On the minimax robustness against correlation and heteroscedasticity of ordinary least squares among generalized least squares estimators of regression
arxiv(2024)
摘要
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.
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