  We provide an efficient unified plug-in approach for estimating symmetric
properties of distributions given $n$ independent samples. Our estimator is
based on profile-maximum-likelihood (PML) and is sample optimal for estimating
various symmetric properties when the estimation error $\epsilon \gg n^{-1/3}$.
This result improves upon the previous best accuracy threshold of $\epsilon \gg
n^{-1/4}$ achievable by polynomial time computable PML-based universal
estimators [ACSS21, ACSS20]. Our estimator reaches a theoretical limit for
universal symmetric property estimation as [Han21] shows that a broad class of
universal estimators (containing many well known approaches including ours)
cannot be sample optimal for every $1$-Lipschitz property when $\epsilon \ll
n^{-1/3}$.
