Orthogonality and D-optimality of the U-type design under general Fourier regression models

Riccomagno et al. [1997. Lattice-based D-optimum design for Fourier regression. Ann. Statist. 25, 2313-2317] gave a comprehensive study on D-optimality under a symmetric Fourier regression model and gave a characterization on orthogonality, complete orthogonality and D-optimality of lattice designs under a symmetrical Fourier model. This paper extends their results to asymmetric Fourier regression models. We give some necessary condition for the orthogonality and complete orthogonality of U-type designs under general Fourier regression models that involve symmetrical and asymmetrical Fourier models and prove that a U-type design, which is completely orthogonal, is not D-optimal for an asymmetrical Fourier model.