The Annals of Statistics

Efficient maximum likelihood estimation in semiparametric mixture models

Aad Van der Vaart

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We consider maximum likelihood estimation in several examples of semiparametric mixture models, including the exponential frailty model and the errors-in-variables model. The observations consist of a sample of size n from the mixture density $\int p_{\theta}(x|z) d \eta(z)$. The mixing distribution is completely unknown. We show that the first component $\hat{\theta}_n$ of the joint maximum likelihood estimator , $(\hat{\theta}_n \hat{\eta}_n)$ is asymptotically normal and asymptotically efficient in the semiparametric sense.

Article information

Ann. Statist., Volume 24, Number 2 (1996), 862-878.

First available in Project Euclid: 24 September 2002

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Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties 62F12: Asymptotic properties of estimators

Maximum likelihood semiparametric model mixture model Donsker class asymptotic efficiency efficient score equation


Van der Vaart, Aad. Efficient maximum likelihood estimation in semiparametric mixture models. Ann. Statist. 24 (1996), no. 2, 862--878. doi:10.1214/aos/1032894470.

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  • ANDERSON, T. W. 1984. Estimating linear statistical relationships. Ann. Statist. 12 1 45. Z.
  • BICKEL, P. J. 1982. On adaptive estimation. Ann Statist. 10 647 671.
  • BICKEL, P. J. and RITOV, Y. 1987. Efficient estimation in the errors in variables model. Ann. Statist. 15 513 540. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semi-Parametric Models. Johns Hopkins Univ. Press. Z.
  • DUDLEY, R. M. 1985. An extended Wichura Theorem, definitions of Donsker classes, and weighted empirical processes. Probability in Banach Spaces V. Lecture Notes in Math. 1153 1306 1326. Springer, New York. Z.
  • GINE, E. and ZINN, J. 1986. Empirical processes indexed by Lipschitz functions. Ann. Probab. ´ 14 1329 1338. Z.
  • GROENEBOOM, P. 1991. Nonparametric maximum likelihood estimators for interval censoring and deconvolution. Technical Report 91-53, Technische Univ. Delft. Z.
  • HECKMAN, J. and SINGER, B. 1984. A method for minimizing the impact of distributional assumptions in economic studies for duration data. Econometrica 52 271 320. Z.
  • KIEFER, J. and WOLFOWITZ, J. 1956. Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist. 27 887 906. Z.
  • KOLMOGOROV, A. N. and TIKHOMIROV, V. M. 1961. Epsilon-entropy and epsilon-capacity of sets in functions spaces. Amer. Math. Soc. Transl. Ser. 2 17 277 364. Z.
  • KUMON, M. and AMARI, S. 1984. Estimation of a structural parameter in the presence of a large number of nuisance parameters. Biometrika 71 445 459. Z.
  • LINDSAY, B. G. 1983a. Efficiency of the conditional score in a mixture setting. Ann. Statist. 11 486 497. Z.
  • LINDSAY, B. G. 1983b. The geometry of mixture likelihoods, I and II. Ann. Statist. 11 86 94 and 783 792. Z.
  • LINDSAY, B. G. 1985. Using empirical Bay es inference for increased efficiency. Ann. Statist. 13 914 931. Z.
  • MURPHY, S. A. 1995. Asy mptotic theory for the frailty model. Ann. Statist. 23 182 198. Z.
  • OSSIANDER, M. 1987. A central limit theorem under metric entropy with L -bracketing. Ann. 2 Probab. 15 897 919. Z.
  • PFANZAGL, J. 1988. Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures. J. Statist. Plann. Inference 19 137 158. Z.
  • PFANZAGL, J. 1990. Estimation in Semiparametric Models. Springer, New York. Z.
  • SEVERINI, T. A. and WONG, W. H. 1992. Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768 1862. Z.
  • STONE, C. J. 1975. Adaptive maximum likelihood estimation of a location parameter. Ann. Statist. 3 267 284. Z.
  • VAN DER VAART, A. W. 1988a. Statistical Estimation in Large Parameter Spaces. CWI Tract 44. Math. Centrum, Amsterdam. Z.
  • VAN DER VAART, A. W. 1988b. Estimating a parameter in incidental and structural models by approximate maximum likelihood. Technical Report 139, Dept. Statistics, Univ. Washington. Z.
  • VAN DER VAART, A. W. 1988c. Estimating a real parameter in a class of semiparametric models. Ann. Statist. 16 1450 1474. Z.
  • VAN DER VAART, A. W. 1993. New Donsker classes. Ann. Probab. To appear. Z.
  • VAN DER VAART, A. W. 1994. Bracketing smooth functions. Stochastic Process. Appl. 52 93 105. Z.
  • WALD, A. 1949. Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595 601.