Admissible experimental designs in multiple polynomial regression

Abstract We give a connection between admissibility of a design in a multiple polynomial setup in v variables and admissibility of subdesigns in multiple setups in less than v variables. This leads to practicable conditions on the support of an admissible design. Especially, for a convex experimental region X an admissible linear regression design is concentrated on the extreme points of X , and an admissible quadratic regression design has at most one support point in the relative interior of an arbitrary face of X . For the ball and the cube we give all admissible (and invariant) linear and quadratic regression designs, proving that the necessary conditions on the support given in Heiligers (1991) are sufficient as well. As demonstrated for both regions, our results substantially reduce the problem of finding optimal designs.

[1]  J. Kiefer,et al.  The Equivalence of Two Extremum Problems , 1960, Canadian Journal of Mathematics.

[2]  Corwin L. Atwood,et al.  Optimal and Efficient Designs of Experiments , 1969 .

[3]  Z. Galil,et al.  Comparison of design for quadratic regression on cubes , 1977 .

[4]  J. Kiefer.  Construction and optimality of generalized Youden designs II , 1975 .

[5]  Sylvain Ehrenfeld,et al.  Complete Class Theorems in Experimental Design , 1956 .

[6]  J. Kiefer.  General Equivalence Theory for Optimum Designs (Approximate Theory) , 1974 .

[7]  J. Kiefer,et al.  Optimum Designs in Regression Problems , 1959 .

[8]  F. Graybill,et al.  Matrices with Applications in Statistics. , 1984 .

[9]  K. Schneider,et al.  Invariant admissible and optimal designs in cubic regression on the v-ball , 1992 .

[10]  F. Pukelsheim.  Information increasing orderings in experimental design theory , 1987 .

[11]  L. Pesotchinsky,et al.  Phi sub p-Optimal Second Order Designs for Symmetric Regions , 1978 .

[12]  Kazumasa Kôno,et al.  OPTIMUM DESIGN FOR QUADRATIC REGRESSION ON k-CUBE , 1962 .

[13]  J. Kiefer.  Optimum Experimental Designs , 1959 .

[14]  R. H. Farrell,et al.  Optimum multivariate designs , 1967 .

[15]  Norbert Gaffke,et al.  Further Characterizations of Design Optimality and Admissibility for Partial Parameter Estimation in Linear Regression , 1987 .

[16]  W. J. Studden,et al.  Optimal Experimental Designs , 1966 .

[17]  Berthold Heiligers,et al.  Admissibility of experimental designs in linear regression with constant term , 1991 .

[18]  W. J. Studden,et al.  Efficient $D_s$-Optimal Designs for Multivariate Polynomial Regression on the $q$-Cube , 1988 .

[19]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[20]  L. L. Pesotchinsky,et al.  D-optimum and quasi-D-optimum second-order designs on a cube , 1975 .

[21]  Z. Galil,et al.  Comparison of Rotatable Designs for Regression on Balls , 1977 .