Sequential nonparametric density estimation

Abstract

Using kernel estimates of the Parzen type, a naive sequential nonparametric density estimation procedure is developed. The asymptotic distribution structure of the stopping variable is examined. The stopping variable is shown to have finite moments of ail order and is shown to be dosed. The stopping variable N depends on some preassigned error \varepsilon , and it is shown that N diverges strongly to \infty as \varepsilon converges to zero. Finally, with \hat{f}_n(x) being a kernel-type estimator, it is shown that \hat{f}_N(X) converges to f(x) , the true density at x , with probability one as \varepsilon converges to zero.