Asymptotically distribution-free goodness-of-fit testing for normality: a log-transformed covariance-driven framework under multiplicative distortion

This paper investigates goodness-of-fit tests for assessing the normality of model errors, employing a novel statistic that is derived from logarithmically transformed observations in conjunction with a k-th power covariance-driven estimator. By strategically selecting the value of k, we ensure that the test is asymptotically distribution-free, thereby establishing its limiting null distribution with a clear and explicit characterization of variance. Importantly, this statistic is independent of model specifications, which enhances its universality and applicability. Extensive simulation studies have confirmed the robustness of this method for normality verification in ultra-large samples, outperforming conventional approaches when sample size is a constraint. Furthermore, we explore the application of this method in the presence of multiplicative distortion measurement errors. Theoretically, the test inherently neutralizes the contamination of distortions in both the response and covariate variables through the mechanism of logarithmic transformation, thereby circumventing the bias amplification that is inherent in classical methods. Numerical experiments have validated the distortion-robust properties of this approach, and an empirical analysis has demonstrated its practical utility and effectiveness.