In this paper we consider the exact $D$-optimal design problem for linear trigonometric regression models with or without intercept on a partial circle. In a recent papper Dette, Melas and Pepelyshev (2001) found explicit solutions of approximate $D$-optimal designs for trigonometric regression models with intercept on a partial circle. The exact optimal designs are determined by means of moment sets of trigonometric functions. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of the design points.

Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a complete solution of the exact D-optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.

This paper discusses the approximate and the exact n-point D-optimal design problems for the common multivariate linear and quadratic polynomial regression on some convex design spaces. For the linear polynomial regression, the design space considered are q-simplex, q-ball and convex hull of a set of finite points. It is shown that the approximate and the exact n-point D-optimal designs are concentrated on the extreme points of the design space. The structure of the optimal designs on regular polygons or regular polyhedra is also discussed. For the quadratic polynomial regression, the design space considered is a q-ball. The configuration of the approximate and the exact n-point D-optimal designs for quadratic model in two variables on a disk are investigated.

Optimal design problems aim at selecting design points optimally with respect to certain statistical criteria. The research of this thesis focuses on optimal design problems with respect to A-, D- and E-optimal criteria, which minimize the trace, determinant and largest eigenvalue of the information matrix, respectively. Semide nite programming (SDP) is concerned with optimizing a linear objective function subject to a linear matrix being positive semide nite. Two powerful MATLAB add-ons, SeDuMi and CVX, have been developed to solve SDP problems e ciently. In this paper, we show in detail how to formulate A- and E-optimal design problems as SDP problems and solve them by SeDuMi and CVX. This technique can be used to construct approximate A-optimal and E-optimal designs for all linear and non-linear models with discrete design spaces. The results can also provide guidance to nd optimal designs on continuous design spaces. For one variable polynomial regression models, we solve the A- and E- optimal designs on the continuous design space by using a two-stage procedure. In the rst stage we nd the optimal moments by casting it as an SDP problem and in the second stage we extract the optimal designs from the optimal moments obtained from the rst stage. Unlike E- and A-optimal design problems, the objective function of D-optimal design problem is nonlinear. So D-optimal design problems cannot be reformulated as an SDP. However, it can be cast as a convex problem and solved by an interior point method. In this thesis we give details on how to use the interior point method to solve D-optimal design problems. Finally several numerical examples for A-, D-, and E-optimal designs along with the MATLAB codes are presented. / Graduate