Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T17:56:46.590Z Has data issue: false hasContentIssue false
  • English
  • Français

MAXIMAL UNIFORM CONVERGENCE RATES IN PARAMETRIC ESTIMATION PROBLEMS

Published online by Cambridge University Press:  18 August 2009

Walter Beckert  and
Daniel L. McFadden
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers parametric estimation problems with independent, identically nonregularly distributed data. It focuses on rate efficiency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion, largely unexplored in parametric estimation. Under mild conditions, the Hellinger metric, defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates. These rates are shown to be attainable in general classes of parametric estimation problems.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 2 , April 2010 , pp. 469 - 500
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are indebted to Masafumi Akahira, Richard Blundell, Andrew Chesher, David Donoho, Hide Ichimura, Oliver Linton, and two anonymous referees for helpful comments and discussions.

References

REFERENCES

Akahira, M. (1991) The amount of information and the bound for the order of consistency for a location parameter family of densitites. In Mammitzsch, V. and Schneeweiss, H. (eds.) Symposia Gaussiana, Conf. B. Gryuter & Co.Google Scholar
Akahira, M. & Takeuchi, K. (1991) A definition of information amount applicable to non-regular cases. Journal of Computing and Information 2, 71–92.Google Scholar
Akahira, M. & Takeuchi, K. (1995) Non-Regular Statistical Estimation. Spinger Verlag.10.1007/978-1-4612-2554-6CrossRefGoogle Scholar
Birgé, L. & Massart, P. (1993) Rates of convergence for minimum contrast estimators. Probability Theory and Related Fields 97, 113–150.CrossRefGoogle Scholar
Chan, K.S. (1993) Consistency and limiting distribution of the least squares estimator of a threshold autoregression model. Annals of Statistics 21, 520–533.CrossRefGoogle Scholar
Chan, K.S. & Tsay, R.S. (1998) Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrika 85, 413–426.CrossRefGoogle Scholar
Cosslett, S.R. (1987) Efficiency bounds for distribution-free estimators of the binary choice and censored regression models. Econometrica 55, 559–585.CrossRefGoogle Scholar
David, H.A. (1997) Order Statistics, 2nd ed. Wiley.Google Scholar
Donoho, D.L. & Liu, R.C. (1991a) Geometrizing rates of convergence, II Annals of Statistics, 19, 633–667.Google Scholar
Donoho, D.L. & Liu, R.C. (1991b) Geometrizing rates of convergence, III. Annals of Statistics 19, 668–701.Google Scholar
Hajek, J. (1970) A characterization of the limiting distributions of regular estimates. Z. Wahrsch. Verw. Gebiete 14, 323–330.CrossRefGoogle Scholar
Hall, P. (1989) On convergence rates in nonparametric problems. International Statistical Review/Revue Internationale de Statistique 57, 45–58.Google Scholar
Hansen, B. (2000) Sample splitting and threshold estimation. Econometrica 68, 575–603.CrossRefGoogle Scholar
Hirano, K. & Porter, J.R. (2003) Efficiency in Asymptotic Shift Experiments. Mimeo, University of Miami and Harvard University.Google Scholar
Hoeffding, W. & Wolfowitz, J. (1958) Distinguishability of sets of distributions. Annals of Mathematical Statistics 3, 700–718.CrossRefGoogle Scholar
Horowitz, J.L. (1993) Optimal rates of convergence of parameter estimators in the binary response model with weak distributional assumptions. Econometric Theory 9, 1–18.10.1017/S0266466600007301CrossRefGoogle Scholar
Ibragimov, I.A. & Has’minskii, R.Z. (1981) Statistical Estimation. Springer.CrossRefGoogle Scholar
Klein, R.W. & Spady, R.H. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61, 387–421.CrossRefGoogle Scholar
LeCam, L.M. (1972) Limits of experiments. In Proceedings of the Sixth Berkeley Symposium of Mathematical Statistics 1, 245–261.Google Scholar
LeCam, L.M. (1986) Asymptotic Methods in Statistical Decision Theory. Springer.Google Scholar
LeCam, L.M. & Yang, G.L. (2000) Asymptotics in Statistics—Some Basic Concepts. Springer.Google Scholar
Matusita, K. (1955) Decision rules based on the distance for problems of fit, two samples and estimation. Annals of Mathematical Statistics 26, 631–640.CrossRefGoogle Scholar
Paarsch, H.J. (1992) A Comparison of Estimators for Empirical Models of Auctions. Working paper No. 9210, University of Western Ontario.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer.10.1007/978-1-4612-5254-2CrossRefGoogle Scholar
Pollard, D. (1989) Asymptotics via empirical processes. Statistical Science 4, 341–354.Google Scholar
Prakasa Rao, B.L.S. (1968) Estimation of the location of the cusp of a continuous density. The Annals of Mathematical Statistics 39, 76–87.Google Scholar
Seo, M. & Linton, O. (2005) A Smoothed Least Squares Estimator for the Threshold Regression Model. Mimeo, London School of Economics.Google Scholar
Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 1348–1360.CrossRefGoogle Scholar
Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10, 1040–1053.CrossRefGoogle Scholar
Van de Geer, S. (1993) Hellinger-consistency of certain nonparametric maximum likelihood estimators. Annals of Statistics 21, 14–44.CrossRefGoogle Scholar
Van de Geer, S. (2000) Empirical Processes in M-Estimation. Cambridge University Press.Google Scholar