Admissible and optimal exact designs for polynomial regression
The authors' interest is to address some classical optimal design questions in the context of the exact design setting for the polynomial regression model. Let \(x_ 0,...,x_ l\) denote the distinct levels at which \(n_ 0,...,n_ l\) observations are to be taken. Here \(n_ 0+...+n_ l=N\). An exact design \(\xi^ N\) is a probability measure on [a,b] which concentrates mass \(n_ i/N\) at each \(x_ i\). Let \(M(\xi^ N)\) denote the information matrix (per observation) of an exact design \(\xi^ N\). An exact design \(\xi^ N\) is said to be admissible if and only if there exists no other exact design \({\tilde \xi}^ N\) such that \(M(\xi^ N)\neq M({\tilde \xi}^ N)\) and \(M(\xi^ N)\leq M({\tilde \xi}^ N)\). Solution of this problem will enable us to restrict our search for an optimal design to the class of admissible designs. A necessary condition for an exact design to be admissible is provided. It is conjectured that this condition is also sufficient and the basis for the conjecture is discussed. It is shown that the Salaevskij's conjecture holds for cubic regression. The interesting forms of G-optimal exact designs are indicated.
scientific article; zbMATH DE number 3999049
scientific article