On estimation and adaptive estimation for locally asymptotically normal families

  • Václav Fabian
  • James Hannan
Article

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.

Keywords

Stochastic Process Probability Theory Mathematical Biology Additional Assumption Normal Family 

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References

  1. 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)Google Scholar
  2. Beran, R.: An efficient and robust adaptive estimator of location. Ann. Statist. 6, 292–313 (1978)Google Scholar
  3. Beran, R.: Asymptotic lower bounds for risk in robust estimation. Ann. Statist. 8, 1252–1264 (1980)Google Scholar
  4. Eggleston, H.G.: Convexity. Cambridge: (1958)Google Scholar
  5. Fabian, V.: On uniform convergence of measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 15, 139–143 (1970)Google Scholar
  6. Fabian, V.: Asymptotically efficient stochastic approximation; the RM case. Ann. Statist. 1, 485–495 (1973)Google Scholar
  7. 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]Google Scholar
  8. Fabian, V.: Local asymptotic minimax properties of some recursive estimates. RM-409. Department of Statistics and Probability, Michigan State University. [Submitted for publication [1980]Google Scholar
  9. 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 1971Google Scholar
  10. Hájek. J.: Local asymptotic minimax and admissibility in estimation. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. I, 175–194. Univ. Calif. Press (1972)Google Scholar
  11. 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)Google Scholar
  12. Ibragimov, I.A., Has'minskij, R.Z.: Local asymptotic normality for non-identically distributed observations. Theory Probab. Appl. 20, 246–260 (1975)Google Scholar
  13. Ibragimov, I.A., Has'minskij, R.Z.: Asymptotic theory of estimation. Nanka, Moskva (1979). (Russian; English translation announced)Google Scholar
  14. Koshevnik, Yu.A., Levit, B.Ya.: On a non-parametric analogue of the information matrix. Theory Probab. Appl. 21, 738–753 (1976)Google Scholar
  15. 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)Google Scholar
  16. Le Cam, L.: Locally asymptotically normal families of distributions. Univ. Calif. Publ. Statist. 3, 37–98 (1960)Google Scholar
  17. Le Cam, L.: Théorie asymptotique de la décision statistique. Séminaire de mathématiques supérieures-été 1968. Les presses de l'Université de Montréal (1969)Google Scholar
  18. Le Cam, L.: On the assumptions used to prove asymptotic normality of maximum likelihood estimates. Ann. Math. Statist. 41, 802–828 (1970)Google Scholar
  19. Le Cam, L.: Limits of experiments. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. I, 245–261. Univ. Calif. Press (1972)Google Scholar
  20. Le Cam, L.: Notes on Asymptotic Methods in Statistical Decision Theory. Centre de Recherches Mathématiques. Universite de Montréal, Montréal (1974)Google Scholar
  21. Le Cam, L.: On a theorem of J. Hájek. Contribution to Statistics, Jaroslav Hájek Memorial Volume, Jana Jurečková, ed. Dordrecht, Holland: Reidel Publ. 1979Google Scholar
  22. Levit, B.: On optimality of some statistical estimates. Proc. Prague Symp. Asympt. Statist. II, 215–238. Universita Karlova. Praha (1974)Google Scholar
  23. Levit, B.Ya.: On the efficiency of a class of non-parametric estimates. Theory Probab. Appl. 20, 723–740 (1975)Google Scholar
  24. Pfanzagl, J.: Investigating the quantile of an unknown distribution. Contrib. to Appl. Statist. (Ziegler, W.J., ed.). Basel: Birkhäuser 1976Google Scholar
  25. Rao, R.R.: Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33, 659–680 (1962)Google Scholar
  26. Stein, C.: Efficient nonparametric testing and estimation. Proc. Third Berkeley Sympos. Math. Statist. and Probab. I, 187–195. Univ. Calif. Press (1956)Google Scholar
  27. Stone, C.J.: Adaptive maximum likelihood estimators of a location parameter. Ann. Statist. 3, 267–284 (1975)Google Scholar
  28. Weiss, L., Wolfowitz, J.: Asymptotically efficient non-parametric estimators of location and scale parameters. Z. Wahrscheinlichkeitstheorie verw. Gebiete 16, 134–150 (1970a)Google Scholar
  29. 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)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Václav Fabian
    • 1
    • 2
  • James Hannan
    • 2
  1. 1.University of BernSwitzerland
  2. 2.Dept. of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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