The Annals of Statistics

The Geometry of Mixture Likelihoods: A General Theory

Bruce G. Lindsay

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Mixture maximum likelihood geometry


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