Uniformly valid inference for partially linear high-dimensional single-index models

Abstract

Uniformly valid inference is obtained for the linear part of a partially linear single-index model with a high-dimensional variable in the single-index part of the model. The linear model part consists of a fixed and limited number of variables, while the control variables in the single-index might consist of a large number of variables that is allowed to grow with the sample size and potentially exceeds the sample size. The unknown function in the nonlinear model part is estimated via B-splines in combination with a ℓ1-regularization to take the high dimensionality into account. We establish uniformly valid inferential properties for the parameters in the linear model part under imperfect selection of the control variables. The numerical results showcase the method’s behavior in practice.