Summary
Strasser (1981) introduced approximately maximum likelihood estimators (AMLE's) and found a condition equivalent to strong consistency of all AMLE's. Here a condition weaker than that of Strasser is proved to be equivalent to the usual consistency of all AMLE's. Under an additional regularity this condition is shown to be doubly equivalent, which means that it is equivalent to consistency, and its contrary is equivalent to inconsistency of all AMLE's. The doubly equivalent conditions are important—we present an example where MLE is strongly consistent but some AMLE's are inconsistent. It is proved that the additional regularity can be reduced to the finiteness of an observations entropy. All results are motivate and illustrated by examples.
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References
Bahadur, R. R. (1960), On the asymptotic efficiency of tests and estimators,Sankya, 22, 229–252.
Cover, T. M. andThomas, J. B. (1991).Elements of Information Theory, Wiley, New York.
Doob, J. L. (1953),Stochastic Processes, Wiley, New York.
Ferguson, T. S. (1982). An inconsistent maximum likelihood estimate,J. Amer. Statist. Assoc. 77, 831–834.
Gallager, R. G. (1968),Information Theory and Reliable Communication, Wiley, New York.
Halmos, P. R. (1964).Measure Theory, Van Nostrand, Princeton.
Hanan, J. (1960), Consistency of maximum likelihood estimation of discrete distributions,Contr. to Probab. and Statist., Essays in Honor of H. Hotelling, Stanford Univ. Press, 249–257.
Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions,Proc. 5th Berkeley Symp. on Math. Statist. and Probab., Vol. 1, 221–233.
Ibragimov, I. A., Hasminskii, R. Z. (1981).Statistical Estimation, Asymptotic Theory, Springer, New York.
Kiefer, J., Wolfowitz, J (1956). Consistency of the maximum likelihood estimation in the presence of infinitely many incidental parameters.Ann. Math. Statist. 27, 887–906.
Le Cam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes estimates,Univ. Calif. Publ. in Statist., 1, 277–330.
Liese, F. andVajda, I. (1987),Convex Statistical Distances, Teubner, Leipzig.
Feranzagl, J. (1969). On the measuraibility and consistency of minimum contrast estimators,Metrika, 14, 249–272.
Perlman, M. D. (1972), On the strong consistency of aproximate maximum likelihood estimators,Proc. VIth Berkeley Symp. Prob. Math. Statist., 263–281.
Reeds, J. (1985). Asymptotic number of roots of Cauchy location likelihood equations,Ann. Statist., 13, 775–784.
Strasser, H. (1981), Consistency of maximum likelihood and Bayes estimates,Ann. Statist., 9, 1107–1113.
Strasser, A. (1985). Mathematical Theory of Statistics, De Guyter, Berlin.
Vajda, I. (1970), Note on discrimination information and variation,Trans. IEEE on Inform. Theory, IT-16, 771–773.
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate,Ann. Math. Statist., 20, 595–601.
Wolfowitz, J. (1949). On Wald's proof of the consistency of the maximum likelhood estimate,Ann. Math. Statist., 20, 601–602.
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Vajda, I. Conditions equivalent and doubly equivalent to consistency of approximate MLE's. J. It. Statist. Soc. 2, 107–125 (1993). https://doi.org/10.1007/BF02589078
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DOI: https://doi.org/10.1007/BF02589078