Consistency in Nonparametric Estimation of the Mode

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.