A Simultaneous Confidence Band for Dense Longitudinal Regression

We present a method of using local linear smoothing to construct simultaneous confidence bands for the mean function of densely spaced functional data. Our approach works well under mild conditions. In addition, the local linear estimator and its accompanying confidence band enjoy semiparametric efficiency in the sense that they are asymptotically equivalent to the counterparts obtained from the random trajectories entirely observed without errors. We illustrate the performance of the proposed confidence band through a simulation study. Furthermore, an application in food science is presented.

[1]  Lijian Yang,et al.  Management Effects on Relationships of Crop Yields with Topography Represented by Wetness Index and Precipitation , 2008 .

[2]  Catherine A. Sugar,et al.  Clustering for Sparsely Sampled Functional Data , 2003 .

[3]  Denis Bosq,et al.  Nonparametric statistics for stochastic processes , 1996 .

[4]  Nancy E. Heckman,et al.  Estimating and depicting the structure of a distribution of random functions , 2002 .

[5]  Jeffrey S. Morris,et al.  Wavelet‐based functional mixed models , 2006, Journal of the Royal Statistical Society. Series B, Statistical methodology.

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

[7]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[8]  Jin-Ting Zhang,et al.  Statistical inferences for functional data , 2007, 0708.2207.

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

[10]  Lijian Yang,et al.  Simultaneous inference for the mean function based on dense functional data , 2012, Journal of nonparametric statistics.

[11]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[12]  H. Muller,et al.  Generalized functional linear models , 2005, math/0505638.

[13]  Jing Wang,et al.  POLYNOMIAL SPLINE CONFIDENCE BANDS FOR REGRESSION CURVES , 2009 .

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

[15]  Lan Xue,et al.  ADDITIVE COEFFICIENT MODELING VIA POLYNOMIAL SPLINE , 2005 .

[16]  Raymond J. Carroll,et al.  A SIMULTANEOUS CONFIDENCE BAND FOR SPARSE LONGITUDINAL REGRESSION , 2012 .

[17]  P. Hall,et al.  Properties of principal component methods for functional and longitudinal data analysis , 2006, math/0608022.

[18]  Wei Biao Wu,et al.  Inference of trends in time series , 2007 .

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

[20]  H. Cardot,et al.  Estimation in generalized linear models for functional data via penalized likelihood , 2005 .

[21]  F. Yao,et al.  Penalized spline models for functional principal component analysis , 2006 .

[22]  Murray D. Burke,et al.  Strong Approximations in Probability and Statistics , 2011, International Encyclopedia of Statistical Science.

[23]  Lijian Yang,et al.  A jump-detecting procedure based on spline estimation , 2011 .

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

[25]  Wei Biao Wu,et al.  CONFIDENCE BANDS IN NONPARAMETRIC TIME SERIES REGRESSION , 2008, 0808.1010.

[26]  Wolfgang Karl Härdle,et al.  Asymptotic maximal deviation of M-smoothers , 1989 .

[27]  James Stephen Marron,et al.  BOOTSTRAP SIMULTANEOUS ERROR BARS FOR NONPARAMETRIC REGRESSION , 1991 .

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

[29]  Fang Yao,et al.  Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data , 2007 .

[30]  John E. Shelton.  People's Republic of China , 1973 .

[31]  Lijian Yang,et al.  Spline confidence bands for variance functions , 2009 .

[32]  Xiaotong Shen,et al.  Local asymptotics for regression splines and confidence regions , 1998 .

[33]  R. Carroll,et al.  Efficient Semiparametric Marginal Estimation for Longitudinal/Clustered Data , 2005 .

[34]  Catherine A. Sugar,et al.  Principal component models for sparse functional data , 1999 .

[35]  D. Degras,et al.  Simultaneous confidence bands for nonparametric regression with functional data , 2009, 0908.1980.

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

[37]  Fang Yao,et al.  Functional Variance Processes , 2006 .

[38]  Xin Zhao,et al.  The functional data analysis view of longitudinal data , 2004 .

[39]  Jianhua Z. Huang,et al.  Identification of non‐linear additive autoregressive models , 2004 .

[40]  Gerda Claeskens,et al.  Bootstrap confidence bands for regression curves and their derivatives , 2003 .

[41]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[42]  Lijian Yang,et al.  Spline-backfitted kernel smoothing of nonlinear additive autoregression model , 2006, math/0612677.

[43]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[44]  Gareth M. James,et al.  Functional Adaptive Model Estimation , 2005 .

[45]  Jianqing Fan,et al.  Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models , 2000 .

[46]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[47]  J. S. Marron,et al.  Analysis of nonlinear modes of variation for functional data , 2007 .

[48]  D. M. Titterington,et al.  On confidence bands in nonparametric density estimation and regression , 1988 .

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

[50]  P. Sarda,et al.  SPLINE ESTIMATORS FOR THE FUNCTIONAL LINEAR MODEL , 2003 .