Supplemental Material to “ Partially Linear Additive Quantile Regression in Ultra - High Dimension
نویسندگان
چکیده
The tables of the appendix provide additional numerical results. Table 1 summarizes simulation results for Q-SCAD, LS-SCAD, Q-MCP, LS-MCP with sample sizes 50, 100 and 200 for modeling the 0.7 conditional quantile for the heteroscedastic error setting described in Section 4 of the main paper. The MCP approaches, Q-MCP and LS-MCP, are the equivalent of Q-SCAD and LS-SCAD with the SCAD penalty function replaced by the MCP penalty function [Zhang (2010)]. Table 2 reports the simulation results for the MCP penalty function for the four simulation settings presented in Section 4 of the main paper. Table 3 is an extension of Table 3 from the main text with results for p = 1200 and 2400. Tables 4 and 5 report numerical results with the MCP penalty function for the real data analysis. Table 6 presents the simulation results using the MCP penalty function for simultaneous variable selection at different quantiles. The selection of the tuning parameter λ for the MCP penalty function uses the modified high-dimensional BIC criterion as we recommended for the SCAD penalty function. The tuning parameter a for MCP is set to 3, which is used as the default in the R package ncvreg for the least squares implementation of the MCP penalty [Breheny and Lee (2015)]. The results with MCP are observed to be similar as those with SCAD.
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