ACF estimation via difference schemes for a semiparametric model with m-dependent errors

In this manuscript, we discuss a class of difference-based estimators of the autocovariance structure in a semiparametric regression model where the signal is discontinuous and the errors are serially correlated. The signal in this model consists of a sum of the function with jumps and an identifiable smooth function. A simpler form of this model has been considered earlier under the name of Nonparametric Jump Regression (NJRM). The estimators proposed allow us to bypass a complicated problem of prior estimation of the mean signal in such a model. We provide finite-sample expressions for biases and variance of the proposed estimators when the errors are Gaussian. Gaussianity in the above is only needed to provide explicit closed form expressions for biases and variances of our estimators. Moreover, we observe that the mean squared error of the proposed variance estimator does not depend on either the unknown smooth function that is a part of the mean signal nor on the values of difference sequence coefficients. Our approach also suggests sufficient conditions for $\sqrt{n}-$ consistency of the proposed estimators.