Title: Regression with genuinely functional errors-in-covariates,
Abstract: Contamination of covariates by measurement error is a classical problem in
multivariate regression, where it is well known that failing to account for
this contamination can result in substantial bias in the parameter estimators.
The nature and degree of this effect on statistical inference is also
understood to crucially depend on the specific distributional properties of the
measurement error in question. When dealing with functional covariates,
measurement error has thus far been modelled as additive white noise over the
observation grid. Such a setting implicitly assumes that the error arises
purely at the discrete sampling stage, otherwise the model can only be viewed
in a weak (stochastic differential equation) sense, white noise not being a
second-order process. Departing from this simple distributional setting can
have serious consequences for inference, similar to the multivariate case, and
current methodology will break down. In this paper, we consider the case when
the additive measurement error is allowed to be a valid stochastic process. We
propose a novel estimator of the slope parameter in a functional linear model,
for scalar as well as functional responses, in the presence of this general
measurement error specification. The proposed estimator is inspired by the
multivariate regression calibration approach, but hinges on recent advances on
matrix completion methods for functional data in order to handle the nontrivial
(and unknown) error covariance structure. The asymptotic properties of the
proposed estimators are derived. We probe the performance of the proposed
estimator of slope using simulations and observe that it substantially improves
upon the spectral truncation estimator based on the erroneous observations,
i.e., ignoring measurement error. We also investigate the behaviour of the
estimators on a real dataset on hip and knee angle curves during a gait cycle.