Abstract
Let\(\mathfrak{P}\) be a general family of probability measures,κ :\(\mathfrak{P} \to \mathbb{R}\) a functional, and\(N_{(0,\sigma ^2 (P))} \) the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of\(\mathfrak{P}\). For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for\(P \in \mathfrak{P}\), as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.
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References
Hewitt, E. and Stromberg, K. (1965).Real and Abstract Analysis, Springer, New York.
Le Cam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates,University of California Publications in Statistics,1, 277–330.
Le Cam, L. (1974). Notes on asymptotic methods in statistical decision theory, Research Memo., Centre de Recherches Mathématiques, Univ. de Montreal.
Pfanzagl, J. (with the assistance of R. Hamböker) (1994).Parametric Statistical Theory, de Gruyter, Berlin.
Pfanzagl, J. (2002). Asymptotic optimality of estimator sequences: “Pointwise” versus “Locally uniform”,Math. Methods Statist.,11, 69–97.
Strasser, H. (1978). Global asymptotic properties of risk functions in estimation.Z. Wahrsch. Verw. Gebiete,45, 35–48.
van der Vaart, A. W. (1997). Superefficiency,Festschrift for Lucien Le Cam, Research Papers in Probability and Statistics (eds. D. Pollard, E. Torgersen and G. L. Yang), 397–410, Springer, New York.
Witting, H. and Müller-Funk, U. (1995).Mathematische Statistik II, Teubner, Stuttgart.
Wolfowitz, J. (1953). The method of maximum likelihood and the Wald theory decision functions,Indag. Math. 15, 114–119.
Wolfowitz, J. (1965). Asymptotic efficiency of the maximum likelihood estimator,Theory Prob. Appl.,X, 247–260.
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Pfanzagl, J. Asymptotic bounds for estimators without limit distribution. Ann Inst Stat Math 55, 95–110 (2003). https://doi.org/10.1007/BF02530487
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DOI: https://doi.org/10.1007/BF02530487