基于稳定化技术的Tsallis熵估计量的渐近正态性与收敛速率
Asymptotic Normality and Convergence Rates for Tsallis Entropy Estimators via Stabilization Techniques
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中文总结 AI 辅助
该研究针对一般度量测度空间上的Tsallis熵估计量，采用稳定化技术建立其渐近正态性与收敛速率，扩展了Shannon和Rényi熵的相关结果，为复杂空间和高维场景的非参数统计推断提供了有效框架。
中文摘要 AI 辅助
我们研究一般度量测度空间上与泊松和二项点过程相关的基于最近邻的Tsallis熵估计量。利用基于灵活的加一成本算子及二阶庞加莱不等式的稳定化技术，我们建立了渐近正态性，并推导了柯尔莫哥洛夫距离的显式收敛速率。我们的分析避免了显式得分函数分解，而是依赖于加一成本的灵活局部化，这简化了高阶项的处理。在自然的稳定化和矩条件下，所得界恢复了经典的正态近似速率\ns^{-1/2}和n^{-1/2}，并扩展了Shannon和Rényi熵估计量的相应结果。我们还通过涉及Tsallis熵泛函、加权k近邻Shannon熵估计量的示例说明了该框架的适用范围，这些示例凸显了稳定化正态近似在复杂空间和高维设置的非参数统计推断中的优势。
英文摘要
We study nearest-neighbor-based estimators of Tsallis entropy associated with Poisson and binomial point processes on general metric measure spaces. Using stabilization techniques based on flexible add-one cost operators together with second-order Poincaré inequalities, we establish asymptotic normality and derive explicit convergence rates for the Kolmogorov distance. Our analysis avoids explicit score-function decompositions and instead relies on flexible localizations of add-one costs, which simplify the treatment of higher-order terms. Under natural stabilization and moment conditions, the resulting bounds recover the classical normal approximation rates \(s^{-1/2}\) and \(n^{-1/2}\) and extend corresponding results for Shannon and Rényi entropy estimators. We further illustrate the scope of the framework through examples involving Tsallis entropy functionals, weighted \(k\)-nearest-neighbor Shannon entropy estimators. The examples provided highlight the benefits of stabilization-based normal approximations for non-parametric statistical inference in complex spatial and high-dimensional settings.
