Tests of Linear Hypotheses
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Basic properties of the multivariate normal distribution are proved. The independence of quadratic forms and the independence of a quadratic form and a linear form in a multivariate normal are characterized. The matrix version of the Cochran’s Theorem is formulated with a succinct proof. The results are applied to the one-way and two-way classification, with and without interaction. The general linear hypothesis is considered and the corresponding F-test is developed. Maximum likelihood estimates of the parameters in a linear model are obtained. The multiple correlation coefficient is defined and its relation with the F-statistic for the significance of the regression coefficients is proved.