A Note on the Estimation of the Mode

Abstract

Let X1, ⋯, Xn be a sample from a unimodal distribution, F, and let an be a sequence converging to zero. A nonparametric estimate of the mode is the center of the interval of length 2an containing the most observations. This estimate is shown to be strongly consistent and conditions on the speed at which an may converge to zero are given. This estimator of the mode is related to the naive density estimator, (Fn(x + an) - Fn(x - an))/2an, where Fn is the empirical distribution function. A simple strong consistency result for this naive density estimator is given. Also other estimators of the mode are discussed briefly and an application of estimators of the mode is mentioned.