1 research outputs found
Minimax Estimation of the Distance
We consider the problem of estimating the distance between two discrete
probability measures and from empirical data in a nonasymptotic and
large alphabet setting. When is known and one obtains samples from ,
we show that for every , the minimax rate-optimal estimator with samples
achieves performance comparable to that of the maximum likelihood estimator
(MLE) with samples. When both and are unknown, we construct
minimax rate-optimal estimators whose worst case performance is essentially
that of the known case with being uniform, implying that 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 case for every and the unknown case. However, the
construction of optimal estimators for requires new techniques and
insights beyond the approximation-based method of functional estimation in Jiao
\emph{et al.} (2015).Comment: to appear on IEEE Transactions on Information Theor