Sequential confidence bands for densities

Abstract

We propose a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and show the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt (1973, Ann. Statist. 1, 1071-1095). Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results. We also present some simulation results and applications of the basic method to stopping rules of interest. Similar result is also obtained for censored data case.