Optimum continuous block designs

Abstract

The incidence matrix of a block design is replaced by a normalized version, N, in which the entries are non-negative numbers whose sum is unity. The so-called C-matrix, the information matrix for estimation of treatm ent contrasts, is similarly replaced by the normalized analogue C(N). We study the set of ordered eigenvalues of all C(N) and give a complete specification for three treatments (rows). For any number of treatm ents we characterize the eigenvalues of an im portant subclass of designs for which the non-zero entries in any given block are equal. It is suggested that the natural ordering between designs is upper weak majorization of the eigenvalues. Using this we show how to improve a given N-matrix and this leads to several optimality statements.