A Correlation Test for Normality Based on the Lévy Characterization

Abstract

A powerful test of fit for normal distributions is proposed. Based on the Lévy characterization, the test statistic is the sample correlation coefficient of normal quantiles and sums of pairs of observations from a random sample. Since the test statistic is location-scale invariant, critical values can be obtained by simulation without estimating any parameters. It is proved that this test is consistent. A power comparison study including some directed tests shows that the proposed test is competitive, it is more powerful than the well-known Jarque–Bera test, and it is comparable to Shapiro–Wilk test against a number of alternatives.