A matrix identity and its applications to equivalent hypotheses in linear models

Abstract

A matrix identity for the decomposition of sum of squares is considered, and used to convert a general linear hypothesis H in a linear model into an equivalent hypothesis involving only estimable functions. This new approach provides an alternative to the one given by Peixoto (1986, Theorem 2.1). The ‘G-testable’ condition of del Rio (1989, Proposition 1) is then extended to yield a number of necessary and sufficient conditions for testability. The matrix identity is further utilized not only to derive a test statistic for H but also to show that the statistic derived is invariant for testing any hypothesis that is equivalent to H. The results are extended to provide a concise and unified treatment for the inference under a constrained linear model.

Keywords:

• constrained linear model
• equivalent hypotheses
• estimable function
• generalized inverse
• sum of squares
• testable hypothesis