Some optimization problems with applications to canonical correlations and sphericity tests

C. G. Khatri

Journal of Multivariate Analysis, 1978, vol. 8, issue 3, 453-467

Abstract: Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X'[Sigma]X)-1[Sigma]Y(Y'[Sigma]Y)-1Y'[Sigma]X, where [Sigma] is positive definite, X and Y are p - r and p - s matrices of ranks r and s (>=r), respectively, and X'Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h1 + [epsilon]1hp, ..., hr + [epsilon]rhp-r+1) and , where h1, ..., hp are the eigenvectors of [Sigma] corresponding to the eigenvalues [lambda]1 >= [lambda]2 >= ... >= [lambda]p > 0, [epsilon]j = +1 or -1 for j = 1,2,..., r and , are linear functions of hr+1,..., hp-r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations [theta]1,...,[theta]r and they are given by expressions (3.8)-(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)-(3.13).

Keywords: Correlationship; eigenvalues; and; eigenvectors; efficiency; canonical; correlations; multiple; correlation; union-intersection; test; procedures; sphericity; geometric; arithmetic; and; harmonic; means; inequalities (search for similar items in EconPapers)
Date: 1978
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