The Linear Hypothesis and Large Sample Theory

Abstract

and is unknown save for the fact that it is known to belong to Q a subset of s-disional Euclidean space R8, and we wish to test whether O0, the true value of 0, belongs to co an s - r dimensional subspace of U. There are two ways of specifying co: either in the form of constraint equations hi(0) = h2(0) = ... = hr(O) = 0, or in the form of freedom equations 0 = 0(e) where o' = (al, a2 , , * * X a,-,), or perhaps by a combination of both constraint and freedom equations. Although to any freedom equation specification there will correspond a constraint equation specification, this relationship is often difficult to derive in practice and therefore the two forms of specification are usually dealt with separately. These alternative methods of specification have lead to the formulation of three methods of testing co: the Wald test (Wald [7]), the Lagrange Multiplier test (Rao [4] and Silvey [6]) and the likelihood ratio test, all of which are described fully in Aitchison and Silvey [2] and Aitchison [1]. The choice of which method to use depends largely on the ease of computation of the test statistic and therefore to some extent on the method of specification of co. In [1], Aitchison considers the problem of testing more than one hypothesis and introduces the notion of separability which is analogous to the idea of orthogonality in linear hypothesis theory (c.f. Darroch and Silvey [3]). He