The inner partial least square: An exploration of the “necessary” dimension reduction

The partial least square (PLS) algorithm retains the combinations of predictors that maximize the covariance with the outcome. Cook et al. (2013) showed that PLS results in a predictor envelope, which is the smallest reducing subspace of predictors’ covariance that contains the coefficient. However, PLS and predictor envelope both target at a space that contains the regression coefficients and therefore they may sometimes be too conservative to reduce the dimension of the predictors. In this paper, we propose a new method that may improve the estimation efficiency of regression coefficients when both PLS and predictor envelope fail to do so. Specifically, our method results in the largest reducing subspace of predictors’ covariance that is contained in the coefficient matrix space. Interestingly, the moment based algorithm of our proposed method can be achieved by changing the max in PLS to min. We define the modified PLS as the inner PLS and the resulting space as the inner predictor envelope space. We provide the theoretical properties of our proposed methods as well as demonstrate their use in China Health and Nutrition Survey.