A Two-Scale Characteristic Function Test for Exponentiality with an Extension to Random Right Censoring

Testing exponentiality is one of the classical problems in the goodness-of-fit literature. In this paper, we study a limitation in the classical minimum characterization of the exponential distribution. If only one scaling identity is applied without an added smoothness condition, some non-exponential distributions with log-periodic cumulative hazards also satisfy this “characterization” and cannot be distinguished from the exponential model. We address this problem by jointly using the minimum identities at scales two and three. Together, these identities characterize the exponential distribution without requiring differentiability at zero. This characterization is used to construct a new test that compares the empirical characteristic function of the studentized observations to the characteristic functions obtained from the sample minima of both pairs and groups of three. We derive the joint Gaussian limit and covariance kernel, including cross-scale dependence and mean estimation. A Monte Carlo study is performed to evaluate the empirical power performance of the test under conventional and specially constructed log-periodic alternatives and to examine the tuning parameter and component weights. The results show satisfactory power performance for several conventional alternatives, but combining the scales can reduce finite-sample power. The procedure is then extended to randomly right-censored data using weights obtained from the Kaplan–Meier estimator. The test is shown to be consistent against fixed alternatives in both the complete-sample and right-censored settings under stated assumptions. Its practical use is illustrated with censored high-voltage endurance data.