Centered Greenwood statistic approach to goodness-of-fit testing for latent variables subject to multiplicative distortion Errors

In this study, we introduce a novel adaptation of the Greenwood statistic-designated as the centered Greenwood statistic-and meticulously apply this statistic to the analysis of data involving distortion measurement errors. We conduct an in-depth analysis of its asymptotic properties, deriving its theoretical distribution and constructing corresponding asymptotic confidence intervals. Furthermore, we extend its application to goodness-of-fit testing for uniformity, normality, and exponentiality of latent variables, thereby showcasing its versatility in distributional hypothesis testing. Three calibrated variants are developed for multiplicative distortion measurement error models, demonstrating asymptotic normality and efficiency while ensuring that distortion functions exert no influence on asymptotic variances. Monte Carlo simulations substantiate finite-sample performance, and an empirical analysis of a real-world dataset underscores its practical efficacy.