as the step-intensity model. It was developed to directly represent the TAAF cycle. These alternatives assume that the time to failure is exponentially distributed, but these distributions have mean times to failure that vary following some prescribed formulation linked to the TAAF process. Sen and Bhattacharya (1993) retained the power law form but gave it an interpretation that was linked more directly to the TAAF process—referred to as exponential reliability growth.

Other alternatives to the power law process exist to handle situations such as upward trends in the failure rate, situations in which the time to first failure can be infinite with positive probability, and failure-rate functions that have a bathtub shape (failure rate initially decreasing and subsequently increasing). A second distinct class of models is derived as solutions to differential equations. The defining equations represent the relationship between cumulative expected time between failures and nonlinear functions of time. Unfortunately, these procedures are complicated to use for purposes of statistical inference. A third distinct class of models makes use of a Bayesian formulation through which the subjective inputs of experts in appropriate disciplines can be elicited, quantified, and included in the analysis. Finally, there are nonparametric approaches to modeling reliability growth that are straightforward applications and generalizations of the Kaplan-Meier estimates used in survival analysis.

Clearly there is a wide variety of reliability growth models from which to choose. No single model is best for all purposes. Parametric models permit extrapolation to areas in which few or no failures have been observed, but they are based on assumptions that need to be validated or evaluated for robustness. Fully nonparametric models are essentially always valid but, for a fixed desired precision, typically require a substantial number of replications; they can also be inefficient relative to parametric alternatives when the relevant assumptions of the latter are approximately true.

Sen argued that it is important for decision makers to be provided with a full representation of the complete evolution of the bottom-line result, instead of a simplistic presentation of a single point estimate or an elementary pass/fail pronouncement. A full representation of the results should include some physical justification or validation of the model’s assumptions (given that a parametric approach is used), together with use of a nonparametric approach for purposes of validation and comparison and, at times, results from leading parametric alternative representations. Agreement of alternative modeling approaches offers important assurance of the stability of the results presented. Disagreement is often indicative of the



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