Nonparametric Maximum Likelihood Estimation of Spatial Patterns

Abstract

Let $\mathbf{X}$ be an absolutely continuous random variable in $\mathbb{R}^k$ with distribution function $F(\mathbf{x})$ and density $f(\mathbf{x})$. Let $\mathbf{X}_1, \cdots, \mathbf{X}_n$ be independent random variables distributed according to $F$. Mapping the spatial distribution of $\mathbf{X}$ normally entails drawing a map of the isopleths, or level curves, of $f$. In this paper, it is shown how to map the isopleths of $f$ nonparametrically according to the criterion of maximum likelihood. The procedure involves specification of a class $\mathscr{L}$ of sets whose boundaries constitute admissible isopleths and then maximizing the likelihood $\prod^n_{i = 1} g(\mathbf{x}_i)$ over all $g$ whose isopleths are boundaries of $\mathscr{L}$-sets. The only restrictions on $\mathscr{L}$ are that it be a $\sigma$-lattice and an $F$-uniformity class. The computation of the estimate is normally straightforward and easy. Extension is made to the important case where $\mathscr{L}$ may be data-dependent up to locational and/or rotational translations. Strong consistency of the estimator is shown in the most general case.