Bayes minimax ridge regression estimators

ABSTRACT

The problem of estimating of the vector β of the linear regression model y = Aβ + ϵ with ϵ ∼ N_p(0, σ²I_p) under quadratic loss function is considered when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.

KEYWORDS:

• Admissible estimation
• Bayes estimation
• Condition number
• Minimax estimation
• Multivariate normal distribution
• Ridge regression.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62C10
• 62C15
• 62C20

Acknowledgments

The author would like to thank the editor and the referee for careful reading and comments which greatly improved the article.

Additional information

Funding

This research was in part supported by a grant from IPM (No. 95620068).