  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.
