Invariant Tests for Symmetry about an Unspecified Point Based on the Empirical Characteristic Function
Date: 07. 06. 2001
This paper considers a flexible class of omnibus affine
invariant tests for the hypothesis that a multivariate
distribution is symmetric about an unspecified point.
The test statistics are weighted integrals involving the
imaginary part of the empirical characteristic function
of suitably standardized given data, and they have an
alternative representation in terms of an $L^2$-distance
of nonparametric kernel density estimators. Moreover,
there is a connection with two measures of multivariate
skewness. The tests are performed via a permutational
procedure that conditions on the data.
Primary MSC: 62G10 Hypothesis testing
Secondary MSC: 62G20 Asymptotic properties
Keywords: Test for symmetry, affine invariance, Mardia's measure of multivariate skewness, skewness in the sense of Mori, Rohatgi and Szekely, empirical characteristic function, permutational limit theorem.