Entropy-Based Goodness-of-Fit Tests for Generalized κ -Gaussian Distributions
Abstract
We introduce this class of multivariate Kappa-Gaussian distributions by maximizing a type of Kaniadakis κ entropy subject to certain first- and second-moment constraints. This differs slightly from the approach of Rényi and Tsallis, because the κ-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 κ→ 0. We then create a k-nearest-neighbor (NN) estimator for κ-entropy and describe it using the difference between two NN functionals involving sums of powers. Next, we establish L2-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 κ-exponential, we derive a closed-form formula for the entropy of the κ-Gaussian distribution itself. From this, we determine the exact threshold value κ< 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 Rényi and Tsallis entropies as a natural direction for future work.
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BibTeX
@misc{cadirci2026entropybased,
title = {{Entropy-Based Goodness-of-Fit Tests for Generalized $κ$ -Gaussian Distributions}},
author = {Mehmet Siddik Cadirci and Konstantinos Zografos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06873},
eprint = {2610.06873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropy-Based Goodness-of-Fit Tests for Generalized κ -Gaussian Distributions
AU - Mehmet Siddik Cadirci
AU - Konstantinos Zografos
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06873
UR - https://arxiv.org/abs/2610.06873
AB - We introduce this class of multivariate Kappa-Gaussian distributions by maximizing a type of Kaniadakis κ entropy subject to certain first- and second-moment constraints. This differs slightly from the approach of Rényi and Tsallis, because the κ-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 κ→ 0. We then create a k-nearest-neighbor (NN) estimator for κ-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 κ-exponential, we derive a closed-form formula for the entropy of the κ-Gaussian distribution itself. From this, we determine the exact threshold value κ< 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 Rényi and Tsallis entropies as a natural direction for future work.
ER -