Functional response regression analysis

Abstract In this paper, we study functional regression with a random response curve and vector covariates. We propose a supervised least squares estimation procedure after utilizing B-spline functions to approximate the unknown functions and establish the asymptotic normality of the proposed estimators. The method has an analytic form and is easily implemented. Compared to existing methods, it does not rely on a normality assumption and can be broadly applied to sparse or non-sparse, equally or non-equally spaced, and balanced or unbalanced observations. We assess the numerical performance of the proposed procedure through simulation experiments and illustrate its performance on a real example.

[1]  Li Ping Yang,et al.  Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data , 1998 .

[2]  Jane-ling Wang,et al.  Functional linear regression analysis for longitudinal data , 2005, math/0603132.

[3]  T. Tony Cai,et al.  Prediction in functional linear regression , 2006 .

[4]  James O. Ramsay,et al.  Functional Data Analysis , 2005 .

[5]  H. Müller,et al.  Functional quadratic regression , 2010 .

[6]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .

[7]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[8]  Huazhen Lin,et al.  Semiparametric Regression Analysis of Longitudinal Skewed Data , 2014 .

[9]  Chin-Tsang Chiang,et al.  Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model With Longitudinal Data , 1998 .

[10]  Tailen Hsing,et al.  On rates of convergence in functional linear regression , 2007 .

[11]  Haipeng Shen,et al.  Supervised Sparse and Functional Principal Component Analysis , 2016 .

[12]  Jane-Ling Wang,et al.  Functional canonical analysis for square integrable stochastic processes , 2003 .

[13]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[14]  G. He,et al.  EXTENDING CORRELATION AND REGRESSION FROM MULTIVARIATE TO FUNCTIONAL DATA , 2000 .

[15]  Tailen Hsing,et al.  DECIDING THE DIMENSION OF EFFECTIVE DIMENSION REDUCTION SPACE FOR FUNCTIONAL AND HIGH-DIMENSIONAL DATA , 2010, 1011.2620.

[16]  Jiguo Cao,et al.  Supervised functional principal component analysis , 2018, Stat. Comput..

[17]  Jane-Ling Wang,et al.  Functional quasi‐likelihood regression models with smooth random effects , 2003 .

[18]  M. Kosorok.  Introduction to Empirical Processes and Semiparametric Inference , 2008 .

[19]  Yuejiao Fu,et al.  Functional mixture regression. , 2011, Biostatistics.

[20]  Xingwei Tong,et al.  Varying coefficient transformation models with censored data , 2010 .

[21]  Kengo Kato,et al.  Estimation in functional linear quantile regression , 2012, 1202.4850.

[22]  Raymond J Carroll,et al.  Generalized Functional Linear Models With Semiparametric Single-Index Interactions , 2010, Journal of the American Statistical Association.

[23]  Jianhua Z. Huang.  Local asymptotics for polynomial spline regression , 2003 .

[24]  Marc Teboulle,et al.  Gradient-based algorithms with applications to signal-recovery problems , 2010, Convex Optimization in Signal Processing and Communications.

[25]  Joseph L Schafer,et al.  A Two-Part Random-Effects Model for Semicontinuous Longitudinal Data , 2001 .

[26]  Jianhui Zhou,et al.  Quantile regression in partially linear varying coefficient models , 2009, 0911.3501.

[27]  H. Müller,et al.  FUNCTIONAL RESPONSE MODELS , 2004 .

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

[29]  Piotr Kokoszka,et al.  Inference for Functional Data with Applications , 2012 .

[30]  Jianhua Z. Huang,et al.  Joint modelling of paired sparse functional data using principal components. , 2008, Biometrika.

[31]  Henry W. Altland,et al.  Applied Functional Data Analysis , 2003, Technometrics.

[32]  P. Sarda,et al.  Functional linear model , 1999 .

[33]  W. Wong,et al.  Convergence Rate of Sieve Estimates , 1994 .

[34]  D. Pollard.  Convergence of stochastic processes , 1984 .

[35]  Fang Yao,et al.  FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS FOR LONGITUDINAL AND SURVIVAL DATA , 2007 .

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

[37]  T. Hsing,et al.  Theoretical foundations of functional data analysis, with an introduction to linear operators , 2015 .

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

[39]  Gareth M. James.  Generalized linear models with functional predictors , 2002 .

[40]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.