Optimal and Quasi-Optimal Designs

Abstract

Optimal design theory deals with the choice of the allocation of the observations to accomplish the estimation of some linear combination of the coefficients in a regression model in an optimal way. Canonical moments provide an elegant framework to the theory of optimal designs. An optimal design for polynomial regression of a given degree r can be fatally inappropriate in case the polynomial degree should in fact be s, and hence when r is unknown it would be preferable to consider designs that show good performance for different values of the polynomial degree. Anderson’s (1962) pathbreaking solution of this multidecision problem has originated many developments, as optimal discriminant designs and optimal robust designs. But once again a design devised for a specific task can be grossly inefficient for a slightly different purpose. We introduce mixed designs; tables for regression of degrees r= 2,3,4 exhibiting the loss of efficiency when the optimal mixed design is used instead of the optimal discriminant or of the optimal robust design show that the loss of efficiency is at most 1% and 2%, respectively, while the loss of efficiency when using a discriminant design instead of a robust design or vice-versa can be as high as 10%. Using recursive relations we compute pseudo-canonical moments for measures with infinite support, showing that such pseudo-canonical moments do not share the good identifiability properties of canonical moments of measures whose support is a subset of a compact interval of the real line.