The Annals of Statistics

The Geometry of Mixture Likelihoods: A General Theory

Bruce G. Lindsay

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Abstract

In this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set. The existence, support size, likelihood equations, and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull. It is shown using geometric techniques that the estimator exists under quite general conditions, with a support size no larger than the number of distinct observations. Analysis of the convex dual of the likelihood set leads to a dual maximization problem. A convergent algorithm is described. The defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures. Sufficient conditions for uniqueness are given. Part II will deal with a special theory for exponential family mixtures.

Article information

Source
Ann. Statist. Volume 11, Number 1 (1983), 86-94.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176346059

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176346059

Mathematical Reviews number (MathSciNet)
MR684866

Zentralblatt MATH identifier
0512.62005

Subjects
Primary: 62A10
Secondary: 62G05: Estimation 52A40: Inequalities and extremum problems

Keywords
Mixture maximum likelihood geometry

Citation

Lindsay, Bruce G. The Geometry of Mixture Likelihoods: A General Theory. Ann. Statist. 11 (1983), no. 1, 86--94. doi:10.1214/aos/1176346059. http://projecteuclid.org/euclid.aos/1176346059.


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See also

  • Part II: Bruce G. Lindsay. The Geometry of Mixture Likelihoods, Part II: The Exponential Family. Ann. Statist., Volume 11, Number 3 (1983), 783--792.