熵型积分泛函的估计
Estimation of entropy-type integral functionals
摘要：
密度的熵型积分泛函在数理统计、信息论和计算机科学中有着广泛的应用。示例包括分布之间的接近度的度量（例如，密度功率发散）和随机变量的不确定性特性（例如，Rényi熵）。本文研究了一类泛函的U-统计量估计。估计量是基于ε-关闭向量观测相应的独立和同分布的样本。我们证明了估计的渐近性质（一致性和渐近正态性）在温和的可积性和光滑性条件下的密度。结果可以应用于数理统计和计算机科学中的各种问题（例如，分布识别问题、随机数据库的近似匹配、双样本问题）。
英文摘要：
Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Rényi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on epsilon-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).
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