Discrete Parameter-Free Zone Distribution and Its Application in Normality Testing

Abstract

In recent research endeavors, discrete models have gained considerable attention, even in cases where the observed variables are continuous. These variables can often be effectively approximated by a normal distribution. Given the prevalence of processes requiring robust quality control, models associated with the normal distribution have found widespread applicability; nevertheless, there remains a persistent need for enhanced accuracy in normality analysis, prompting the exploration of novel and improved solutions. This paper introduces a discrete parameter-free distribution linked to the normal distribution, derived from a quality control methodology rooted in the renowned ‘3-sigma’ rule. The development of a novel normality test, based on this distribution, is presented. A comprehensive examination encompasses mathematical derivation, distribution tables generated through Monte Carlo simulation studies, properties, power analysis, and comparative analysis, all with key features illustrated graphically. Notably, the proposed normality test surpasses conventional methods in performance. Termed the ‘Zone distribution’, this newly introduced distribution, along with its accompanying ‘Zone test’, demonstrates superior efficacy through illustrative examples. This research contributes a valuable tool to the field of normality analysis, offering a robust alternative for applications requiring precise and reliable assessments.