关于Chernoff信息的估计
On the Estimation of Chernoff Information
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中文总结 AI 辅助
针对Chernoff信息非参数估计的难题，通过导数条件重构优化并结合k近邻法与二分法，证明了导数估计量的L₂一致性，提出了Chernoff信息的非参数估计方法。
中文摘要 AI 辅助
Chernoff信息是一种基础的散度度量，用于刻画贝叶斯二元假设检验中的最优误差指数，可应用于信息融合、时间序列分析和统计学习理论。不过，仅在简单参数族中存在闭式表达式，非参数估计仍很困难，因为该量被定义为关于其阶数的未归一化Rényi散度的优化。我们通过导数条件重新构建该优化，其零点对应最优混合参数，并使用k近邻方法直接估计该导数。我们在密度及其定义域的温和正则性条件下，证明了导数估计量的L₂一致性。结合二分法（可将最优参数定位至任意精度），这就得到了Chernoff信息的一个估计量。
英文摘要
Chernoff information is a fundamental divergence measure characterizing the optimal error exponent in Bayesian binary hypothesis testing, with applications in information fusion, time-series analysis, and statistical learning theory. However, closed-form expressions exist only for simple parametric families, and nonparametric estimation remains difficult because the quantity is defined as an optimization of the unnormalized Rényi divergence over its order. We reformulate this optimization via a derivative condition, whose zero locates the optimal mixture parameter, and estimate the derivative directly using a $k$-nearest-neighbor method. We prove the $L_2$-consistency of the derivative estimator under mild regularity conditions on the densities and their domain. Coupled with a bisection procedure that locates the optimal parameter up to arbitrary precision, this yields an estimator for Chernoff information.
