A note on optimal designs in weighted polynomial regression for the classical efficiency functions

In this note we consider the D-optimal design problem for the heteroscedastic polynomial regression model. Karlin and Studden (Ann. Math. Statist. 37 (1966a) 783) found explicit solutions for three types of efficiency functions. We introduce two "new" functions to model the heteroscedastic structure, for which the D-optimal designs can also be found explicitly. The optimal designs have equal masses at the roots of generalized Bessel- and Jacobi-polynomials with complex parameters. It is also demonstrated that there exist no other efficiency functions such that the supporting polynomial of the D-optimal design satisfies a generalized Rodriguesʹ formula.