A truncated Cramér–von Mises test of normality

Abstract

A new test of normality with unknown parameters is proposed in this article. We introduce a Cramér–von Mises type statistic with weight function equal to the inverse of the standard normal density function supported in the interval $[-a_n,a_n]$ depending on the sample size n. The sequence $\left\{a_n\right\}$ is chosen so that the statistic goes to infinity and after subtracting the mean, a suitable test statistic is obtained, with the same asymptotic law as the well-known Shapiro–Wilk statistic. It is shown that the performance of the new test in many cases improves that of other well-known tests of normality.

Keywords:

• Cramér–von Mises statistic
• empirical process
• normality tests

Keywords:

• 2010 MATHEMATICS SUBJECT CLASSIFICATIONS 60G10

Acknowledgments

I want to deeply Enrique Cabaña for his invaluable help during long years of advice.