On the existence of search designs with continuous factors

Abstract

Consider the search linear model defined as follows. Lety(N×1) be a vector ofN observations such that

$$E(y) = A_1 \xi _1 + A_2 \xi _2 ,V(y) = \sigma ^2 I_N$$

((1))

whereσ ² may or may not be known,A ₁(N × υ ₁) andA ₂(N ×υ ₂) are known matrices, ξ₁(υ ₁ × 1) is unknown and ξ₂(υ ₂ × 1) is partly known in the following sense. We known that at mostk elements of ξ₂ are non zero but we do not know particularly which these nonzero elements are. The problem is to make inferences about the elements of ξ₁ and, furthermore, to search the nonzero elements of ξ₁ and make inferences about them. We wanty to be such that the above problem can be resolved with certainty whenσ ²=0; the underlying design corresponding toy is then called a search design. It has been shown in earlier work that for a search design, we must haveN≧υ ₁+2k. In this paper, we consider the special case of search linear models, when the object of the experiment is to fit an appropriate response surface. We establish a basic result, namely, that when the true response surface is representable by a polynomial, then search designs exist for whichN=υ ₁+2k, irrespective of the value ofυ ₂.