OPTIMAL DESIGNS IN MULTIVARIATE POLYNOMIAL REGRESSION AND POLYNOMIAL SPLINE REGRESSION

For the polynomial regression model in q variables, of degree (LESSTHEQ) n on the q-cube, D-optimal designs are hard to get theoretically. Farrell, et al (1967) have discussed the difficulty with characterizing admissible designs and found that the elegant ideas of Guest (1958) and Hoel (1958) do not apply to q \u3e 1 cases. For n = 3 with q = 3 and n = 4, 5 with q = 2, symmetric D-optimal designs are given through numerical methods. Also the D-optimal design among the class of product designs is found by using canonical moments theory for arbitary n and q. For n = 3 on the q-simplex, the D-optimal design is discussed. Secondly designs are considered for situations where the mean response consists of a general model together with any number of 2 level factors and suitable interactions. The D-optimal criterion is shown to be equivalent to a type of weighted model selection. Thirdly for the spline polynomial regression model, the uniqueness of the optimal design under the strict concave optimality criterion is shown and various properties of the D-optimal design are discussed. It is conjectured that the D-optimal design is saturated. Also D-optimal designs for some cases are found by numerical methods. Finally Salaveskii\u27s conjecture for exact D-optimal designs is shown to be true for cubic regression model