Adaptive Probability Density Estimation in Lower Dimensions using Random Tessellations

Abstract : This paper presents a class of non-parametric density estimators on a low dimensional space. The support of these estimators is defined by the convex hull of the set of observations. A random sample from the set of observations is used to tessellate the interior of the convex hull. The attribution of empirical probability mass to the tiles resulting from the tessellation produces a density estimate. With a set of appropriate linear constraints on the attribution of mass, the estimator is shown to be a conditional maximum likelihood estimator. Repeating this procedure, and averaging these density estimates within tiles, produces a bootstrap estimate of the density function. The results of this resampling and density estimation process are presented in graphic form.