for different h∗ and τ1 when x0 = 0, x1 = 0.2, x2 = 0.6, x3 =
1.0, β0 = 1 and θ1 = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References
Alhadeed, A. A. and Yang, S. S. (2005). Optimal simple step-stress plan for cumulative
exposure model using log-normal distribution. IEEE Transactions on Reliability, 54,
64–68.
Bai, D. S. and Kim, M. S. (1993). Optimum simple step-stress accelerated life tests for
Weibull distribution and Type I censoring. Naval Research Logistics, 40, 193–210.
Bai, D. S., Kim, M. S. and Lee, S. H. (1989). Optimum simple step-stress accelerated life
test with censoring. IEEE Transactions on Reliability, 38, 528–532.
Balakrishnan, N. (2009). A synthesis of exact inferential results for exponential step-stress
models and associated optimal accelerated life-tests. Metrika, 69, 351–396.
Balakrishnan, N. and Han, D. (2009). Optimal step-stress testing for progressively Type-
I censored data from exponential distribution. Journal of Statistical Planning and
Inference, 139, 1782–1798.
Balakrishnan, N. and Kundu, D. (2013). Hybrid censoring: Models, inferential results and applications. Computational Statistics & Data Analysis, 57, 166–209.
Balakrishnan, N., Kundu, D., Ng, H. K. T. and Kannan, N. (2007). Point and interval
estimation for a simple step-stress model with Type-II censoring. Journal of Quality
Technology, 39, 35–47.
Balakrishnan, N., Zhang, L. and Xie, Q. (2009). Inference for a simple step-stress model
with Type-I censoring and lognormally distributed lifetimes. Communications in
Statistics–Theory and Methods, 38, 1690–1709.
Chung, S. W. and Bai, D. S. (1998). Optimal designs of simple step-stress accelerated life
tests for lognormal lifetime distributions. International Journal of Reliability, Quality
and Safety Engineering, 5, 315–336.
Chung, S. W., Seo, Y. S. and Yun, W. Y. (2006). Acceptance sampling plans based on
failure-censored step-stress accelerated tests for Weibull distributions. Journal of
Quality in Maintenance Engineering, 12, 373–396.
Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987). Minimizing multi-modal
functions of continuous variables with the “simulated annealing” algorithm. ACM
Transactions on Mathematical Software, 13, 262–280.
Dharmadhikari, A. D. and Rahman, M. M. (2003). A model for step-stress accelerated life
testing. Naval Research Logistics 50, 841–868.
Fard, N. and Li, C. (2009). Optimal simple step stress accelerated life test design for relia-
bility prediction. Journal of Statistical Planning and Inference, 139, 1799–1808.
Gouno, E., Sen, A. and Balakrishnan, N. (2004). Optimal step-stress test under progressive
Type-I censoring. IEEE Transactions on Reliability, 53, 383–393.
Han, D., Balakrishnan, N., Sen, A. and Gouno, E. (2006). Corrections on “Optimal step-
stress test under progressive Type-I censoring”. IEEE Transactions on Reliability,
55, 613–614.
Kateri, M. and Balakrishnan, N. (2008). Inference for a simple step-stress model with type-II
censoring, and weibull distributed lifetimes. IEEE Transactions on Reliability, 57,
616–626.
Khamis, I. H. (1997). Comparison between constant and step-stress tests forWeibull models.
International Journal of Quality, Reliability and Management, 14, 74–81.
Khamis, I. H. and Higgins, J. J. (1996). Optimum 3-step step-stress tests. IEEE Transac-
tions on Reliability, 45, 341–345.
Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory).
The Annals of Statistics, 2, 849–879.
Ling, L., Xu, W. and Li, M. (2011). Optimal bivariate step-stress accelerated life test for
Type-I hybrid censored data. Journal of Statistical Computation and Simulation 81,
1175–1186.
Ma, H. and Meeker, W. Q. (2008). Optimum step-stress accelerated life test plans for log-
location-scale distributions. Naval Research Logistics, 55, 551–562.
Miller, R. and Nelson, W. B. (1983). Optimum simple step-stress plans for accelerated life
testing. IEEE Transactions on Reliability, 32, 59–65.
Nelson, W. B. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis.
John Wiley & Sons, New York.
Nelson, W. B. (2005a). A bibliography of accelerated test plans, Part I–Overview. IEEE Transactions on Reliability, 54, 194–197.
Nelson, W. B. (2005b). A bibliography of accelerated test plans, Part II–References. IEEE
Transactions on Reliability, 54, 370–373.
Nelson, W. B. and Kielpinski, T. J. (1976). Theory for optimum censored accelerated life
tests for normal and lognormal life distributions. Technometrics, 18, 105–114.
Srivastava, P.W. and Shukla, R. (2008). A log-logistic step-stress model. IEEE Transactions
on Reliability, 57, 431–434.
Tang, L. C., Sun, Y. S., Goh, T. N. and Ong, H. L. (1996). Analysis of Step-Stress
Accelerated-Life-Test Data: A New Approach. IEEE Transactions on Reliability,
45, 69–74.
Wu, S. J., Lin, Y. P. and Chen, S. T. (2008). Optimal step-stress test under Type-I pro-
gressive group-censoring with random removals. Journal of Statistical Planning and
Inference, 138, 817–826.
Wu, S. J., Lin, Y. P. and Chen, Y. J. (2006). Planning step-stress life test with progressively
Type-I group-censored exponential data. Statistica Neerlandica, 60, 46–56.
Yeo, K. P. and Tang, L. C. (1999). Planning step-stress life-test with a target acceleration-
factor. IEEE Transactions on Reliability, 54, 370–373.