Abstract
This paper studies estimation in partial functional linear quantile regression in which the dependent variable is related to both a vector of finite length and a function-valued random variable as predictor variables. The slope function is estimated by the functional principal component basis. The asymptotic distribution of the estimator of the vector of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. It is showed that this rate is optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. The convergence rate of the mean squared prediction error for the proposed estimators is also be established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.
Similar content being viewed by others
References
Cai T T, Hall P. Predictionin functional linear regression. Ann Statist, 2006, 34: 2159–2179
Cai Z, Xu X. Nonparametric quantile estimations for dynamic smooth coefficient models. J Amer Statist Assoc, 2009, 104: 371–383
Cardot H, Crambes C, Sarda P. Quantile regression when the covariates are functions. J Nonparametric Statist, 2005, 17: 841–856
Cardot H, Ferraty F, Sarda P. Functional linear model. Statist Probab Lett, 1999, 45: 11–22
Cardot H, Ferraty F, Sarda P. Spline estimators for the functional linear model. Statist Sin, 2003, 13: 571–591
Chen K, Müller H G. Conditional quantile analysis when covariates are functions, with application to growth data. J R Statist Soc B, 2012, 74: 67–89
Crambes C, Kneip A, Sarda P. Smoothing splines estimators for functional linear regression. Ann Statist, 2009, 37: 35–72
Hall P, Horowitz J L. Methodology and convergence rates for functional linear regression. Ann Statist, 2007, 35: 70–91
Hofmann B, Scherzer O. Local ill-posedness and source conditions of operator equations in Hilbert space. Inverse Problems, 1998, 14: 1189–1206
Hu Y, He X M, Tao J, et al. Modeling and prediction of children’s growth data via functional principal component analysis. Sci China Ser A, 2009, 52: 1342–1350
Kai B, Li R, Zou H. New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann Statist, 2011, 39: 305–332
Kengo Kato. Estimation in functional linear quantile regression. Ann Statist, 2012, 40: 3108–3136
Kocherginsky M, He X, Mu Y. Practical confidence intervals for regression quantiles. J Comput Graph Statist, 2005, 14: 41–55
Koenker R. Quantile Regression. Cambridge: Cambridge University Press, 2005
Koenker R, Machado J A. Goodness of fit and related inference processes for quantile regression. J Amer Statist Assoc, 1999, 94: 1296–1310
Lee S. Efficient semiparametric estimation of a partially linear quantile regression model. Econometric Theory, 2003, 19: 1–31
Li Y, Hsing T. On rates of convergence in functional linear regression. J Multivariate Anal, 2007, 98: 1782–1804
Ramsay J O, Silverman B W. Applied Functional Data Analysis: Methods and Case Studies. New York: Springer, 2002
Ramsay J O, Silverman B W. Functional Data Analysis. New York: Springer, 2005
Reiss P T, Ogden R T. Functional generalized linear models with images as predictors. Biometrics, 2010, 66: 61–69
Shin H. Partial functional linear regression. J Statist Plann Inference, 2009, 139: 3405–3418
Shin H, Lee M H. On prediction rate in partial functional linear regression. J Multi Anal, 2012, 103: 93–106
Tian M Z. Robust estimation in inverse problems via quantile coupling. Sci China Math, 2012, 55: 1029–1041
Tuddenham R, Snyder M. Physical growth of California boys and girls from birth to age 18. Calif Publ Chld Dev, 1954, 1: 183–364
van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes: With Applications to Statistics. New York: Springer, 1996
Wang H, Zhu Z, Zhou J. Quantile regression in partially linear varing coefficient models. Ann Statist, 2009, 37: 3841–3866
Wei Y, He X. Conditional growth charts. Ann Statist, 2006, 34: 2069–2097
Yao F, Müller H G, Wang J L. Functional linear regression analysis for longitudinal data. Ann Statist, 2005, 33: 2873–2903
Zhang D, Lin X, Sowers M F. Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome. Biometrics, 2007, 63: 351–362
Zhu L P, Li R Z, Cui H J. Robust estimation for partially linear models with large-dimensional covariates. Sci China Math, 2013, 56: 2069–2088
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tang, Q., Cheng, L. Partial functional linear quantile regression. Sci. China Math. 57, 2589–2608 (2014). https://doi.org/10.1007/s11425-014-4819-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4819-x
Keywords
- partial functional linear quantile regression
- quantile estimator
- functional principal component analysis
- convergence rate