Robust designs and optimality of least squares for regression problems (Q795449)

Considered is the linear regression \(y_ i=a_ 0+a^ Tx_ i+\psi(x_ i)+\epsilon_ i\) where the errors \(\epsilon_ i\) are uncorrelated with mean zero and variance \(\sigma^ 2\), \(\psi\) is an unknown ''contamination function'' bounded by a known convex function \(\phi\), \(x_ i\in X\subset {\mathbb{R}}^ k\) and \(a_ 0\), \(a^ T\) are the parameters to be estimated. The mean square error matrix and \(\Phi_ p\)-design-optimality criteria are combined to find for the minimax approach w.r.t. the design the (linear) estimator and to \(\psi\) a best pair of estimator and design. Sufficient conditions are given that in case \(k=1\) the least squares estimator and a two point design are a best pair for all \(\Phi_ p\)- criteria and - in case k arbitrary - that the least squares estimator and a uniform design measure on a sphere whose radius depends on \(\sigma^ 2\) and \(\phi\) are a best pair for \(\Phi_ 1\) and \(\Phi_{\infty}\).