Minimax Estimation of the $L_1$ Distance

We consider the problem of estimating the $L_1$ distance between two discrete probability measures $P$ and $Q$ from empirical data in a nonasymptotic and large alphabet setting. We construct minimax rate-optimal estimators for $L_1(P,Q)$ when $Q$ is either known or unknown, and show that the performance of the optimal estimators with $n$ samples is essentially that of the Maximum Likelihood Estimators (MLE) with $n\ln n$ samples. Hence, the \emph{effective sample size enlargement} phenomenon, identified in Jiao \emph{et al.} (2015), holds for this problem as well. However, the construction of optimal estimators for $L_1(P,Q)$ requires new techniques and insights beyond the \emph{Approximation} methodology of functional estimation in Jiao \emph{et al.} (2015).