On D-optimality of exact linear regression designs with minimum support

Abstract

For some classical linear regression models, such as polynomial regression on an interval or multiple linear or quadratic regression on a simplex, D-optimal approximate designs ξ ∗ with minimum support are well known. In the exact design setting with a given total number n of observations the designs ξ n ∗ obtained from ξ ∗ by rounding off its weights to integral multiples of 1 n are good candidates for D-optimality. A modified condition from the equivalence theorem of Kiefer and Wolfowitz is proved to be sufficient for D-optimality of the exact designs ξ n ∗ . This is applied to some regression models studied in the literature. For the more special situation that the experimental region is an interval and the regression function is twice differentiable, another condition is proved to be necessary for D-optimality of a given exact design. For polynomial regression this yields that the exact designs ξ n ∗ with minimum support are not always D-optimal. Thus, a conjecture of Šalaevskiǐ is disproved.