On the Best Obtainable Asymptotic Rates of Convergence in Estimation of a Density Function at a Point

Abstract

Estimation of the value $f(0)$ of a density function evaluated at 0 is studied, $f: \mathbb{R}_m \rightarrow \mathbb{R}, 0 \in \mathbb{R}_m$. Sequences of estimators $\{\gamma_n, n \geqq 1\}$, one estimator for each sample size, are studied. We are interested in the problem, given a set $C$ of density functions and a sequence of numbers $\{a_n, n \leqq 1\}$, how rapidly can $a_n$ tend to zero and yet have $\lim\inf_{n\rightarrow\infty} \inf_{f\in C}P_f(|\gamma_n(X_1,\cdots, X_n) - f(0)|\leqq a_n) > 0?$ In brief, by "rate of convergence" we will mean the rate which $a_n$ tends to zero. For a continuum of different choices of the set $C$ specified by various Lipschitz conditions on the $k$th partial derivatives of $f, k \geqq 0$, lower bounds for the possible rate of convergence are obtained. Combination of these lower bounds with known methods of estimation give best possible rates of convergence in a number of cases.