Consistency in Nonparametric Estimation of the Mode

Abstract

Let $X$ be an absolutely continuous real-valued random variable with additional restrictions to be imposed later. Venter (1967) ("On estimation of the mode," Ann. Math. Statist. 37 1446-1455) estimated the mode of $X$ by a point from the shortest interval containing a specified number $r = r(n)$ of observations. Venter demonstrated that such an estimator is strongly consistent under appropriate conditions on the distribution of $X$ and on $r(n)$. It is the purpose of this paper to show that strong consistency actually holds under very general conditions on the distribution of $X$. Convergence rates are also obtained which are, in some cases, much faster than those reported by Venter.