A Note on the Estimation of the Mode

Abstract

Let $X_1, \cdots, X_n$ be a sample from a unimodal distribution, $F$, and let $\{a_n\}$ be a sequence converging to zero. A nonparametric estimate of the mode is the center of the interval of length $2a_n$ containing the most observations. This estimate is shown to be strongly consistent and conditions on the speed at which $a_n$ may converge to zero are given. This estimator of the mode is related to the naive density estimator, $(F_n(x + a_n) - F_n(x - a_n))/2a_n$, where $F_n$ 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.