Estimation and variable selection in single-index composite quantile regression

ABSTRACT In this article, a new composite quantile regression estimation approach is proposed for estimating the parametric part of single-index model. We use local linear composite quantile regression (CQR) for estimating the nonparametric part of single-index model (SIM) when the error distribution is symmetrical. The weighted local linear CQR is proposed for estimating the nonparametric part of SIM when the error distribution is asymmetrical. Moreover, a new variable selection procedure is proposed for SIM. Under some regularity conditions, we establish the large sample properties of the proposed estimators. Simulation studies and a real data analysis are presented to illustrate the behavior of the proposed estimators.

[1]  Yan Yu,et al.  Single-index quantile regression , 2010, J. Multivar. Anal..

[2]  A. G. Fisher,et al.  Generalized body composition prediction equations for men using simple measurement techniques , 1985 .

[3]  Keith Knight,et al.  Limiting distributions for $L\sb 1$ regression estimators under general conditions , 1998 .

[4]  Runze Li,et al.  Tuning parameter selectors for the smoothly clipped absolute deviation method. , 2007, Biometrika.

[5]  Hua Liang,et al.  Estimation and Variable Selection for Semiparametric Additive Partial Linear Models (SS-09-140). , 2011, Statistica Sinica.

[6]  Kjell A. Doksum,et al.  On average derivative quantile regression , 1997 .

[7]  B. Silverman,et al.  Weak and strong uniform consistency of kernel regression estimates , 1982 .

[8]  Tao Huang,et al.  Penalized least squares for single index models , 2011 .

[9]  Ker-Chau Li,et al.  Sliced Inverse Regression for Dimension Reduction , 1991 .

[10]  Wei-Min Qian,et al.  Single-index composite quantile regression , 2012 .

[11]  H. Tong,et al.  Article: 2 , 2002, European Financial Services Law.

[12]  H. Zou.  The Adaptive Lasso and Its Oracle Properties , 2006 .

[13]  Jian Huang,et al.  SCAD-penalized regression in high-dimensional partially linear models , 2009, 0903.5474.

[14]  Riquan Zhang,et al.  Quantile regression and variable selection for the single-index model , 2014 .

[15]  Ker-Chau Li,et al.  On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma , 1992 .

[16]  Hohsuk Noh,et al.  Model Selection via Bayesian Information Criterion for Quantile Regression Models , 2014 .

[17]  H. Zou,et al.  Composite quantile regression and the oracle Model Selection Theory , 2008, 0806.2905.

[18]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[19]  D. Pollard,et al.  Asymptotics for minimisers of convex processes , 2011, 1107.3806.

[20]  W. Härdle,et al.  Semi-parametric estimation of partially linear single-index models , 2006 .

[21]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[22]  H. Tong,et al.  An adaptive estimation of dimension reduction , 2002 .

[23]  Runze Li,et al.  NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS. , 2011, Annals of statistics.

[24]  R. Koenker,et al.  Regression Quantiles , 2007 .

[25]  P. Cabilio,et al.  A simple test of symmetry about an unknown median , 1996 .

[26]  Thomas M. Stoker,et al.  Investigating Smooth Multiple Regression by the Method of Average Derivatives , 2015 .

[27]  Riquan Zhang,et al.  B spline variable selection for the single index models , 2017 .

[28]  Hu Yang,et al.  Penalized weighted composite quantile estimators with missing covariates , 2016 .

[29]  Yujie Gai,et al.  Weighted local linear composite quantile estimation for the case of general error distributions , 2013 .