Test Criteria for Hypotheses of Symmetry of a Regression Matrix

Abstract

Hotelling's [1] theoretical findings in mathematical economics on the rational behavior of buyers in maximizing their net profit indicate that the matrix of the first partial derivatives of a set of related demand functions would be symmetric and negative definite. It is the object of this paper to determine whether the assumption of symmetry will be tenable in the light of the particular set of observations. The study of test functions for the property of definiteness as a whole will form the subject of a forthcoming paper. The present investigation assumes that the demand functions are regression functions and, therefore, results obtained in this paper do not cover all types of demand functions. The test function $U$ proposed here for the hypothesis of symmetry is invariant under all contragredient transformations. The distribution of $U$ depends on unknown nuisance parameters. The likelihood ratio under the hypothesis of symmetry leads to a multilateral matric equation which represents $\frac{1}{2} p(p + 1)$ equations of the third degree in $\frac{1}{2} p(p + 1)$ unknown regression coefficients for the $p$-variate case. It has not been possible to establish the existence of a nontrivial solution of this equation, and it is, therefore, not being given here.