Truncated Cram\'er-von Mises test of normality

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.