Minimax linear smoothers (Q2722230)

Consider the standard model for a random vector \(y=(y_{1},\dots,y_{n})\) such that NEWLINE\[NEWLINEy_{i}=\mu_{i}+\epsilon_{i},\tag{1}NEWLINE\]NEWLINE \(i=1,\dots,\) where \((\epsilon_{i})\) are i.i.d. zero mean random variables with \(Var \epsilon_{i}=\sigma^{2}<\infty.\) The interest focuses on estimating \(\mu=(\mu_{1},...,\mu_{n})\) subject to the condition NEWLINE\[NEWLINE\mu'B\mu\leq\rho, \tag{2}NEWLINE\]NEWLINE where \(B\) is symmetric. The standard estimation method for this problem is penalized least squares which minimizes \(\|y-\mu\|^{2}\) subject to the previous inequality or its Lagrangian form NEWLINE\[NEWLINE\min_{\mu}\{\|y-\mu\|^{2}+k\mu'B\mu\},NEWLINE\]NEWLINE where \(k\) depends on \(\rho.\) The purpose of this paper is to study an alternative estimation method based on the minimax principle. An estimator \(\hat\mu\) is minimax (relative to the parameter space \(\Theta\)) if NEWLINE\[NEWLINE\inf_{\mu^{*}}\sup_{\mu\in\Theta}E\|\mu^{*}-\mu\|^{2}= \sup_{\mu\in\Theta}E\|\hat\mu-\mu\|^{2},NEWLINE\]NEWLINE where the \(\inf\) is taken over all estimators \(\mu^{*}.\) The paper provides a complete characterization of minimax linear estimators for the model (1) when (2) is fulfilled. Based on Mallows' \(C_{L}\)- statistic, a way of constructing adaptive estimators for unknown \(\rho\) and \(\sigma^{2}\) is given which are asymptotically minimax. A comparison of minimax risk with the risk of penalized least squares estimators in a special case motivated by nonparametric regression is contained.