Let y=A\beta+\epsilon, where y is an N\times1 vector of observations, \betais a p\times1 vector of unknown regression coefficients, A is an N\times pdesign matrix and \epsilon is a spherically symmetric error term with unknownscale parameter \sigma. We consider estimation of \beta under general quadraticloss 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 adaptiveminimax estimators (which are, under the normality assumption, also generalizedBayes) of \beta, which have greater numerical stability (i.e., smallercondition number) than the usual least squares estimator. In particular, wegive a subclass of such estimators which, surprisingly, has a very simple form.We also show that under certain conditions the generalized Bayes minimaxestimators in the normal case are also generalized Bayes and minimax in thegeneral case of spherically symmetric errors.
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机译：设y = A \ beta + \ epsilon，其中y是观测值的N \ times1向量，未知回归系数的\ betais ap \ times1向量，A是N \ times pdesign矩阵，\ epsilon是具有未知标度的球对称误差项参数\ sigma。我们考虑在一般的二次损失函数下对\ beta的估计，特别是扩展了Strawderman的工作[J。美国统计学家。副会长73（1978）623-627]和Casella [Ann。统计员。 8（1980）1036-1056，J。Amer。统计员。副会长80（1985）753-758]通过找到\ beta的自适应极小极大估计量（在正态假设下也是广义贝叶斯），其数值稳定性（即较小的条件数）比通常的最小二乘估计量大。特别是，此类估计量的一个子类令人惊讶地具有非常简单的形式。我们还表明，在某些情况下，正常情况下的广义贝叶斯极小估计量在球对称误差的一般情况下也是广义贝叶斯和极小极大值。
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