Estimation of a Multivariate Mode

Abstract

Consider a random sample from an absolutely continuous multivariate distribution. Let $\mathscr{J}$ be a class of sets which are not too long and thin. A point $\mathbf{\theta}_n$ chosen from a minimum volume set $S_n \in \mathscr{J}$ containing at least $r = r(n)$ of the data may be used as an estimate of the mode of the distribution. In this paper, it is shown that $\mathbf{\theta}_n$ converges almost surely to the true mode under very minor conditions on $\{r(n)\}$ and the distribution. Convergence rates are also obtained. Extensions to estimation of local and/or multiple modes are noted. Finally, computational simplifications resulting from choosing $S_n$ from spheres or cubes centered at observations are discussed.