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 x0. For two classes of density estimators fn, 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 fN(d)(TN(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. 1976 Springer-Verlag.
