This thesis is principally concerned with test criteria for testing different hypotheses for the multivariate regression. In this preface a brief summary of each of the succeeding chapters is given. In Chapter 1 the problem of testing the equality of two population multiple correlation coefficients in identical regression experiments has been studied. The author's results are extentions to those of Schuman and Bradley. In Chapter 2 the results of Chapter 1 are extended to the multivariate case, in other words, the author has constructed tests in order to test the equality of two population generalized multiple correlation matrices. In Chapter 3 the author shows that the Ridge Regression, Principal Components and Shrunken estimators yield the same central t and F statistics as the ordinary least square estimator. In Chapter 4 using the results of Aitken, simultaneous tests for the Cp-criterion of Mallows are constructed. Some comments on extrapolation and prediction are made. In Chapter 5 the Ridge and Principal components residuals are studied. Their use for detecting outliers, when multi-collinearity is present, is examined.

Includes bibliography.