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Estimation of the mean and covariance parameters for functional data is a critical task, with local linear smoothing being a popular choice. In recent years, many scientific domains are producing multivariate functional data for which $$p$$, the number of curves per subject, is often much larger than the sample size $$n$$. In this setting of high-dimensional functional data, much of developed methodology relies on preliminary estimates of the unknown mean functions and the auto- and cross-covariance functions. This paper investigates the convergence rates of local linear estimators in terms of the maximal error across components and pairs of components for mean and covariance functions, respectively, in both $L^2$ and uniform metrics. The local linear estimators utilize a generic weighting scheme that can adjust for differing numbers of discrete observations $$N_{ij}$$ across curves $$j$$ and subjects $$i$$, where the $$N_{ij}$$ vary with $$n$$. Particular attention is given to the equal weight per observation (OBS) and equal weight per subject (SUBJ) weighting schemes. The theoretical results utilize novel applications of concentration inequalities for functional data and demonstrate that, similar to univariate functional data, the order of the $$N_{ij}$$ relative to $$p$$ and $$n$$ divides high-dimensional functional data into three regimes (sparse, dense, and ultra-dense), with the high-dimensional parametric convergence rate of $$\left\{\log(p)/n\right\}^{1/2}$$ being attainable in the latter two.
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Basis expansions for functional snippets
Summary Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. We investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly, and often much, shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed estimator enjoys a convergence rate that is adaptive to the smoothness of the underlying covariance function, and has superior finite-sample performance in simulation studies.
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- Award ID(s):
- 1914917
- PAR ID:
- 10287093
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 108
- Issue:
- 3
- ISSN:
- 0006-3444
- Page Range / eLocation ID:
- 709 to 726
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/biomet/asaa088
