Summary
Locally asymptotically minimax (LAM) estimates are constructed for locally asymptotically normal (LAN) families under very mild additional assumptions. Adaptive estimation is also considered and a sufficient condition is given for an estimate to be locally asymptotically minimax adaptive. Incidently, it is shown that a well known lower bound due to Hájek (1972) for the local asymptotic minimax risk is not sharp.
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Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, 170–176 (1955)
Beran, R.: An efficient and robust adaptive estimator of location. Ann. Statist. 6, 292–313 (1978)
Beran, R.: Asymptotic lower bounds for risk in robust estimation. Ann. Statist. 8, 1252–1264 (1980)
Eggleston, H.G.: Convexity. Cambridge: (1958)
Fabian, V.: On uniform convergence of measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 15, 139–143 (1970)
Fabian, V.: Asymptotically efficient stochastic approximation; the RM case. Ann. Statist. 1, 485–495 (1973)
Fabian, V., Hannan, J.: Sufficient conditions for local asymptotic normality. RM-403. Department of Statistics and Probability, Michigan State University. [Submitted to Z. Wahrscheinlichkeitstheorie verw. Gebiete. 1980]
Fabian, V.: Local asymptotic minimax properties of some recursive estimates. RM-409. Department of Statistics and Probability, Michigan State University. [Submitted for publication [1980]
Hájek, J.: Limiting properties of likelihoods and inference. Foundations of Statistical Inference (Godambe, V.P. and D.A. Sprott eds.) 142–162. New York: Holt, Rinehart and Winston 1971
Hájek. J.: Local asymptotic minimax and admissibility in estimation. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. I, 175–194. Univ. Calif. Press (1972)
Has'minskij, R.Z., Ibragimov, I.A.: On the nonparametric estimation of functionals. Proc. Second Prague Symp. Asympt. Statist. 41–51. Charles University, Prague (1979)
Ibragimov, I.A., Has'minskij, R.Z.: Local asymptotic normality for non-identically distributed observations. Theory Probab. Appl. 20, 246–260 (1975)
Ibragimov, I.A., Has'minskij, R.Z.: Asymptotic theory of estimation. Nanka, Moskva (1979). (Russian; English translation announced)
Koshevnik, Yu.A., Levit, B.Ya.: On a non-parametric analogue of the information matrix. Theory Probab. Appl. 21, 738–753 (1976)
Le Cam, L.: On the asymptotic theory of estimation and testing hypotheses. Proc. Third Berkeley Sympos. Math. Statist. Probab. I, 129–156. Univ. Calif. Press (1956)
Le Cam, L.: Locally asymptotically normal families of distributions. Univ. Calif. Publ. Statist. 3, 37–98 (1960)
Le Cam, L.: Théorie asymptotique de la décision statistique. Séminaire de mathématiques supérieures-été 1968. Les presses de l'Université de Montréal (1969)
Le Cam, L.: On the assumptions used to prove asymptotic normality of maximum likelihood estimates. Ann. Math. Statist. 41, 802–828 (1970)
Le Cam, L.: Limits of experiments. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. I, 245–261. Univ. Calif. Press (1972)
Le Cam, L.: Notes on Asymptotic Methods in Statistical Decision Theory. Centre de Recherches Mathématiques. Universite de Montréal, Montréal (1974)
Le Cam, L.: On a theorem of J. Hájek. Contribution to Statistics, Jaroslav Hájek Memorial Volume, Jana Jurečková, ed. Dordrecht, Holland: Reidel Publ. 1979
Levit, B.: On optimality of some statistical estimates. Proc. Prague Symp. Asympt. Statist. II, 215–238. Universita Karlova. Praha (1974)
Levit, B.Ya.: On the efficiency of a class of non-parametric estimates. Theory Probab. Appl. 20, 723–740 (1975)
Pfanzagl, J.: Investigating the quantile of an unknown distribution. Contrib. to Appl. Statist. (Ziegler, W.J., ed.). Basel: Birkhäuser 1976
Rao, R.R.: Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33, 659–680 (1962)
Stein, C.: Efficient nonparametric testing and estimation. Proc. Third Berkeley Sympos. Math. Statist. and Probab. I, 187–195. Univ. Calif. Press (1956)
Stone, C.J.: Adaptive maximum likelihood estimators of a location parameter. Ann. Statist. 3, 267–284 (1975)
Weiss, L., Wolfowitz, J.: Asymptotically efficient non-parametric estimators of location and scale parameters. Z. Wahrscheinlichkeitstheorie verw. Gebiete 16, 134–150 (1970a)
Weiss, L., Wolfowitz, J.: Asymptotically efficient estimation of non-parametric regression coefficients. Statistical Decision Theory (S.S. Gupta and J. Yaeckel, eds.) 29–39 (1970b)
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Research partially supported by NSF grants no. MCS 78-02846 and MCS 77-03493-01
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Fabian, V., Hannan, J. On estimation and adaptive estimation for locally asymptotically normal families. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 459–478 (1982). https://doi.org/10.1007/BF00532803
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DOI: https://doi.org/10.1007/BF00532803