F-test is the most popular test in the general linear model. However, there is few discussions on the robustness of F-test under the singular linear model. In this paper, the necessary and sufficient conditions of robust F-test statistic are given under the general linear models or their partition models, which allows that the design matrix has deficient rank and the covariance matrix of error is a nonnegative definite matrix with parameters. The main results obtained in this paper include the existing findings of the general linear model under the definite covariance matrix. The usage of the theorems is illustrated by an example.
This paper proposes the corrected likelihood ratio test （LRT） and large-dimensional trace criterion to test the independence of two large sets of multivariate variables of dimensions P1 and P2 when the dimensions P = P1 ＋ P2 and the sample size n tend to infinity simultaneously and proportionally. Both theoretical and simulation results demonstrate that the traditional X2 approximation of the LRT performs poorly when the dimension p is large relative to the sample size n, while the corrected LRT and large-dimensional trace criterion behave well when the dimension is either small or large relative to the sample size. Moreover, the trace criterion can be used in the case of p 〉 n, while the corrected LRT is unfeasible due to the loss of definition.
