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The curse of dimensionality is a recognized challenge in nonparametricestimation. This paper develops a new L0-norm regularization approach to theconvex quantile and expectile regressions for subset variable selection. Weshow how to use mixed integer programming to solve the proposed L0-normregularization approach in practice and build a link to the commonly usedL1-norm regularization approach. A Monte Carlo study is performed to comparethe finite sample performances of the proposed L0-penalized convex quantile andexpectile regression approaches with the L1-norm regularization approaches. Theproposed approach is further applied to benchmark the sustainable developmentperformance of the OECD countries and empirically analyze the accuracy in thedimensionality reduction of variables. The results from the simulation andapplication illustrate that the proposed L0-norm regularization approach canmore effectively address the curse of dimensionality than the L1-normregularization approach in multidimensional spaces.

Variable selection in convex quantile regression: L1-norm or L0-norm regularization?