  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
(MLE) 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 \emph{effective sample size
enlargement} phenomenon, identified in Jiao \emph{et al.} (2015), 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 in Jiao
\emph{et al.} (2015).
