FUNCTIONAL ADDITIVE QUANTILE REGRESSION

We investigate functional additive quantile regression that models the conditional quantile of a scalar response by nonparametric effects of a functional predictor. We model the nonparametric effects of the principal component scores as additive components which are approximated by B-splines. We also select the relevant components using a nonconvex SCAD penalty. We establish that, when the relevant components are known, the convergence rate of the estimator using the estimated principal component scores is the same as that using the true scores. We also show that the estimator based on relevant components is a local solution of the SCAD penalized quantile regression problem. The practical performance of the proposed method is illustrated via simulation studies and an empirical application to the corn yield data.

[1]  Yang Feng,et al.  Nonparametric Independence Screening in Sparse Ultra-High-Dimensional Additive Models , 2009, Journal of the American Statistical Association.

[2]  Zhongjun Qu,et al.  Nonparametric estimation and inference on conditional quantile processes , 2015 .

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

[4]  Raymond K. W. Wong,et al.  Partially Linear Functional Additive Models for Multivariate Functional Data , 2018, Journal of the American Statistical Association.

[5]  P. Shi,et al.  Convergence rate of b-spline estimators of nonparametric conditional quantile functions ∗ , 1994 .

[6]  Ali Gannoun,et al.  Support vector machine quantile regression approach for functional data: Simulation and application studies , 2013, J. Multivar. Anal..

[7]  Runze Li,et al.  Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension , 2012, Journal of the American Statistical Association.

[8]  Kengo Kato,et al.  Group Lasso for high dimensional sparse quantile regression models , 2011, 1103.1458.

[9]  H. Müller,et al.  Functional Data Analysis for Sparse Longitudinal Data , 2005 .

[10]  Hongtu Zhu,et al.  Quantile regression for functional partially linear model in ultra-high dimensions , 2019, Comput. Stat. Data Anal..

[11]  H. Zou,et al.  One-step Sparse Estimates in Nonconcave Penalized Likelihood Models. , 2008, Annals of statistics.

[12]  D. Lobell,et al.  On the use of statistical models to predict crop yield responses to climate change , 2010 .

[13]  Jian Huang,et al.  Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space , 2018 .

[14]  A. Gannoun,et al.  Modelling functional additive quantile regression using support vector machines approach , 2014 .

[15]  C. J. Stone,et al.  Additive Regression and Other Nonparametric Models , 1985 .

[16]  R. Koenker.  Additive models for quantile regression: Model selection and confidence bandaids , 2010 .

[17]  J. Horowitz,et al.  VARIABLE SELECTION IN NONPARAMETRIC ADDITIVE MODELS. , 2010, Annals of statistics.

[18]  Fan Wang,et al.  Regularized partially functional quantile regression , 2017, J. Multivar. Anal..

[19]  Kehui Chen,et al.  Conditional quantile analysis when covariates are functions, with application to growth data , 2012 .

[20]  Lan Wang,et al.  Partially linear additive quantile regression in ultra-high dimension , 2016, 1601.06000.

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

[22]  Fang Yao,et al.  Functional Additive Models , 2008 .

[23]  H. Lian.  Semiparametric Estimation of Additive Quantile Regression Models by Two-Fold Penalty , 2012 .

[24]  Fang Yao,et al.  Structured functional additive regression in reproducing kernel Hilbert spaces , 2014, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[25]  V. Chernozhukov,et al.  QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING , 2007, 0704.3649.

[26]  T. Hsing,et al.  Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data , 2010, 1211.2137.