发表机构
Cumhuriyet University; University of Ioannina（Cumhuriyet大学; 约阿尼纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过最大化Kaniadakis $\kappa$ 熵引入广义 $\kappa$ 高斯分布，构造 $k$ 近邻熵估计器并建立其理论性质，推导熵的闭式公式，确定有限方差阈值，进而构建并模拟验证非参数拟合优度检验。
AI 中文摘要
我们通过在某些一阶和二阶矩约束下最大化一类Kaniadakis $\kappa$ 熵，引入了这类多元Kappa-高斯分布。这与Rényi和Tsallis的方法略有不同，因为$\kappa$ 熵源于对指数函数和对数函数的相对论一致性变形，并产生一族尾部服从幂律的分布，当 $\kappa \to 0$ 时，该族实际上回归到标准高斯分布。然后，我们为 $\kappa$ 熵构造了一个 $k$ 近邻（NN）估计器，并通过两个涉及幂和的NN泛函之差来描述它。接下来，我们利用众所周知的次可加欧几里得泛函和矩展开框架，建立了 $L^2$ 一致性和渐近正态性。此外，利用 $\kappa$ 指数的缩放恒等式，我们推导了 $\kappa$ 高斯分布本身熵的闭式公式。由此，我们确定了分布保持有限方差的精确阈值 $\kappa < 2/(m+2)$；这类似于Student-$t$ 分布中的“自由度 $>2$”条件。利用这些组成部分，我们构建了一个非参数拟合优度检验，并通过模拟对其进行了验证。最后，我们提及Sharma--Mittal熵，它结合了Rényi和Tsallis熵，作为未来工作的一个自然方向。
英文摘要
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 $κ\to 0$. 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.
Comments26 pages, 12 figures
