A distribution-based LASSO for a general single-index model

A general single-index model with high-dimensional predictors is considered. Additive structure of the unknown link function and the error is not assumed in this model. The consistency of predictor selection and estimation is investigated in this model. The index is formulated in the sufficient dimension reduction framework. A distribution-based LASSO estimation is then suggested. When the dimension of predictors can diverge at a polynomial rate of the sample size, the consistency holds under an irrepresentable condition and mild conditions on the predictors. The new method has no requirement, other than independence from the predictors, for the distribution of the error. This property results in robustness of the new method against outliers in the response variable. The conventional consistency of index estimation is provided after the dimension is brought down to a value smaller than the sample size. The importance of the irrepresentable condition for the consistency, and the robustness are examined by a simulation study and two real-data examples.