Some Contributions to Modeling and Inference for Censored Lifetime Data from Exponential Distribution Under Step-stress Accelerated Life Tests

Bai, Tianyu
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In this work, we investigate several modeling and inferential issues regarding the censored lifetime data from exponential distribution under the step-stress accelerated life tests (SSALT). In SSALT, the stress levels are increased discretely at pre-fixed time points and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. For reasons of time constraint and cost reduction, censored sampling is also commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing removal of test units at non-terminal time points.

First, we discuss the point estimation of the regression parameters for interval monitored failure data from the general k-level constant-stress and step-stress ALT. The expectation-maximization (EM) algorithm is proposed to derive the maximum likelihood estimates (MLE) of the parameters, and their asymptotic variances and covariances are also calculated using the principle of missing information. Then, the MLE of the mean time to failure at each stress level and their conditional distribution functions are derived explicitly for the general k-level SSALT under interval inspection. Using the exact distributions of the estimators, the confidence intervals for the respective parameters are obtained along with the bootstrap and approximate confidence intervals. For the bootstrap confidence intervals, two smoothed justifications are suggested for the discrete estimators. Monte Carlo simulation studies were conducted to assess the performances of various confidence intervals as well as the precisions of the MLE under continuous and interval monitorings.

Finally, the design optimization of a general k-level SSALT under progressive Type-I censoring is considered with a uniform step duration under the interval inspection. Also, the design optimization of a simple SSALT under progressive Type-I censoring was studied with non-uniform, flexible step durations with a given test length. Allowing the intermediate censoring at each stress change time point, the optimal step duration is determined under various design criteria including D-optimality, T-optimality, C-optimality, A-optimality, and E-optimality. The existence of these optimal designs is investigated in detail for exponential lifetimes with a single stress variable, and the design efficiency is compared to the case of continuous inspection with engineering applications to device reliability studies.

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EM Algorithm, Exact Inference, Interval Estimation, Interval Monitoring, Optimal Design, Step-stress
Management Science and Statistics