Optimal designs for estimating individual coefficients in polynomial regression : a functional approach

In this paper the optimal design problem for the estimation of the individual coefficients in a polynomial regression on an arbitrary interval [a, b] (- inf. < a < b < inf) is considered. Recently, Sahm (2000) demonstrated that the optimal design is one of four types depending on the symmetry parameter s = (a + b) / (a-b) and the specific coefficient which has to be estimated. In the same paper the optimal design was identified explicitly in three cases. It is the basic purpose of the present paper to study the remaining open fourth case. It will be proved that in this case the support points and weights are real analytic functions of the boundary points of the design space. This result is used to provide a Taylor expansion for the weights and support points as functions of the parameters a and b, which can easily be used for the numerical calculation of the optimal designs in all cases, which were not treated by Sahm (2000).