Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T05:15:45.287Z Has data issue: false hasContentIssue false
  • English
  • Français

SHRINKAGE EFFICIENCY BOUNDS

Published online by Cambridge University Press:  02 October 2014

Bruce E. Hansen
Bruce E. Hansen*
Affiliation:
University of Wisconsin
*
*Address correspondence to Bruce Hansen, Department of Economics, 1180 Observatory Drive, University of Wisconsin, Madison, WI 53706, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is an extension of Magnus (2002, Econometrics Journal 5, 225–236) to multiple dimensions. We consider estimation of a multivariate normal mean under sum of squared error loss. We construct the efficiency bound (the lowest achievable risk) for minimax shrinkage estimation in the class of minimax orthogonally invariate estimators satisfying the sufficient conditions of Efron and Morris (1976, Annals of Statistics 4, 11–21). This allows us to compare the regret of existing orthogonally invariate shrinkage estimators. We also construct a new shrinkage estimator which achieves substantially lower maximum regret than existing estimators.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 860 - 879
Copyright
Copyright © Cambridge University Press 2014 

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

Research supported by the National Science Foundation. I thank a referee for a thorough reading of the paper and helpful comments.

References

REFERENCES

Alam, K. (1973) A family of admissible minimax estimators of the mean of a multivariate normal distribution. Annals of Statistics 1, 517–525.CrossRefGoogle Scholar
Baranchick, A.J. (1964) Multiple Regression and Estimation of the Mean of a Multivariate Normal Distribution. Technical report no. 51, Department of Statistics, Stanford University.Google Scholar
Baranchick, A.J. (1970) A family of minimax estimators of the mean of a multivariate normal distribution. Annals of Mathematical Statistics 41, 642–645.Google Scholar
Belloni, A., Chen, D., Chernozhukov, V., & Hansen, C. (2012) Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica 80, 2369–2429.Google Scholar
Belloni, A. & Chernozhukov, V. (2011) L1-penalized quantile regression in high-dimensional sparse models. Annals of Statistics 39, 82–130.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., & Fernandez-Val, I. (2013) Program Evaluation with High-dimensional Data. Working paper.Google Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2014) Inference on treatment effects after selection amongst high-dimensional controls. Review of Economic Studies 81, 608–650.CrossRefGoogle Scholar
Berger, J.O. (1976a) Admissible minimax estimation of a multivariate normal mean with arbitrary quadratic loss. Annals of Statistics 4, 223–226.Google Scholar
Berger, J.O. (1976b) Tail minimaxality in location vector problems and its applications. Annals of Statistics 4, 33–50.CrossRefGoogle Scholar
Bock, M.E. (1975) Minimax estimators of the mean of a multivariate normal distribution. Annals of Statistics 3, 209–218.CrossRefGoogle Scholar
Caner, M. (2009) Lasso-type GMM estimator. Econometric Theory 25, 1–23.CrossRefGoogle Scholar
DasGupta, A. & Strawderman, W.E. (1997) All estimates with a given risk, Riccati differential equations and a new proof of a theorem of Brown. Annals of Statistics 25, 1208–1221.Google Scholar
de Boor, C. (2000) A Practical Guide to Splines, rev. ed. Springer.Google Scholar
Efron, B.E. & Morris, C. (1976) Families of minimax estimators of the mean of a multivariate normal distribution. Annals of Statistics 4, 11–21.CrossRefGoogle Scholar
Faith, R.E. (1978) Minimax Bayes estimators of a multivariate normal mean. Journal of Multivariate Analysis 8, 372–379.CrossRefGoogle Scholar
Fourdrinier, D., Strawderman, W.E., & Wells, M.T. (1998) On the construction of Bayes minimax estimators. Annals of Statistics 26, 660–671.Google Scholar
Guo, Y.Y. & Pal, N. (1992) A sequence of improvements over the James-Stein estimator. Journal of Multivariate Analysis 42, 302–317.CrossRefGoogle Scholar
Hansen, B.E. (2014) Efficient Shrinkage in Parametric Models. Working paper.Google Scholar
James, W. & Stein, C.M. (1961) Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1, 361–380.Google Scholar
Kubokawa, T. (1991) An approach to improving the James-Stein estimator. Journal of Multivariate Analysis 36, 121–126.CrossRefGoogle Scholar
Kuriki, S. & Takemura, A. (2000) Shrinkage estimation towards a closed convex set with a smooth boundary. Journal of Multivariate Analysis 75, 79–111.Google Scholar
Lehmann, E.L. & Casella, G. (1998) Theory of Point Estimation, 2nd ed. Springer.Google Scholar
Lehmann, E.L. & Romano, J.P. (2005) Testing Statistical Hypotheses, 3rd ed. Springer.Google Scholar
Li, T.F. & Kuo, K.H. (1982) Generalized James-Stein estimators. Communications in Statistics - Theory and Methods 11, 2249–2257.Google Scholar
Liao, Z. (2013) Adaptive GMM shrinkage estimation with consistent moment selection. Econometric Theory 25, 1–23.Google Scholar
Liao, Z. & Phillips, P.C.B. (2014) Automated estimation of vector error correction models. Econometric Theory, forthcoming.Google Scholar
Magnus, J.R. (2002) Estimation of the mean of a univariate normal distribution with known variance. Econometrics Journal 5, 225–236.CrossRefGoogle Scholar
Magnus, W., Oberhettinger, F., & Soni, R.P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer.CrossRefGoogle Scholar
Maruyama, Y. (1998) A unified and broadened class of admissible minimax estimators of a multivariate mean. Journal of Multivariate Analysis 64, 196–205.CrossRefGoogle Scholar
Maruyama, Y. (2004) Stein’s idea and minimax admissible estimation of a multivariate normal mean. Journal of Multivariate Analysis 88, 320–334.CrossRefGoogle Scholar
Maruyama, Y. (2007) Some notes on improving upon the James-Stein estimator. Journal of Statistical Studies 26, 77–84.Google Scholar
Phillips, P.C.B. (1984) The exact distribution of the Stein-rule estimator. Journal of Econometrics 25, 123–131.CrossRefGoogle Scholar
Shao, P.Y.-S. & Strawderman, W.E. (1994) Improving on the James-Stein positive-part estimator. Annals of Statistics 22, 1517–1538.CrossRefGoogle Scholar
Stein, C.M. (1973) Estimation of the mean of a multivariate normal distribution. Proceedings of the Prague Symposium on Asymptotic Statistics, 345–381.Google Scholar
Stein, C.M. (1981) Estimation of the mean of a multivariate normal distribution. Annals of Statistics 9, 1135–1151.CrossRefGoogle Scholar
Strawderman, W.E. (1971) Proper Bayes minimax estimators of the multivariate normal distribution. Annals of Mathematical Statistics 42, 385–388.CrossRefGoogle Scholar
Tibshirani, R. (1996) Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society, Series B 58, 267–288.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar