On the Estimation of Chernoff Information

Abstract

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 R\'enyi 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.