Entropy-based test for generalized Gaussian distributions

In this paper, we provide the proof of $L^2$ consistency for the $k$th nearest neighbour distance estimator of the Shannon entropy for an arbitrary fixed $k\geq 1.$ We construct the non-parametric test of goodness-of-fit for a class of introduced generalized multivariate Gaussian distributions based on a maximum entropy principle. The theoretical results are followed by numerical studies on simulated samples.