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December, 1982 Nonparametric Maximum Likelihood Estimation of Spatial Patterns
Thomas W. Sager
Ann. Statist. 10(4): 1125-1136 (December, 1982). DOI: 10.1214/aos/1176345978


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.


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Thomas W. Sager. "Nonparametric Maximum Likelihood Estimation of Spatial Patterns." Ann. Statist. 10 (4) 1125 - 1136, December, 1982.


Published: December, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0506.62025
MathSciNet: MR673648
Digital Object Identifier: 10.1214/aos/1176345978

Primary: 62G05
Secondary: 60F15 , 62H99

Keywords: $\sigma$-lattice , consistency , Density estimation , isopleth , isotonic regression , maximum likelihood estimation , nonparametric estimation

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 4 • December, 1982
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