The Annals of Statistics

Binomial mixtures: geometric estimation of the mixing distribution

G. R. Wood

Full-text: Open access


Given a mixture of binomial distributions, how do we estimate the unknown mixing distribution? We build on earlier work of Lindsay and further elucidate the geometry underlying this question, exploring the approximating role played by cyclic polytopes. Convergence of a resulting maximum likelihood fitting algorithm is proved and numerical examples given; problems over the lack of identifiability of the mixing distribution in part disappear.

Article information

Ann. Statist., Volume 27, Number 5 (1999), 1706-1721.

First available in Project Euclid: 23 September 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section
Secondary: 62P15: Applications to psychology 52B12: Special polytopes (linear programming, centrally symmetric, etc.)

Binomial mixture mixing distribution geometry moment curve cyclic polytope nearest point least squares weighted least squares maximum likelihood Kullback-Leibler distance


Wood, G. R. Binomial mixtures: geometric estimation of the mixing distribution. Ann. Statist. 27 (1999), no. 5, 1706--1721. doi:10.1214/aos/1017939148.

Export citation


  • [1] Alfsen,E. M. (1971). Compact Convex Sets and Boundary Integrals. Springer, Berlin.
  • [2] Bohning,D. (1995). A review of reliable maximum likelihood algorithms for semiparametric mixture models. J. Statist. Plann. Inference 47 5-28.
  • [3] Choquet,G. (1969). Lectures on Analysis (J. Marsden, T. Lance and S. Gelbart, eds.). Benjamin, New York.
  • [4] Cressie,N. (1979). A quickand easy empirical Bayes estimate of true scores. Sankhy ¯a Ser. B 41 101-108.
  • [5] Karlin,S. and Shapley,L. S. (1953). Geometry of moment spaces. Mem. Amer. Math. Soc. 12.
  • [6] Karlin,S. and Studden,W. J. (1966). Tchebycheff Systems: with applications in analysis and statistics. In Pure and Applied Mathematics (R. Courant, L. Bers, J. J. Stoker, eds.) 15. Interscience, New York.
  • [7] Kelley,J. L. (1955). General Topology. Van Nostrand, Princeton.
  • [8] Laird,N. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. J. Amer. Statist. Assoc. 73 805-811.
  • [9] Lesperance,M. L. and Kalbfleisch,J. D. (1992). An algorithm for computing the nonparametric MLE of a mixing distribution. J. Amer. Statist. Assoc. 87 120-126.
  • [10] Lindsay,B. G. (1981). Properties of the maximum likelihood estimator of a mixing distribution. In Statistical Distributions in Scientific Work (C. Taillie, ed.) 5 95-109. Reidel, Dordrecht.
  • [11] Lindsay,B. G. (1983). The geometry of mixture likelihoods: a general theory. Ann. Statist. 11 86-94.
  • [12] Lindsay,B. G. (1983). The geometry of mixture likelihoods II: the exponential family. Ann. Statist. 11 783-792.
  • [13] Lindsay,B. G. (1986). Exponential family mixture models (with least squares estimators). Ann. Statist. 14 124-137.
  • [14] Lindsay,B. G. (1995). Mixture Models: Theory, Geometry and Applications. IMS, Hayward, CA.
  • [15] Lindsay,B. G.,Clogg,C. C. and Grego,J. (1991). Semi-parametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. J. Amer. Statist. Assoc. 86 96-107.
  • [16] Lindsay,B. G. and Lesperance,M. L. (1995). A review of semiparametric mixture models. J. Statist. Plann. Inference 47 29-39.
  • [17] Lord,F. M. (1969). Estimating true-score distributions in psychological testing (an empirical Bayes estimation problem). Psychometrika 34 259-299.
  • [18] Lord,F. M. and Cressie,N. (1975). An empirical Bayes procedure for finding an interval estimate. Sankhy ¯a Ser. B 37 1-9.
  • [19] Mattheiss,T. H. and Rubin,D. S. (1980). A survey and comparison of methods for finding all vertices of convex polyhedral sets. Math. Oper. Res. 5 167-185.
  • [20] McMullen,P. and Shephard,G. C. (1971). Convex Polytopes and the Upper Bound Conjecture. Cambridge Univ. Press.
  • [21] Papadopoulou,S. (1982). Stabile konvexe Mengen. Jahresber. Deutsch. Math.Verein. 84 92-106.
  • [22] Phelps,R. R. (1966). Lectures on Choquet's Theorem. Van Nostrand, Princeton.
  • [23] Sampson,A. R. and Smith,R. L. (1982). Assessing risks through the determination of rare event probabilities. Oper. Res. 30 839-866.
  • [24] Saville,D. J. and Wood,G. R. (1991). Statistical Methods: The Geometric Approach. Springer, New York.
  • [25] Sivaganesan,S. and Berger,J. (1993). Robust Bayesian analysis of the binomial empirical Bayes problem. Canad. J. Statist. 21 107-119.
  • [26] Turnbull,B. W. (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data. J. Roy. Statist. Soc. Ser. B 38 290-295.
  • [27] Wood,G. R. (1992). Binomial mixtures and finite exchangeability. Ann. Probab. 20 1167- 1173.