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. When Q is known and one obtains n samples from P, we show that for every Q, the minimax rate-optimal estimator with n samples achieves performance comparable to that of the maximum likelihood estimator with n ln n samples. When both P and Q are unknown, we construct minimax rate-optimal estimators, whose worst case performance is essentially that of the known Q case with Q being uniform, implying that Q being uniform is essentially the most difficult case. The effective sample size enlargement phenomenon, identified by Jiao et al., holds both in the known Q case for every Q and the Q unknown case. However, the construction of optimal estimators for ∥P - Q∥
₁
requires new techniques and insights beyond the approximation-based method of functional estimation by Jiao et al.