Entropy-Based Goodness-of-Fit Tests for Generalized $\kappa$ -Gaussian Distributions

Abstract

We introduce this class of multivariate Kappa-Gaussian distributions by maximizing a type of Kaniadakis $\kappa$ entropy subject to certain first- and second-moment constraints. This differs slightly from the approach of R\'enyi and Tsallis, because the $\kappa$-entropy arises from a relativity-consistent deformation of the exponential and logarithmic functions and yields a family of distributions whose tails follow a power law, a family that actually reverts to the standard Gaussian distribution as $\kappa \to 0$. We then create a $k$-nearest-neighbor (NN) estimator for $\kappa$-entropy and describe it using the difference between two NN functionals involving sums of powers. Next, we establish $L^2$-consistency and asymptotic normality, using the well-known subadditive Euclidean functional and moment expansion framework for NN entropy estimators. Furthermore, using a scaling identity for the $\kappa$-exponential, we derive a closed-form formula for the entropy of the $\kappa$-Gaussian distribution itself. From this, we determine the exact threshold value $\kappa<2/(m+2)$ at which the distribution preserves finite variance; this is analogous to the ``degrees of freedom $>2$''condition in the Student's-$t$ distribution. With these components, we construct a non-parametric goodness-of-fit test and then verify it through simulations. Finally, we mention the Sharma--Mittal entropy, which combines the R\'enyi and Tsallis entropies as a natural direction for future work.