Truncated Cram\'er-von Mises test of normality

TL;DR

This paper introduces a new normality test based on a weighted empirical process with a specific asymptotic distribution, and compares its performance with existing tests through extensive power analysis.

Contribution

It proposes a novel Cramér-von Mises type normality test with adaptive weighting and analyzes its asymptotic behavior and power performance against other established tests.

Findings

The new test has an asymptotic distribution similar to Shapiro-Wilk.

It performs favorably in power against various alternatives.

The test adapts to sample size through the interval $[-a_n,a_n]$.

Abstract

A new test of normality based on a standardised empirical process is introduced in this article. The first step is to introduce a Cram\'er-von Mises type statistic with weights equal to the inverse of the standard normal density function supported on a symmetric interval $[-a_n,a_n]$ depending on the sample size $n.$ The sequence of end points $a_n$ tends to infinity, and is chosen so that the statistic goes to infinity at the speed of $\ln \ln n.$ After substracting the mean, a suitable test statistic is obtained, with the same asymptotic law as the well-known Shapiro-Wilk statistic. The performance of the new test is described and compared with three other well-known tests of normality, namely, Shapiro-Wilk, Anderson-Darling and that of del Barrio-Matr\'an, Cuesta Albertos, and Rodr\'{\i}guez Rodr\'{\i}guez, by means of power calculations under many alternative hypotheses.