Quantile regression in partially linear varying coefficient models

Semiparametric models are often considered for analyzing longitudinal data for a good balance between flexibility and parsimony. In this paper, we study a class of marginal partially linear quantile models with possibly varying coefficients. The functional coefficients are estimated by basis function approximations. The estimation procedure is easy to implement, and it requires no specification of the error distributions. The asymptotic properties of the proposed estimators are established for the varying coefficients as well as for the constant coefficients. We develop rank score tests for hypotheses on the coefficients, including the hypotheses on the constancy of a subset of the varying coefficients. Hypothesis testing of this type is theoretically challenging, as the dimensions of the parameter spaces under both the null and the alternative hypotheses are growing with the sample size. We assess the finite sample performance of the proposed method by Monte Carlo simulation studies, and demonstrate its value by the analysis of an AIDS data set, where the modeling of quantiles provides more comprehensive information than the usual least squares approach.

[1]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[2]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[3]  S. Portnoy.  Asymptotic Behavior of $M$-Estimators of $p$ Regression Parameters when $p^2/n$ is Large. I. Consistency , 1984 .

[4]  S. Portnoy.  Asymptotic behavior of M-estimators of p regression parameters when p , 1985 .

[5]  J. Phair,et al.  The Multicenter AIDS Cohort Study: rationale, organization, and selected characteristics of the participants. , 1987, American journal of epidemiology.

[6]  P. Hall,et al.  On the Distribution of a Studentized Quantile , 1988 .

[7]  R. Koenker,et al.  Hierarchical Spline Models for Conditional Quantiles and the Demand for Electricity , 1990 .

[8]  Roger Koenker,et al.  Tests of linear hypotheses based on regression rank scores , 1993 .

[9]  R. Koenker.  Confidence Intervals for Regression Quantiles , 1994 .

[10]  S. Lipsitz,et al.  Quantile Regression Methods for Longitudinal Data with Drop‐outs: Application to CD4 Cell Counts of Patients Infected with the Human Immunodeficiency Virus , 1997 .

[11]  M. C. Jones,et al.  Local Linear Quantile Regression , 1998 .

[12]  Q. Shao,et al.  On Parameters of Increasing Dimensions , 2000 .

[13]  Chin-Tsang Chiang,et al.  Smoothing Spline Estimation for Varying Coefficient Models With Repeatedly Measured Dependent Variables , 2001 .

[14]  Jianhua Z. Huang,et al.  Varying‐coefficient models and basis function approximations for the analysis of repeated measurements , 2002 .

[15]  Zhongyi Zhu,et al.  Estimation in a semiparametric model for longitudinal data with unspecified dependence structure , 2002 .

[16]  R. Koenker.  Quantile regression for longitudinal data , 2004 .

[17]  Toshio Honda,et al.  Quantile regression in varying coefficient models , 2004 .

[18]  Qi Li,et al.  Efficient estimation of a semiparametric partially linear varying coefficient model , 2005, math/0504510.

[19]  Jianqing Fan,et al.  Profile likelihood inferences on semiparametric varying-coefficient partially linear models , 2005 .

[20]  Hulin Wu,et al.  Semiparametric Time‐Varying Coefficients Regression Model for Longitudinal Data , 2005 .

[21]  Xuming He,et al.  Conditional growth charts , 2006 .

[22]  Hui-xia,et al.  Detecting Differential Expressions in GeneChip Microarray Studies: A Quantile Approach , 2006 .

[23]  Runze Li,et al.  Quadratic Inference Functions for Varying‐Coefficient Models with Longitudinal Data , 2006, Biometrics.

[24]  Runze Li,et al.  Analysis of Longitudinal Data With Semiparametric Estimation of Covariance Function , 2007, Journal of the American Statistical Association.

[25]  Mi-Ok Kim,et al.  Quantile regression with varying coefficients , 2007, 0708.0471.

[26]  Zongwu Cai,et al.  Nonparametric Quantile Estimations for Dynamic Smooth Coefficient Models , 2008 .

[27]  Ying Wei,et al.  A DYNAMIC QUANTILE REGRESSION TRANSFORMATION MODEL FOR LONGITUDINAL DATA , 2009 .