  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.
