Mathematics – Statistics Theory
Scientific paper
2005-08-16
Annals of Statistics 2005, Vol. 33, No. 4, 1753-1770
Mathematics
Statistics Theory
Published at http://dx.doi.org/10.1214/009053605000000327 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
10.1214/009053605000000327
Let y=A\beta+\epsilon, where y is an N\times1 vector of observations, \beta is a p\times1 vector of unknown regression coefficients, A is an N\times p design matrix and \epsilon is a spherically symmetric error term with unknown scale parameter \sigma. We consider estimation of \beta 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 \beta, 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.
Maruyama Yuzo
Strawderman William E.
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