On Estimating $L_2^2$ Divergence
This work addresses a fundamental statistical estimation problem for researchers in machine learning and statistics, offering rigorous theoretical tools for divergence estimation.
The paper tackles the problem of estimating the L2^2 divergence between continuous distributions by providing a nonparametric estimator with theoretical guarantees, showing it is √n-consistent, asymptotically normal, and minimax optimal in smooth regimes.
We give a comprehensive theoretical characterization of a nonparametric estimator for the $L_2^2$ divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is $\sqrt{n}$-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.