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

Estimation of the value $f(0)$ of a density function evaluated at 0 is studied, f: Rₘ → R, 0 ∈ Rₘ. Sequences of estimators \γₙ, n 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 ₙ, n 1\, how rapidly can aₙ tend to zero and yet have liminfn→∞ inff∈ CPf(|γₙ(X₁,⋯, Xₙ) - f(0)| aₙ) > 0? In brief, by "rate of convergence" we will mean the rate which aₙ tends to zero. For a continuum of different choices of the set C specified by various Lipschitz conditions on the kth partial derivatives of f, k 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.