An optimal projection test for zero multiple correlation coefficient in high-dimensional normal data

Abstract

Testing the hypothesis of zero multiple correlation coefficient is of interest in wide variety of applications including multiple regression analysis. In high-dimensional data, traditional testing procedures to test this hypothesis become practically infeasible due to the singularity of the sample covariance matrix. To deal with this problem, an optimal projection test with a computationally simple and efficient algorithm for implementation is proposed, which can also be used in low-dimensional data. Some simulations are performed to evaluate the performance of the proposed test in high-dimensional normal data as well as to compare the proposed test with the classical exact test in low-dimensional normal data. Lastly, the experimental validation of the proposed approach is carried out on mice tumor volumes data.

Keywords:

• Multiple correlation coefficient
• multivariate normal distribution
• high-dimensional data
• hypothesis testing
• optimal projection test

Notes

1 *The reader should be aware that “projection” is used in a loose sense here, since $\mathbf{\Gamma }$ usually is not a projection matrix.