On the distribution of the left singular vectors of a random matrix and its applications

In several dimension reduction techniques, the original variables are replaced by a smaller number of linear combinations. The coefficients of these linear combinations are typically the elements of the left singular vectors of a random matrix. We derive the asymptotic distribution of the left singular vectors of a random matrix that has a normal limit distribution. This result is then used to develop a Wald-type test for testing variable importance in Sliced Inverse Regression (SIR) and Sliced Average Variance Estimation (SAVE), two popular sufficient dimension reduction methods.