Tests of parameter matrix in multivariate general linear model with large dimensions

A statistic for testing the significance of parameter matrix in a multivariate general linear model is proposed when the matrix of regressors is random, and its dimension (number of columns) is large, possibly exceeding the sample size (number of rows). Taking advantage of the random regressors, the regression testing problem is equivalently formulated as a correlation testing problem. This strategy helps avoid estimating several large unknown parameters and construct a simple test statistic which is composed of computationally highly efficient estimators. The proposed theory is valid under a general multivariate model, with multivariate normality as a special case, and a few mild assumptions. A few special cases are discussed, with particular focus on testing for univariate general linear model. The accuracy of the statistics is shown through extensive simulations. Applications on several real data sets are demonstrated.