在一般的壽命測試和衰變測試中，若在實驗期間無法看到足夠的產品失效，則無法對於產品壽命提供足夠的資訊，此時便會進而考慮重複量測的加速衰變測試。在非線性的退化模型假設下，本文處理此種測試規劃的最佳化問題，同時，測試單元間變異亦以非線性之隨機項存在於模型中。

由於模型的非線性型態，概似函數與費雪訊息矩陣，以及退化模型所衍生出的壽命分佈之推算有其難處。因此，本文以蒙地卡羅積分方法計算測試規劃最佳化之準則，即參數估計量的漸進變異數，並以格子點搜索的方式找到最佳的測試規劃。為了減輕計算上的負擔，本文將最佳化問題考慮在一些限制條件下進行處理，同時並使用擬蒙地卡羅法及一些R套件來進行有效率的統計計算。本文亦將所提出之方法應用在一集成電路裝置之實例上。

Repeated measures accelerated degradation tests (RMADTs) can provide more information about the product reliability
when one would expect few or even no failures during a study. In this paper, we deal with the optimal test planning problem under the non-linear Arrhenius acceleration model. Further, unit-to-unit variability is described by the random effects, which is also non-linear in time.

Due to the assumption of non-linear mixed effect model (NLMEM), the analytical calculation of the likelihood and some related functions such as Fisher information matrix are difficult. Therefore, we perform the Monte Carlo integration to calculate the asymptotic variance of estimators as the optimality criterion. Grid search procedures are conducted to find the optimum test plan under some constraints. Quasi-random low-discrepancy sequences and some R packages for efficient computation are used to relief the computational burden. The method is illustrated with an application of an integrated circuit device example.

Comets, E., Lavenu, A., and Lavielle, M. (2011), “SAEMIX, an R version of
the SAEM algorithm”, 20th meeting of the Population Approach Group in
Europe, Athens, Greece.
Das, S., Spall, J. C., and Ghanem, R. (2010), “Efficient monte carlo computation
of fisher information matrix using prior information”, Computational
Statistics & Data Analysis, 54(2), 272–289.
Guan, Q. and Tang, Y. (2013), “Optimal design of accelerated degradation
test based on gamma process models”, Chinese Journal of Applied Probability
and Statistics, 29(2).
Kim, S. J. and Bae, S. J. (2013), “Cost-effective degradation test plan for a
nonlinear random-coefficients model”, Reliability Engineering and System
Safety, 110, 68–79.
Krykova, I. (2003), “Evaluating of path-dependent securities with low discrepancy
methods”, Master’s thesis, Worcester Polytechnic Institute.
Kuhn, E. and Lavielle, M. (2005), “Maximum likelihood estimation in nonlinear
mixed effects models”, Computational Statistics & Data Analysis,
49, 1020–1038.
Lim, H. and Yum, B. J. (2011), “Optimal design of accelerated degradation
tests based on wiener process models”, Journal of Applied Statistics, 38(2),
309–325.
Meeker, W. Q. and Escobar, L. A. (1998), Statistical methods for reliability
data, volume 78, Wiley New York.
Nguyen, T. T. and Mentré, F. (2014), “Evaluation of the fisher information
matrix in nonlinear mixed effect models using adaptive gaussian quadrature”,
Computational Statistics & Data Analysis, 80, 57–69.
Niederreiter, H. (1992), Random Number Generation and Quasi Monte Carlo
Methods, SIAM, Philadelphia.
Pillai, G. C., Mentre, F., and Steimer, J. L. (2005), “Non-linear mixed effects
modeling - from methodology and software development to driving implementation
in drug development science”, Journal of Pharmacokinetics and
Pharmacodynamics, 32(2), 161–83.
Plan, E. L., Maloney, A., Mentre, F., Karlsson, M. O., and Bertrand, J.
(2012), “Performance comparison of various maximum likelihood nonlinear
mixed-effects estimation methods for dose-response models”, The
AAPS Journal, 14(3), 420–32.
Robert, C. P. and Casella, G. (1999), Monte Carlo Statistical Method,
Springer.
Spall, J. C. (2005), “Monte carlo computation of the fisher information matrix
in nonstandard settings”, Journal of Computational and Graphical Statistics,
14(4), 889–909.
(2008), “Improved methods for monte carlo estimation of the fisher
information matrix”, in American Control Conference, 2008, 2395–2400.
Tang, S., Guo, X., Yu, C., Xue, H., and Zhou, Z. (2014), “Accelerated degradation
tests modeling based on the nonlinear wiener process with random
effects”, Mathematical Problems in Engineering, 2014, 1–11.
Tsai, C. C., Tseng, S. T., and Balakrishnan, N. (2012), “Optimal design for
degradation tests based on gamma processes with random effects”, IEEE
Transactions on Reliability, 61(2), 604–613.
Weaver, B. P. and Meeker, W. Q. (2014), “Methods for planning repeated
measures accelerated degradation tests”, Applied Stochastic Models in
Business and Industry, 30(6), 658–671.
Weaver, B. P., Meeker, W. Q., Escobar, L. A., and Wendelberger, Joanne
(2013), “Methods for planning repeated measures degradation studies”,
Technometrics, 55(2), 122–134.
Weisstein, E. W. (1999), “Leibniz integral rule. From MathWorld – A Wolfram
Web Resource”, URL http://mathworld.wolfram.com/
LeibnizIntegralRule.html.
Ye, Z. S., Chen, L. P., Tang, L. C., and Xie, M. (2014), “Accelerated degradation
test planning using the inverse gaussian process”, IEEE Transactions
on Reliability, 63(3), 750–763.
Yuan, X. X. and Pandey, M. D. (2009), “A nonlinear mixed-effects model
for degradation data obtained from in-service inspections”, Reliability Engineering
& System Safety, 94(2), 509–519.