Abstract
In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.
Similar content being viewed by others
References
Akahira, M. (1975). Asymptotic theory for estimation of location in non-regular cases, I: Order of convergence of consistent estimators. Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 22, 8–26.
Akahira, M. (1986). The Structure of Asymptotic Deficiency of Estimators, Queen's Papers in Pure and Appl. Math., No. 75, Queen's University Press, Kingston, Ontario, Canada.
Akahira, M. (1988a). Second order asymptotic properties of the generalized Bayes estimators for a family of non-regular distributions, Statistical Theory and Data Analysis II (ed. K. Matusita), 87–100, North-Holland, Amsterdam.
Akahira, M. (1988b). Second order asymptotic bounds for the concentration probability of estimators in a family of truncated distributions, Proceedings of Symposium, Research Institute of Mathematical Sciences, 645, 37–51, Kyoto Univ. (in Japanese).
Akahira, M. and K. Takeuchi (1979). Remarks on asymptotic efficiency and inefficiency of maximum probability estimators. Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 26, 132–138.
Akahira, M. and Takeuchi, K. (1981). Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency, Lecture Notes in Statist., 7, Springer, New York.
Ghosh, J. K., Sinha, B. K. and Wieand, H. S. (1980). Second order efficiency of the mle with respect to any bounded bowl-shaped loss function, Ann. Statist., 8, 506–521.
Pfanzagl, J. and Wefelmeyer, W. (1978). A third order optimum property of the maximum likelihood estimator, J. Multivariate Anal., 8, 1–29.
Weiss, L. and Wolfowitz, J. (1967). Maximum probability estimators, Ann. Inst. Statist. Math., 19, 193–206.
Weiss, L. and Wolfowitz, J. (1974). Maximum Probability Estimators and Related Topics, Lecture Notes in Math., 424, Springer, Berlin.
Author information
Authors and Affiliations
Additional information
Research supported by University of Tsukuba Project Research.
About this article
Cite this article
Akahira, M. The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases. Ann Inst Stat Math 43, 181–195 (1991). https://doi.org/10.1007/BF00116477
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00116477