Regression with Functional Errors-in-Predictors: A Generalized Method-of-Moments Approach
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Functional regression is an important topic in functional data analysis. Traditionally, one often assumes that samples of the functional predictor are independent realizations of an underlying stochastic process, and are observed over a grid of points contaminated by independent and identically distributed measurement errors. In practice, however, the dynamical dependence across different curves may exist and the parametric assumption on the measurement error covariance structure could be unrealistic. In this paper, we consider functional linear regression with serially dependent functional predictors, when the contamination of predictors by the measurement error is "genuinely functional" with fully nonparametric covariance structure. Inspired by the fact that the autocovariance operator of observed functional predictors automatically filters out the impact from the unobservable measurement error, we propose a novel autocovariance-based generalized method-of-moments estimate of the slope parameter. The asymptotic properties of the resulting estimators under different functional scenarios are established. We also demonstrate that our proposed method significantly outperforms possible competitors through intensive simulation studies. Finally, the proposed method is applied to a public financial dataset, revealing some interesting findings.
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