|Performance of Parameter-Estimates in Step-Stress Accelerated Life-Tests With Various Sample-Sizes
McSorley, E.O., Lu, J.C., and Li, C.S
IEEE Transactions on Reliability. 2002. 51(3):271-277.
In accelerated life test (ALT) studies, the maximum likelihood (ML) method is commonly used in estimating model parameters, and its asymptotic variance is the key quantity used in searching for the optimum design of ALT plans and in making statistical inferences. This paper uses simulation techniques to investigate the required sample size for using the large sample Gaussian approximation s-confidence interval and the properties of the ML estimators in the finite sample situation with different fitting models. Both the likelihood function and its second derivatives needed for calculating the asymptotic variance are very complicated. This paper shows that a sample size of 100 is needed in practice for using large-sample inference procedures. When the model is Weibull with a constant shape parameter, fitting exponential models can perform poorly in large-sample cases, and fitting Weibull models with a regression function of shape parameters can give undesirable results in small-sample situations. When small fractions of the product-life distribution are used to establish warranties and service polices, the interests of the producer and consumer should be balanced with the resources available for conducting life tests. From the standpoint of safety, an unrealistically long projected service life could harm the consumer. Establishing a short warranty period protects the producer but hurts revenue. An overly generous warranty period could cost the producer in terms of product replacement. To estimate life-distribution parameters well, use more than 100 samples in for LSCI and life-testing plans derived from the asymptotic theory. When the model becomes complicated, as in this paper, monitor the convergence of the parameter estimation algorithm, and do not trust the computer outputs blindly. In many applications, the likelihood ratio test inverted confidence interval performs better than the usual approximate confidence interval in small samples; thus its performance in the step-stress ALT studies should be in future research
Keywords: life testing maximum likelihood estimation statistical analysis Gaussian approximation s-confidence interval Weibull model asymptotic variance constant shape parameter cumulative exposure model exponential models fitting finite sample situation fitting models generous warranty period life-distribution parameters estimation likelihood ratio test inverted confidence interval maximum likelihood method optimum design parameter estimation algorithm convergence parameter-estimates performance product replacement product-life distribution regression function safety sample-sizes service polices short warranty period simulation techniques statistical inferences step-stress accelerated life-tests unrealistically long projected service life warranties IEEE TERMS Acceleration Gaussian approximation Life estimation Life testing Maximum likelihood estimation Parameter estimation Protection Safety Shape Warranties