A general approach to D-optimal designs for weighted univariate polynomial regression models

We study the D-optimal design problem for the common weighted univariate polynomial regression model with efficiency function λ. We characterize the efficiency functions for which an explicit solution of the D-optimal design problem is available based on a differential equation for the logarithmic derivative of the efficiency function. In contrast to the common approach which starts with a given efficiency function and derives a differential equation for the supporting polynomial of the D-optimal design, we derive a differential equation for the efficiency function, such that an explicit solution of the D-optimal design problem is possible. The approach is illustrated for various convex design spaces and is depicted in several new examples. Also, this concept incorporates all classical efficiency functions discussed in the literature so far.