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Extension to Markov processes of a result by A. Wald about the consistency of the maximum likelihood estimate

  • George G. Roussas
Article

Summary

In this note the proof of the consistency of a maximum likelihood estimate (MLE) obtained by Wald in [7] in the case of independent and identically distributed random variables is extended to the case of Markov processes.

There is an extensive literature about the existence of a MLE and its consistency, most of which includes the assumption of the existence of derivatives of the densities with respect to the parameter involved. (See, for example, [2] and other references cited there.) Even under the rather strong assumption of pointwise differentiability of densities, and other additional regularity conditions, the problem of existence and consistency of a MLE has not been solved satisfactorily. (See, for example, [1], [2], [4], [6].) On the other hand, there have appeared papers like [3], where the consistency of a MLE is proved for processes with dependent random variables, and without the usual differentiability assumptions. The conditions used in the present paper are, however, of a different nature from those imposed in [3], and also are slightly different from Wald's assumptions in [7]. To our knowledge, a proof of consistency of a MLE under conditions similar to the ones used here has not appeared in the literature.

Keywords

Stochastic Process Probability Theory Maximum Likelihood Estimate Likelihood Estimate Markov Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Basu, D.: An inconsistency of the method of maximum likelihood. Ann. math. Statistics 26, 144–146 (1955).Google Scholar
  2. [2]
    Billingsley, P.: Statistical Inferences for Markov Processes. The University of Chicago Press (1961).Google Scholar
  3. [3]
    Kraft, C.: Some conditions of consistency and uniform consistency of statistical procedures. University of California Publications in Statistics 2: 6, 125–242 (1955).Google Scholar
  4. [4]
    —, and L. Lecam: A remark on the roofs of the maximum likelihood equation. Ann. math. Statistics 27, 1174–1176 (1956).Google Scholar
  5. [5]
    Loève, M.: Probability Theory, 3rd ed., Princeton, N. J.: Van Nostrand 1963.Google Scholar
  6. [6]
    Neyman, J., and E. Scott: Consistent estimates based on partially consisted observations. Econometrica 16, 1–32 (1948).Google Scholar
  7. [7]
    Wald, A.: Note on the consistency of the maximum likelihood estimate. Ann. math. Statistics 20, 595–601 (1949).Google Scholar

Copyright information

© Springer-Verlag 1965

Authors and Affiliations

  • George G. Roussas
    • 1
  1. 1.Mathematics DepartmentSan Jose State CollegeSan Jose 14

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