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

Abstract

The curse of dimensionality is a recognized challenge in nonparametric estimation. This paper develops a new L 0 -norm regularization approach to the convex quantile and expectile regressions for subset variable selection. We show how to use mixed integer programming to solve the proposed L 0 -norm regularization approach in practice and build a link to the commonly used L 1 -norm regularization approach. A Monte Carlo study is performed to compare the finite sample performances of the proposed L 0 -penalized convex quantile and expectile regression approaches with the L 1 -norm regularization approaches. The proposed approach is further applied to benchmark the sustainable development performance of the OECD countries and empirically analyze the accuracy in the dimensionality reduction of variables. The results from the simulation and application illustrate that the proposed L 0 -norm regularization approach can more effectively address the curse of dimensionality than the L 1 -norm regularization approach in multidimensional spaces.

Partial frontiers are not quantiles

“…Furthermore, while the estimated quantile function, Qy i , is always unique in the CER estimation, the feasible set of problem (4) could be unbounded. That is, there may exist multiple combinations of shadow prices βij ) leading to the same optimal value of the objective function (Dai, 2021). The non-unique estimates in both CQR and CER may further cause a longstanding problem of quantile crossing in quantile estimation (Dai et al, 2022).…”

Forward-looking assessment of the GHG abatement cost: Application to China

“…By deriving an explicit piece-wise linear characterization for regression function, Kuosmanen (2008) introduces convex nonparametric least squares (CNLS), a multivariate convex regression approach, which later attracts great interest in econometrics, statistics, operations research, and machine learning (e.g., Magnani & Boyd, 2009;Seijo & Sen, 2011;Lim & Glynn, 2012;Hannah & Dunson, 2013;Yagi et al, 2020;Bertsimas & Mundru, 2021). The recent studies by Dai (2021) and Dai et al (2022) impose additional regularization on convex regression to address its various intrinsic problems.…”

“…Second, the curse of dimensionality is an acknowledged challenge in nonparametric estimation. To mitigate the effects of dimensionality, Dai (2021) introduces two different penalty functions, L 1 and L 0 norms, to CQR and CER. The L 1 -CQR approach is formulated as…”

“…It is worth highlighting two of them here, namely, the CNLS-G algorithm proposed by Lee et al (2013) and the cutting-plane algorithm proposed by Balázs et al (2015) and extended by Bertsimas & Mundru (2021). Although both algorithms use the relaxed Afriat inequality constraint set and iteratively introduce new ones as necessary, the cutting-plane algorithm is more efficient to solve the convex regression problems due to its fast identification of violating constraints (Dai, 2021).…”

“…Recently, incorporate contextual variables to the CER estimator to evaluate the performance for English National Health Service hospitals in combating COVID-19 pandemic. OECD countries Provincial regions in China Dai et al (2020) The applications in productivity analysis mainly include three aspects: full frontier productivity analysis (e.g., Kuosmanen, 2013), quantile performance evaluation (e.g., Dai, 2021), and index decomposition analysis (e.g., . Similar to the DEA Malmquist index, the StoNED Malmquist index also relies on the full frontier to gauge productivity growth.…”