Life Testing in Variably Scaled Environments

Abstract

Typically, accelerated life-testing models postulate a specific functional relationship between the stress level at which an experiment is performed and the parameters of the assumed family of lifetime distributions. These models, and the statistical analyses that accompany them, are often criticized on the basis of the dubious validity of the assumed functional relationship and of the uncertainty involved in the extrapolation of experimental results to low stress levels at which little or no data have been obtained. This study focuses on an exponential factorial model for accelerated life tests that postulates that the lifetime distributions of different component types tested under varying environmental conditions are linked via environmental or component-related scale changes. Necessary and sufficient conditions are given for the identifiability of model parameters. For both censored and complete data, the derivation and properties of maximum likelihood estimates of these parameters are discussed in detail. Under the conditions that guarantee identifiability, the existence and the uniqueness of the maximum likelihood estimators are demonstrated, and their computation and large-sample behavior are discussed. In the final section, the model is fitted to published data from an accelerated life-testing experiment.

KEY WORDS:

• Accelerated life testing
• Exponential distributions
• Factorial experiments
• Gauss-Seidel iterations
• Identifiability
• Likelihood ratio test
• Maximum likelihood estimation