Robust Regression Designs when the Design Space Consists of Finitely Many Points

Abstract

We consider the nearly linear regression problem when the assumed first degree model is contaminated by some small constant, and the design space consists of finitely many points, symmetrically distributed on the interval $\lbrack -1/2, 1/2\rbrack$. Under the usual squared loss function for the estimation of the slope and the intercept, and with the use of the least squares estimators, the problem is to find the designs which are optimal in the sense of minimizing the maximum risk among symmetric designs. The results turn out to be quite different from those obtained by Li and Notz (1981) in a setup that is similar except that the design space is the whole interval $\lbrack -1/2, 1/2\rbrack$. In many cases the optimal solution has a support containing more than two points.