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Skewness and kurtosis as locally best invariant tests of normality

Akimichi Takemura·Muneya Matsui·Satoshi Kuriki
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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.

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Akimichi Takemura, Muneya Matsui, Satoshi Kuriki. 2006-08-20. Skewness and kurtosis as locally best invariant tests of normality. https://arxiv.org/abs/math/0608499

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