Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications

Abstract This study introduces a novel nonparametric hypothesis test designed to evaluate exponentiality against the New Better than Renewal Used in Moment Generating Function (NBRU mgf ) class. The test employs the Laplace transform to construct a scale-invariant statistic, facilitating the efficient analysis of both complete and right-censored data. We derive the asymptotic distribution of the test statistic and calculate Pitman’s asymptotic efficiency under common alternatives, including Weibull, Gamma, and Makeham distributions. Extensive Monte Carlo simulations are conducted to determine the critical values and to demonstrate the test's power properties across a wide range of scenarios. The application of this test to real datasets of physical, engineering, and medical relevance corroborates the validity of our method. The test demonstrates superior efficiency and stability compared to existing methodologies, thereby providing a valuable tool for reliability estimation and statistical inference.