Goodness-of-fit tests based on Mellin statistics for reliability data

The selection and validation of probabilistic models for reliability data, often characterized by positive support and asymmetric shapes, are crucial for practical applications in quality and reliability engineering. This article introduces a family of three complementary goodness-of-fit tests based on an adaptation of Hotelling’s T2 statistic applied to vectors of sample log-cumulants. The first test, T(2,3)2, targets log-scale shape (dispersion and skewness); the second, T(1,2,3)2, adds the entropy-related first log-cumulant, making it sensitive to multiplicative shifts; and the third, T(1,…,6)2, exploits the full higher-order log-cumulant characterization, enabling discrimination between distributions such as the log-normal and log-logistic. For each statistic, the correct asymptotic null distribution is derived, and parametric bootstrap p-values are implemented for reliable finite-sample inference. The methodology covers six distributions widely used in reliability: Weibull, Fréchet, gamma, inverse-gamma, log-normal, and log-logistic. A comprehensive simulation study demonstrates good size control and competitive power against shape-based alternatives. The tests are applied to nine real reliability datasets, where they provide targeted, complementary information relative to the Anderson-Darling and Cramér-von Mises tests and the Akaike information criterion.