A limit theorem of D-optimal designs for weighted polynomial regression

Consider the D-optimal designs for the dth-degree polynomial regression model with a continuous weight function on a compact interval. As the degree of the model goes to infinity, we derive the asymptotic value of the logarithm of the determinant of the D-optimal design. If the weight function is equal to 1, we derive the formulae of the values of the D  -criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J1/2,1/2J1/2,1/2 density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs is investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.