On Estimating L_2^2 Divergence

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}[\latex]-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.