Skewness and kurtosis as locally best invariant tests of normality

TL;DR

This paper establishes that sample skewness and kurtosis serve as the most powerful invariant tests for normality against asymmetric and symmetric alternatives respectively, under mild regularity conditions.

Contribution

It proves that skewness and kurtosis are the locally best invariant tests for normality against broad classes of asymmetric and symmetric distributions, respectively.

Findings

Skewness is the LBI test against asymmetric families.

Kurtosis is the LBI test against symmetric families.

Results extend to some multivariate cases.

Abstract

Consider testing normality against a one-parameter family of univariate distributions containing the normal distribution as the boundary, e.g., the family of $t$-distributions or an infinitely divisible family with finite variance. We prove that under mild regularity conditions, the sample skewness is the locally best invariant (LBI) test of normality against a wide class of asymmetric families and the kurtosis is the LBI test against symmetric families. We also discuss non-regular cases such as testing normality against the stable family and some related results in the multivariate cases.