On Minimally-supported D-optimal Designs for Polynomial Regression with Log-concave Weight Function

Abstract

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, $(1-x)^{\alpha+1}(1+x)^{\beta+1}(-1\le x\le 1,\alpha\ge -1,\beta\ge -1),x^{\alpha+1}\exp(-x) (x\ge 0,\alpha\ge -1)$ and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.