Analytic Basis Expansions for Functional Snippets
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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. In this paper, we investigate mean and covariance estimations 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 an analytic basis representation of the covariance function. The proposed approach allows one to consistently estimate an infinite-rank covariance function from functional snippets. Moreover, the convergence rate is shown to be nearly parametric. This unusually high convergence rate is attributed to the analytic assumption on the covariance function, which is imposed to identify the covariance function. We also illustrate the finite-sample performance via simulation studies and an application to spinal bone mineral density.
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