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

  • George G. Roussas


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.


Stochastic Process Probability Theory Maximum Likelihood Estimate Likelihood Estimate Markov Process 
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    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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