Optimal designs for estimating individual coefficients in Fourier regression models

Abstract

In the common trigonometric regression model, we investigate the optimal design problem for the estimation of the individual coefficients, where the explanatory variable varies in the interval $[-a,a]$, $0 <a \le \pi$. It is demonstrated that the structure of the optimal design depends sensitively on the size of the design space. For many important cases, optimal designs can be found explicitly, where the complexity of the solution depends on the value of the parameter a and the order of the term, for which the corresponding coefficient has to be estimated. The main tool of our approach is the reduction of the problem for the trigonometric regression model to a design problem for a polynomial regression. In particular, we determine the optimal designs for estimating the parameters corresponding to the cosine terms explicitly, if the design space is sufficiently small, and prove that under this condition all optimal designs for estimating the parameters corresponding to the sine terms are supported at the same points.