Minimax linear smoothers

We consider the problem of estimating the mean of a multivariate distributi on. As a general alternative to penalized least squares estimators, we cons ider minimax estimators for squared error over a restricted parameter space where the restriction is determined by the penalization term. For a quadra tic penalty term, the minimax estimator among linear estimators can be foun d explicitly. It is shown that all symmetric linear smoothers with eigenval ues in the unit interval can be characterized as minimax linear estimators over a certain parameter space where the bias is bounded. The minimax linea r estimator depends on smoothing parameters that must be estimated in pract ice. Using results in Kneip (1994), this can be done using Mallows' C-L-sta tistic and the resulting adaptive estimator is now asymptotically minimax l inear. The minimax estimator is compared to the penalized least squares est imator both in finite samples and asymptotically.