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Abstract:
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In fitting linear regression models for functional data, a complicating factor with regressors as random curves is that regression estimators have complex distributions, due to issues in bias and scaling. Bias arises because the target slope function is infinite-dimensional, while finite-sample estimators necessarily involve truncations. To approximate sampling distributions, we develop a residual bootstrap method. Despite the parametric regression problem, the bootstrap for functional data requires a development that resembles resampling for nonparametric regression with multivariate regressors. Essentially, original- and bootstrap-data estimators require coordination in the truncation levels to remove bias (akin to tuning parameter choices). The resulting bootstrap has wide applicability for constructing both confidence and prediction regions at target regressor points, and with coverage properties even holding conditionally on data regressors; the method also extends to simultaneous regions. Establishment of the bootstrap further involves correcting and generalizing a foundational central limit theorem for functional linear regression.
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