The Annals of Statistics

Monte Carlo likelihood inference for missing data models

Yun Ju Sung and Charles J. Geyer

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We describe a Monte Carlo method to approximate the maximum likelihood estimate (MLE), when there are missing data and the observed data likelihood is not available in closed form. This method uses simulated missing data that are independent and identically distributed and independent of the observed data. Our Monte Carlo approximation to the MLE is a consistent and asymptotically normal estimate of the minimizer θ* of the Kullback–Leibler information, as both Monte Carlo and observed data sample sizes go to infinity simultaneously. Plug-in estimates of the asymptotic variance are provided for constructing confidence regions for θ*. We give Logit–Normal generalized linear mixed model examples, calculated using an R package.

Article information

Ann. Statist., Volume 35, Number 3 (2007), 990-1011.

First available in Project Euclid: 24 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 65C05: Monte Carlo methods

Asymptotic theory Monte Carlo maximum likelihood generalized linear mixed model empirical process model misspecification


Sung, Yun Ju; Geyer, Charles J. Monte Carlo likelihood inference for missing data models. Ann. Statist. 35 (2007), no. 3, 990--1011. doi:10.1214/009053606000001389.

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  • Aliprantis, C. D. and Border, K. C. (1999). Infinite Dimensional Analysis. A Hitchhiker's Guide, 2nd ed. Springer, Berlin.
  • Attouch, H. (1984). Variational Convergence for Functions and Operators. Pitman, Boston.
  • Aubin, J.-P. and Frankowska, H. (1990). Set-Valued Analysis. Birkhäuser, Boston.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Booth, J. G. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 265–285.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. J. Amer. Statist. Assoc. 88 9–25.
  • Coull, B. A. and Agresti, A. (2000). Random effects modeling of multiple binomial responses using the multivariate binomial logit–normal distribution. Biometrics 56 73–80.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38.
  • Ferguson, T. S. (1996). A Course in Large Sample Theory. Chapman and Hall, London.
  • Gelfand, A. E. and Carlin, B. P. (1993). Maximum-likelihood estimation for constrained- or missing-data models. Canad. J. Statist. 21 303–311.
  • Geyer, C. J. (1994). On the asymptotics of constrained $M$-estimation. Ann. Statist. 22 1993–2010.
  • Geyer, C. J. (1994). On the convergence of Monte Carlo maximum likelihood calculations. J. Roy. Statist. Soc. Ser. B 56 261–274.
  • Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Amer. Statist. Assoc. 90 909–920.
  • Guo, S. W. and Thompson, E. A. (1994). Monte Carlo estimation of mixed models for large complex pedigrees. Biometrics 50 417–432.
  • Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. Probab. 1 221–233. Univ. California Press, Berkeley.
  • Karim, M. R. and Zeger, S. L. (1992). Generalized linear models with random effects: Salamander mating revisited. Biometrics 48 631–644.
  • Kong, A., Liu, J. S. and Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89 278–288.
  • Lange, K. and Sobel, E. (1991). A random walk method for computing genetic location scores. Amer. J. Human Genetics 49 1320–1334.
  • Liang, K. Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 13–22.
  • Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data, 2nd ed. Wiley, Hoboken, NJ.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London.
  • McCulloch, C. E. (1997). Maximum likelihood algorithms for generalized linear mixed models. J. Amer. Statist. Assoc. 92 162–170.
  • Moyeed, R. A. and Baddeley, A. J. (1991). Stochastic approximation of the MLE for a spatial point pattern. Scand. J. Statist. 18 39–50.
  • Ott, J. (1979). Maximum likelihood estimation by counting methods under polygenic and mixed models in human pedigrees. Amer. J. Human Genetics 31 161–175.
  • Penttinen, A. (1984). Modelling interaction in spatial point patterns: Parameter estimation by the maximum likelihood method. Jyväskylä Studies in Computer Science, Economics and Statistics No. 7. Univ. Jyväskylä, Finland.
  • Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, Berlin.
  • Sung, Y. J. (2003). Model misspecification in missing data. Ph.D. dissertation, Univ. Minnesota.
  • Thompson, E. A. (2003). Linkage analysis. In Handbook of Statistical Genetics, 2nd ed. (D. J. Balding, M. Bishop and C. Cannings, eds.) 893–918. Wiley, Chichester.
  • Thompson, E. A. and Guo, S. W. (1991). Evaluation of likelihood ratios for complex genetic models. IMA J. Mathematics Applied in Medicine and Biology 8 149–169.
  • Torrie, G. M. and Valleau, J. P. (1977). Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. J. Comput. Phys. 23 187–199.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595–601.
  • Wei, G. C. G. and Tanner, M. A. (1990). A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms. J. Amer. Statist. Assoc. 85 699–704.
  • White, H. A. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.
  • Wijsman, R. A. (1964). Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70 186–188.
  • Wijsman, R. A. (1966). Convergence of sequences of convex sets, cones and functions. II. Trans. Amer. Math. Soc. 123 32–45.
  • Younes, L. (1988). Estimation and annealing for Gibbsian fields. Ann. Inst. H. Poincaré Probab. Statist. 24 269–294.