The problem of optimal estimation of linear functional ${{A}_{N}}\xi =\sum\limits_{k=0}^{N}{a(k)\xi (k)}\,$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary $n$-th increments from observations of the sequence $\xi (k)$ at points $k=-1,-2,\ldots $ and of the sequence $\xi (k)+\eta (k)$ at points of time $k=N+1,N+2,\ldots $ is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences $\xi (m)$ and $\eta (m)$ are exactly known. Minimax (robust) method of estimation is applied in the case where the spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some specific sets of admissible densities.

翻译：本文研究线性泛函 ${{A}_{N}}\xi =\sum\limits_{k=0}^{N}{a(k)\xi (k)}\,$ 的最优估计问题，该泛函依赖于具有 $n$ 阶平稳增量的随机序列 $\xi (m)$ 的未知值。观测数据包括：序列 $\xi (k)$ 在时刻 $k=-1,-2,\ldots $ 的取值，以及序列 $\xi (k)+\eta (k)$ 在时刻 $k=N+1,N+2,\ldots $ 的取值。在谱确定性条件下（即序列 $\xi (m)$ 和 $\eta (m)$ 的谱密度精确已知），本文提出了计算最优线性估计均方误差及谱特征的公式。当谱密度未能精确已知而仅给定容许谱密度集合时，采用极小极大（鲁棒）估计方法。针对若干特定容许密度集合，本文给出了确定最不利谱密度与极小极大谱特征的公式。