A nonlinear multi-dimensional variable selection method for high dimensional data: Sparse MAVE

Traditional variable selection methods are model based and may suffer from possible model misspecification. On the other hand, sufficient dimension reduction provides us with a way to find sufficient dimensions without a parametric model. However, the drawback is that each reduced variable is a linear combination of all the original variables, which may be difficult to interpret. In this paper, focusing on the sufficient dimensions in the regression mean function, we combine the ideas of sufficient dimension reduction and variable selection to propose a shrinkage estimation method, sparse MAVE. The sparse MAVE can exhaustively estimate dimensions in the mean function, while selecting informative covariates simultaneously without assuming any particular model or particular distribution on the predictor variables. Furthermore, we propose a modified BIC criterion for effectively estimating the dimension of the mean function. The efficacy of sparse MAVE is verified through simulation studies and via analysis of a real data set.