On sequential density estimation

Abstract

We consider the problem of sequential estimation of a density function f at a point x0 which may be known or unknown. Let Tn be a sequence of estimators of x 0 . For two classes of density estimators f n , namely the kernel estimates and a recursive modification of these, we show that if N(d) is a sequence of integer-valued random variables and n(d) a sequence of constants with N(d)/n(d)→ 1 in probability as d → 0, then f N(d) (T N(d) -f(x0) is asymptotically normally distributed (when properly normed). We also propose two new classes of stopping rules based on the ideas of fixed-width interval estimation and show that for these rules, N(d)/n(d) → 1 almost surely and EN(d)/n(d) → 1 as d → 0. One of the stopping rules is itself asymptotically normally distributed when properly normed and yields a confidence interval for f(x0) of fixed-width and prescribed coverage probability.