正则变化占用模型下插件式香农熵估计量的渐近偏差
Asymptotic bias of the plug-in Shannon entropy estimator under a regularly varying occupancy model
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中文总结 AI 辅助
本文针对可数无限支撑离散分布的香农熵估计难题，在正则变化占用模型下推导了插件式熵估计量的渐近偏差关系，明确了幂律频率分布下偏差的渐近行为由基础分布尾特征决定。
中文摘要 AI 辅助
估计具有可数无限支撑的离散分布的香农熵是一个具有挑战性的问题。本文研究了在频率序列具有尾指数α∈(0,1)的正则变化的占用模型下，香农熵H(p)的插件式估计量Ĥₙ的偏差。利用泊松化和正则变化理论，我们建立了渐近关系|E[Ĥₙ] - H(p)| ~ n^(α-1)L(n)C_α，其中L是慢变函数，C_α是仅依赖于α的显式常数，具有积分表示。该结果表明，幂律频率分布下插件式估计量偏差的渐近行为由基础分布的尾行为决定。
英文摘要
Estimating the Shannon entropy of discrete distributions with countably infinite support is a challenging problem. In this paper, we investigate the bias of the plug-in estimator $\hat{H}_n$ for the Shannon entropy $H(\boldsymbol{p})$ under an occupancy model whose frequency sequence exhibits regular variation with tail index $α\in(0,1)$. Using Poissonization and the theory of regular variation, we establish the asymptotic relation $|\mathsf{E}[\hat H_n] - H(\boldsymbol{p})| \sim n^{α-1}L(n)C_α$, where $L$ is a slowly varying function and $C_α$ is an explicit constant depending only on $α$ that admits an integral representation. Our result shows that the asymptotic behavior of the bias of the plug-in estimator under power-law frequency distributions is determined by the tail behavior of the underlying distribution.
