On the use of the bootstrap for estimating functions with functional data

The bootstrap methodology for functional data and functional estimation target is considered. A Monte Carlo study analyzing the performance of the bootstrap confidence bands (obtained with different resampling methods) of several functional estimators is presented. Some of these estimators (e.g., the trimmed functional mean) rely on the use of depth notions for functional data and do not have received yet much attention in the literature. A real data example in cardiology research is also analyzed. In a more theoretical aspect, a brief discussion is given providing some insights on the asymptotic validity of the bootstrap methodology when functional data, as well as a functional parameter, are involved.

[1]  Ricardo Fraiman,et al.  An anova test for functional data , 2004, Comput. Stat. Data Anal..

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

[3]  Frédéric Ferraty,et al.  The Functional Nonparametric Model and Application to Spectrometric Data , 2002, Comput. Stat..

[4]  R. Dudley.  Nonlinear Functionals of Empirical Measures and the Bootstrap , 1990 .

[5]  John A. Rice,et al.  FUNCTIONAL AND LONGITUDINAL DATA ANALYSIS: PERSPECTIVES ON SMOOTHING , 2004 .

[6]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[7]  V. V. Yurinskii.  Exponential inequalities for sums of random vectors , 1976 .

[8]  J. E. Yukich.  Uniform exponential bound for the normalized empirical process , 1986 .

[9]  ScienceDirect.  Computational statistics & data analysis , 1983 .

[10]  R. Gill.  Non- and semi-parametric maximum likelihood estimators and the Von Mises method , 1986 .

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

[12]  A. Cuevas,et al.  Cluster analysis: a further approach based on density estimation , 2001 .

[13]  David Garcia-Dorado,et al.  Cariporide preserves mitochondrial proton gradient and delays ATP depletion in cardiomyocytes during ischemic conditions. , 2003, American journal of physiology. Heart and circulatory physiology.

[14]  E. Giné,et al.  Bootstrapping General Empirical Measures , 1990 .

[15]  B. Presnell,et al.  Nonparametric estimation of the mode of a distribution of random curves , 1998 .

[16]  P. Hall.  Theoretical Comparison of Bootstrap Confidence Intervals , 1988 .

[17]  R. Laha.  Probability Theory , 1979 .

[18]  R. Fraiman,et al.  Kernel-based functional principal components ( , 2000 .

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

[20]  Christina Gloeckner,et al.  Modern Applied Statistics With S , 2003 .

[21]  W. Parr.  The bootstrap: Some large sample theory and connections with robustness , 1985 .

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

[23]  A. Cuevas,et al.  On the Bootstrap Methodology for Functional Data , 2004 .

[24]  Jon A. Wellner,et al.  Uniform Donsker Classes of Functions , 1992 .

[25]  R. Fraiman,et al.  Trimmed means for functional data , 2001 .

[26]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[27]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[28]  D. Freedman,et al.  Some Asymptotic Theory for the Bootstrap , 1981 .