Calibration-enhanced estimation and asymptotically distortion-free testing under multiplicative measurement errors

The validation of the Rayleigh distribution for unobserved variables poses significant challenges when subjected to multiplicative measurement error contamination, as conventional estimation and goodness-of-fit testing methodologies exhibit diminished reliability under such conditions. This study bridges a critical gap in the literature by introducing a comprehensive framework for parameter estimation and hypothesis testing in the presence of multiplicative distortion errors. Specifically, we propose a suite of estimators for the Rayleigh density parameter, encompassing maximum likelihood, moment-based, and logarithmic transformation-based approaches, and rigorously evaluate their asymptotic efficiency. Our analysis reveals a persistent fixed fator attributable to multiplicative distortions within their asymptotic variance structures. Furthermore, we introduce nine asymptotically distortion-free test statistics to evaluate the adherence of unobserved variables to the Rayleigh distribution, demonstrating that these tests asymptotically nullify the impact of multiplicative distortions. Monte Carlo simulations validate the effectiveness of our proposed estimators and test statistics. These methodologies are subsequently applied to an empirical dataset to illustrate their practical utility.