Autocovariance Estimation in Regression with a Discontinuous Signal and m-Dependent Errors: A Difference-Based Approach

Abstract

We discuss a class of difference-based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m-dependent process. These estimators circumvent the particularly challenging task of pre-estimating such an unknown regression function. We provide finite-sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased-optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for math formula-consistency; this result is extended to piecewise Hölder regression with non-Gaussian errors. We combine our biased-optimized autocovariance estimates with a projection-based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.