We present a result according to which certain functions of covariance matrices are maximized at scalar multiples of the identity matrix. This is used to show that the ordinary least squares (OLS) estimate of regression is minimax, in the class of generalized least squares estimates, when the maximum is taken over certain classes of error covariance structures and the loss function possesses a natural monotonicity property. We then consider regression models in which the response function is possibly misspecified, and show that OLS is no longer minimax. We argue that the gains from a minimax estimate are however often outweighed by the simplicity of OLS. We also investigate the interplay between minimax precision matrices and minimax designs. We find that the design has by far the major influence on efficiency and that, when the two are combined, OLS is generally at least 'almost' minimax, and often exactly so.

翻译：我们提出一个结果，即协方差矩阵的某些函数在标量倍数单位矩阵处达到最大值。该结果用于证明，在广义最小二乘估计类中，当最大值在误差协方差结构的特定类别上取得且损失函数具有自然单调性时，回归的普通最小二乘（OLS）估计是极小极大的。随后，我们考虑响应函数可能被错误指定的回归模型，并证明此时OLS不再是极小极大的。我们认为，极小极大估计带来的优势通常被OLS的简洁性所抵消。我们还研究了极小极大精度矩阵与极小极大设计之间的相互作用。发现设计对效率具有主导性影响，且当两者结合时，OLS通常至少是“近似”极小极大的，且往往严格成立。