On optimality of designs with three distinct eigenvalues

Abstract

Let Dv,b,k denote the family of all connected block designs with v treatments and b blocks of size k. Let d∈Dv,b,. The replication of a treatment is the number of times it appears in the blocks of d. The matrix C(d)=R(d)-1/kN(d)N(d)⊤ is called the information matrix of d where N(d) is the incidence matrix of d and R(d) is a diagonal matrix of the replications. Since d is connected, C(d) has v-1 nonzero eigenvalues μ1(d),...,μv-1(d). Let D be the class of all binary designs of Dv,b,k. We prove that if there is a design d*∈D such that (i) C(d*) has three distinct eigenvalues, (ii) d* minimizes trace of C(d)2 over d∈D, (iii) d* maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of C(d) over d∈D, then for all p>0, d* minimizes over d∈D. In the context of optimal design theory, this means that if there is a design d*∈D such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that d* is E- and D-optimal in D, then d* is Φp-optimal in D for all p>0. As an application, we demonstrate the Φp-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming.