The Annals of Statistics

A new class of generalized Bayes minimax ridge regression estimators

Yuzo Maruyama and William E. Strawderman

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Abstract

Let y=Aβ+ɛ, where y is an N×1 vector of observations, β is a p×1 vector of unknown regression coefficients, A is an N×p design matrix and ɛ is a spherically symmetric error term with unknown scale parameter σ. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623–627] and Casella [Ann. Statist. 8 (1980) 1036–1056, J. Amer. Statist. Assoc. 80 (1985) 753–758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which have greater numerical stability (i.e., smaller condition number) than the usual least squares estimator. In particular, we give a subclass of such estimators which, surprisingly, has a very simple form. We also show that under certain conditions the generalized Bayes minimax estimators in the normal case are also generalized Bayes and minimax in the general case of spherically symmetric errors.

Article information

Source
Ann. Statist., Volume 33, Number 4 (2005), 1753-1770.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1123250228

Digital Object Identifier
doi:10.1214/009053605000000327

Mathematical Reviews number (MathSciNet)
MR2166561

Zentralblatt MATH identifier
1078.62006

Subjects
Primary: 62C20: Minimax procedures 62C15: Admissibility 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62A15

Keywords
Ridge regression minimaxity generalized Bayes condition number

Citation

Maruyama, Yuzo; Strawderman, William E. A new class of generalized Bayes minimax ridge regression estimators. Ann. Statist. 33 (2005), no. 4, 1753--1770. doi:10.1214/009053605000000327. https://projecteuclid.org/euclid.aos/1123250228


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References

  • Baranchik, A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution. Ann. Math. Statist. \bf41 642–645.
  • Belsley, D. A., Kuh, E. and Welsch, R. E. (1980). Regression Diagnostics. Wiley, New York.
  • Berger, J. (1976). Admissible minimax estimation of a multivariate normal mean with arbitrary quadratic loss. Ann. Statist. \bf4 223–226.
  • Berger, J. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean. Ann. Statist. \bf8 716–761.
  • Berger, J. and Srinivasan, C. (1978). Generalized Bayes estimators in multivariate problems. Ann. Statist. \bf6 783–801.
  • Bock, M. E. (1975). Minimax estimators of the mean of a multivariate normal distribution. Ann. Statist. \bf3 209–218.
  • Casella, G. (1980). Minimax ridge regression estimation. Ann. Statist. \bf8 1036–1056.
  • Casella, G. (1985). Condition numbers and minimax ridge regression estimators. J. Amer. Statist. Assoc. \bf80 753–758.
  • Efron, B. and Morris, C. (1976). Families of minimax estimators of the mean of a multivariate normal distribution. Ann. Statist. \bf4 11–21.
  • Faith, R. E. (1978). Minimax Bayes estimators of a multivariate normal mean. J. Multivariate Anal. \bf8 372–379.
  • Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics \bf12 55–67.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361–379. Univ. California Press, Berkeley.
  • Kubokawa, T. and Srivastava, M. S. (2001). Robust improvement in estimation of a mean matrix in an elliptically contoured distribution. J. Multivariate Anal. \bf76 138–152.
  • Lin, P. and Tsai, H. (1973). Generalized Bayes minimax estimators of the multivariate normal mean with unknown covariance matrix. Ann. Statist. \bf1 142–145.
  • Maruyama, Y. (2003). A robust generalized Bayes estimator improving on the James–Stein estimator for spherically symmetric distributions. Statist. Decisions \bf21 69–77.
  • Robert, C. (1994). The Bayesian Choice. Springer, New York.
  • Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197–206. Univ. California Press, Berkeley.
  • Strawderman, W. E. (1971). Proper Bayes minimax estimators of the multivariate normal mean. Ann. Math. Statist. \bf42 385–388.
  • Strawderman, W. E. (1978). Minimax adaptive generalized ridge regression estimators. J. Amer. Statist. Assoc. \bf73 623–627.