🤖 AI Summary
This paper addresses the problem of testing whether a multivariate distribution is elliptical. The proposed nonparametric test is based on the Kullback–Leibler (KL) divergence, leveraging the characterization that an elliptical distribution is equivalent to independence between radial length and direction, with the latter uniformly distributed on the unit sphere. The method constructs a test statistic via k-nearest-neighbor estimation of the KL divergence. To establish first-order asymptotic normality—bypassing stringent assumptions on functional Fréchet differentiability—it employs sample splitting, truncation, and spherical projection. Furthermore, bias correction and variance inflation techniques are introduced to handle degeneracy of the influence function, enabling unified treatment under both known and unknown mean/covariance settings. Numerical experiments demonstrate that the proposed test significantly outperforms state-of-the-art methods in both size control and statistical power.
📝 Abstract
We conduct a KL-divergence based procedure for testing elliptical distributions. The procedure simultaneously takes into account the two defining properties of an elliptically distributed random vector: independence between length and direction, and uniform distribution of the direction. The test statistic is constructed based on the $k$ nearest neighbors ($k$NN) method, and two cases are considered where the mean vector and covariance matrix are known and unknown. First-order asymptotic properties of the test statistic are rigorously established by creatively utilizing sample splitting, truncation and transformation between Euclidean space and unit sphere, while avoiding assuming Fréchet differentiability of any functionals. Debiasing and variance inflation are further proposed to treat the degeneration of the influence function. Numerical implementations suggest better size and power performance than the state of the art procedures.