Adaptive calibrated testing strategies for multiplicatively distorted exponential variables

In this paper, our primary focus lies in conducting goodness-of-fit tests for the multiplicative distortion model, where the foundational random variable adheres to an exponential distribution yet remains unobserved as it undergoes multiplicative distortion caused by an observed confounding variable. To address this, we introduce several testing approaches: a transformation test based on U-statistics, a test rooted in negative cumulative extropy, and data-adaptive tests specifically designed for scenarios with unknown scale parameters. We further extend certain of these tests to account for multiplicative distortions. Our research findings indicate that, among the tests developed, two calibrated U-statistics tests do not demonstrate efficiency, whereas four data-adaptive calibrated tests not only show asymptotic efficiency but also remain unaffected by distortion effects. Additionally, we derive the asymptotic null distributions of the data-adaptive calibrated test statistics, complete with known asymptotic variances. To assess the effectiveness of these proposed test statistics, we carry out Monte Carlo simulation experiments and, for illustrative purposes, apply these methods to the analysis of four real-world datasets.