On Rao's inequalities for arrays of strength d
Abstract
The purpose of this paper is twofold: to carry out and elucidate a proof, suggested by Rao, of his inequalities for arbitrary orthogonal arrays, and to prove that an arbitrary fractional factorial design of strength d has resolution d + 1. The proof of the inequalities adapts Rao's original proof for sym metric arrays. The Erdös-Ko-Rado Theorem shows that the "odd" inequality is the best one can get with this approach. We include a discussion of Rao's implicit assumption that the array have no repeated columns. The proof concerning resolution likewise does not assume that the factorial experiment be symmetric, nor that the fraction have a particular form, and so is completely general. Both proofs make use of a mapping, introduced by Tjur, from the partition lattice of a finite set to the lattice of subspaces of a vector space.
