Advertisement

Computational Statistics

, Volume 17, Issue 4, pp 517–524 | Cite as

On Single Versus Multiple Imputation for a Class of Stochastic Algorithms Estimating Maximum Likelihood

  • Edward H. Ip
Article
First Online:

Summary

We discuss a special class of stochastic versions of the EM algorithms. The advantage of the single imputation procedure in non-exponential family applications is highlighted. We prove that ergodic properties of the stochastic algorithms are dependent not on the multiplicity of the imputation scheme but rather on the stability of the deterministic component of an underlying stochastic difference equation.

Keywords

EM algorithm non-exponential family Markov chain stochastic difference equation 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgement

This paper is a modified version of a chapter from the author’s doctoral dissertation. The author thanks both Jean Diebolt and Ingram Olkin for their guidance, and for suggestions to this paper.

References

  1. Celeux, G. and Diebolt, J. (1985), The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem, Comp. Statist. Quart., 2, 73–82.Google Scholar
  2. Chauveau, D. and Diebolt, J. (1999), An automatic stopping rule for MCMC convergence assessment, Comp. Statist., 14, 419–442.CrossRefGoogle Scholar
  3. Chan, K. (1990), Deterministic stability, stochastic stability and ergodicity, in: Tong, H., Non-linear Time Series (Oxford University Press, New York), pp. 448.Google Scholar
  4. Delyon, B., Lavielle, M. and Moulines, E. (1999), Convergence of a stochastic approximation version of the EM algorithm, Ann. Statist., 27, 94–128.MathSciNetCrossRefGoogle Scholar
  5. Dempster, A., Laird, N., and Rubin, D. (1977), Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. R. Statist. Soc., B39, 1–38.zbMATHGoogle Scholar
  6. Diebolt, J., and Ip, E.H. (1996), Stochastic EM: method and application, in: Gilks, W., Richardson, S., and Spiegelhalter, D. J., eds. Markov Chain Monte Carlo in Practice (Chapman and Hall, London) pp. 259–273.Google Scholar
  7. Kifer, Y. (1988), Random Perturbations of Dynamical Systems(Birkhauser, Boston).CrossRefGoogle Scholar
  8. Lehmann, E. (1983). Theory of Point Estimation (Wadsworth and Brooks/(California).CrossRefGoogle Scholar
  9. Meyn, S. P., and Tweedie, R. L. (1993), Markov Chain and Stochastic Stability (Springer-Verlag, New York).CrossRefGoogle Scholar
  10. Ruud, P.A. (1991), Extensions of estimation methods using the EM algorithm, J. Economet., 49, 305–341.MathSciNetCrossRefGoogle Scholar
  11. Tweedie, R. (1975), Sufficient conditions for ergodicity and recurrence of Markov chain on a general state space, Stoch. Proc. Appl., 3, 385–403.MathSciNetCrossRefGoogle Scholar
  12. Tweedie, R. (1983), The existence of moments for stationary Markov chains, J. Appl. Prob., 20, 191–196.MathSciNetCrossRefGoogle Scholar
  13. Wang, N. and Robins, J.M. (1998), Large-sample theory for parametric multiple imputation procedures, Biometrika, 85, 699–704.MathSciNetzbMATHGoogle Scholar
  14. 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. Am. Statist. Assoc. 85, 699–704.CrossRefGoogle Scholar

Copyright information

© Physica-Verlag 2002

Authors and Affiliations

  • Edward H. Ip
    • 1
  1. 1.Marshall School of BusinessUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations

Cite article

Buy options