On minimally-supported D-optimal designs for polynomial regression with log-concave weight function

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 <= x <= 1, alpha >= -1, beta >= -1, x(alpha+1) exp(-x) (x >= 0, alpha >= -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.