Goodness-of-fit tests for imperfect maintenance models based on Martingale residuals, varentropy, and probability integral transform

Abstract

In recent years, various goodness-of-fit tests have been developed to identify the underlying distribution of failure data. In this paper, we extend the application of such tests to evaluate the adequacy of imperfect maintenance models for engineering systems. Specifically, we investigate and compare three types of test statistics: those based on martingale residuals, the probability integral transform, and varentropy—a concept derived from information theory. The null hypothesis assumes that the failure times follow the ARA∞ model with a power law process (PLP) as the initial hazard rate. To evaluate the performance of the proposed tests, we conduct extensive simulation studies under different alternative maintenance models (e.g., ARA1, ARA∞–Log Linear Process(LLP)) and varying parameter settings. Our findings show that the power of the tests varies depending on the nature of the alternatives, and varentropy-based statistics outperform others under certain conditions. Finally, we apply the proposed methods to a real dataset (Ambassador vehicle failure times) to assess their practical relevance. The results confirm the validity of the fitted model and demonstrate the usefulness of varentropy-based approaches for detecting subtle deviations in maintenance patterns.