Exact rates of almost sure convergence of a recursive kernel estimate of a probability densiy function: Application to regression and hazard rate estimation

Let X1…, X n be a random sample of R d -valued random variables from the probability density function f, and let f n *(x) be a recursive kernel estimate of f(x) based on this random sample. Conditions are given under which the exact rates of almost sure convergence of f n *(x) to f(x) are determined. This is done both for d≦1 and also for the special case d = 1. These results are suitably used to establish exact rates for almost sure convergence of a recursive estimate m n *(x) of the regression function m(x). A further application yields exact rates of almost sure convergence of a recursive estimate r n (x) of the hazard rate r(x).