We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. Further, we provide sufficient conditions for ‐consistency this result is extended to piecewise Hölder regression with non‐Gaussian errors. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. 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. Finally, to cater for the demands of the application, we have developed a unified R package, named VarED, that integrates the existing difference-based estimators and the unified estimators in nonparametric regression and have made it freely available in the R statistical program. Using both theory and simulations, we recommend to use the ordinary difference sequence in the unified framework, no matter if the sample size is small or if the signal-to-noise ratio is large. More importantly, the unified framework has also provided a smart way to solve the challenging difference sequence selection problem that remains a long-standing controversial issue in nonparametric regression for several decades. The unified framework has greatly enriched the existing literature on variance estimation that includes most existing estimators as special cases. In this paper, we propose a unified framework for variance estimation that combines the linear regression method with the higher-order difference estimators systematically. Difference-based methods do not require estimating the mean function in nonparametric regression and are therefore popular in practice.
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