A Note On Marginal Dimension Test With Directional Regression

Dimension reduction for regression is aimed at seeking a few linearly linear combina-tions of X, say β1’x,...,βd’x, such that all the information of regression is contained in the d linear combinations. Sufficient dimension reduction(SDR) intends to find the smallest dimension d under mild assumption which is X and Y are independent conditioning on βT X. The primary goals of sufficient dimension reduction are the identifying the central subspace, determining the dimension d, and estimating β1’x,...,βd’x.Sliced inverse regression (SIR) and Sliced average variance estimation (SAVE) are among the most commonly used SDR estimators, which also have well-known limitations. Direction regression naturally synthesizes the moment based estimator based on condi-tional moments, such as SIR and SAVE, and achieves the combined advantages of these methods.Determining the dimension d is approached as a testing problem. Sequential tests have been frequently used in the literature. In this paper, we proposed a new sequential test based on directional regression (DR). As demonstrated, the test asymptotically has chi-squared distribution given normally distributed predictors. A simulation study was conducted to show the new method is more powerful over a wide range of models, as we compare the performance of our new test with that of other existing sequential tests. Finally, we apply our test method on Mussels’ muscles data to determine its structural dimension, concerning the identification of structural dimension.