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.

翻译：本文讨论了一类基于差分的自协方差结构估计量，适用于信号不连续且误差序列相关的半参数回归模型。该模型中的信号由含跳跃的函数与可识别光滑函数之和构成。该模型的简化形式此前已被研究，称为非参数跳跃回归模型（NJRM）。所提出的估计量能够避免此类模型中均值信号先验估计这一复杂问题。我们给出了误差服从高斯分布时估计量偏差与方差的有限样本表达式。上述高斯性假设仅为获得估计量偏差与方差的显式闭式解所需。此外，我们观察到方差估计量的均方误差既不依赖于作为均值信号组成部分的未知光滑函数，也不受差分序列系数值的影响。我们的方法还提出了所提估计量达到$\sqrt{n}$-相合性的充分条件。