  We present \emph{Local Moment Matching (LMM)}, a unified methodology for
symmetric functional estimation and distribution estimation under Wasserstein
distance. We construct an efficiently computable estimator that achieves the
minimax rates in estimating the distribution up to permutation, and show that
the plug-in approach of our unlabeled distribution estimator is "universal" in
estimating symmetric functionals of discrete distributions. Instead of doing
best polynomial approximation explicitly as in existing literature of
functional estimation, the plug-in approach conducts polynomial approximation
implicitly and attains the optimal sample complexity for the entropy, power sum
and support size functionals.
