Simultaneous Canonization of Linear Models

Canonical form plays a similar role in linear models to spectral decomposition in matrix analysis. Let X = (X ₁,…, X _n )′ be a random vector with expectation Aβ and the variance–covariance matrix σV, where V is positive definite and let rank(A) = r. Then there exists a nonsingular linear transformation from X to T = (T ₁,…, T _n )′, such that ET _i  = η _i , for i = 1,…, r and zero for i > r, while cov(T _i , T _j ) = δ _ij σ. This canonical form, introduced by Ko?odziejczyk (1935 Ko?odziejczyk , S. ( 1935 ). On an important class of statistical hypotheses . Biometrika 27 : 161 – 190 .Crossref] , Google Scholar]), was used, among others, by Scheffé (1959 Scheffé , H. ( 1959 ). Analysis of Variance . New York : Wiley . Google Scholar]) and by Lehmann (1959, 1986 Lehmann , E. L. (1959, 1986 ). Testing Statistical Hypotheses . New York : Wiley . Google Scholar]). This technique is extended here for arbitrary (possibly singular) V and for simultaneous canonization of two models of this type.