Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data

By Yehua Li, T. Hsing. 2010

321 citations recorded by the source corpus.

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Local Linear Forests (2018, methodology): “Local linear regression has been particularly well-studied for longitudinal data, as in Li and Hsing (2010) and Yao, Muller, and Wang (2005).”
Time-Varying Additive Models for Longitudinal Data (2013, background): “The latter case is in fact a functional data setting and its theory can be developed as in Hall, Müller, and Wang (2006), and Li and Hsing (2010). Our theoretical coverage also excludes the case where Tij are”
Modeling and forecasting electricity spot prices: A functional data perspective (2013, methodology): “…Chapter 8 in Ramsay and Silverman (2005) and the nonparametric methods for computing the empirical covariance function as proposed in Staniswalis and Lee (1998), Yao, Müller and Wang (2005), Hall, Müller and Wang (2006) and Li and Hsing (2010) are important methodological references for this paper.”
Functional Data Analysis (2016, methodology): “An alternative approach employed in Li & Hsing (2010) is to assign equal weights to each subject.”
Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis (2014, background): “Condition (F20) regards uniform convergence rates of the covariance estimator (see also Li & Hsing, 2010).”
Modelling function‐valued stochastic processes, with applications to fertility dynamics (2017, background): “5) requires nonzero eigengaps for the first P leading components and is widely adopted in the literature (Hall et al. 2006; Li and Hsing 2010).”
Testing for stationarity of functional time series in the frequency domain (2017, background): “Differences for these two cases have been worked out in Li & Hsing (2010).”
Multivariate functional principal component analysis: A normalization approach (2014, methodology): “Here (C5) and (C6) are required for each λh1 > 2 and for each λh2 > 2, unlike the conditions in Li and Hsing (2010). For any bandwidth h1 and h2, let”
A Confidence Corridor for Sparse Longitudinal Data Curves (2011, background): “The consistency of f̂ (x) and σ̂Y (x) is proved e.g. in Li and Hsing (2010), Yao et al. (2005a).”
Functional Regression (2014, unknown): “Li & Hsing (2010) and Degras (2011) dealt with densely sampled functions and used local linear kernel approaches, and Bunea et al. (2011) used orthonormal basis functions.”
On the optimal reconstruction of partially observed functional data (2017, unknown): “Related results can be found in Yao, Müller and Wang (2005a), Li and Hsing (2010), and Zhang and Wang (2016).”
Partially Linear Functional Additive Models for Multivariate Functional Data (2018, methodology): “The most widely used approach is to project functional data into a low-rank functional subspace and use the projections as predictors in a nonlinear model (James and Silverman 2005; Li and Hsing 2010a; Yao, Lei, and Wu 2015).”
A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data (2014, background): “The consistency of f̂ (x) and σ̂Y (x) is proved, for example, in Li and Hsing (2010), Yao, Müller, and Wang (2005a).”