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Consistency of D-estimators. (English) Zbl 0599.62038
In ibid. 20, 189-208 (1984; Zbl 0558.62026) the author introduced three classes of D-estimators. In the present paper Fisher’s and strong consistency of these estimators are established. It is shown in particular that the standard D-estimators estimate the parameters from compact spaces strongly consistently for all discrete sample-generating families with the probabilities continuously depending on the parameter. Strong consistency of weak and directed D-estimators of location and scale is established for a wide variety of sample-generating models including irregular ones. Analogous results concerning abstract parametric spaces are presented too.

62F12 Asymptotic properties of parametric estimators
62F10 Point estimation
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[1] D. Basu: An inconsistency of the method of maximum likelihood. Ann. Math. Statist. 26 (1955), 144-145. · Zbl 0064.14006
[2] D. D. Boos: Minimum distance estimators for location and goodness of fit. J. Amer. Statist. Assoc. 76 (1981), 663-670. · Zbl 0475.62030
[3] H. E. Daniels: The asymptotic efficiency of a maximum likelihood estimator. Proc. 4th Berkeley Symp. Math. Statist. Prob. Vol. 1, 151-164. Berkeley Univ. Press, Berkeley 1961. · Zbl 0166.14802
[4] J. Hannan: Consistency of maximum likelihood estimation of discrete distributions. Contributions to Probability and Statistics (Essays in Honor of Harold Hotelling). Stanford Univ. Press, Stanford 1960, 249-257.
[5] F. R. Hampel: The change of variance curve and optimal redescending M-estimators. J. Amer. Statist. Assoc. 76 (1981), 643 - 648. · Zbl 0476.62040
[6] I. A. Ibragimov, R. Z. Chasminskij: Asymptotic Theory of Estimation. (in Russian). Nauka, Moskva 1979.
[7] J. Jurečková: Asymptotic representation of M-estimators of location. Math. Operatiorsforsch. Statist. Ser. Statist. 11 (1980), 61 - 73.
[8] J. Jurečková: Asymptotic behavior of M-estimators in nonregular cases. Statistics and Decisions 1 (1983), 323-340. · Zbl 0548.62025
[9] L. Le Cam: On the asymptotic theory of estimation and testing. Proc. 3rd Berkeley Symp. Math. Statist. Prob. Vol. 1, 129-156. Berkeley Univ. Press, Berkeley 1965.
[10] R. Š. Lipcer, A. N. Širjajev: Statistics of Random Process. (in Russian). Nauka, Moskva 1974.
[11] I. Vajda: Motivation, existence and equivariance of D-estimators. Kybernetika 20 (1984), 3, 189-208. · Zbl 0558.62026
[12] S. Zacks: The Theory of Statistical Inference. J. Wiley, New York 1971.
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