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The problem of optimal estimation of a linear functional A N ξ = N k=0 a(k)ξ(k) that depends on unknown values of a stochastic sequence {ξ(m), m ∈ Z} with stationary increments of order n by observations of the sequence at points m ∈ Z \ {0, 1,. .. , N} is considered. Formulas for calculating the mean square error and spectral characteristic of the optimal linear estimator of the above functional are derived in the case where the spectral density is known. In the case where the spectral density is not known, but a set of admissible spectral densities is given, the minimax-robust approach is applied to the problem of optimal estimation of a linear functional. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for a given set of admissible spectral densities.

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We consider a stochastic sequence ξ(m) with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional Aξ = ∞ k=0 a(k)ξ(−k). Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.

The problem of optimal estimation of the linear functionals Aξ = ∑ ∞ k=0 a(k)ξ (k) andwhich depend on the unknown values of a stochastic sequence ξ(m) with stationary nth increments is considered. Estimates are obtained which are based on observations of the sequence ξ(m) + η(m) at points of time m = −1, −2, . . ., where the sequence η(m) is stationary and uncorrelated with the sequence ξ(m). Formulas for calculating the mean-square errors and the spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences ξ(m) and η(m) are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities.

The problem of optimal estimation of the linear functional A = ∑ ∞ k=0 a(k) (−k) depending on the unknown values of a stochastic sequence (m) with nth stationary increments from observations of the sequence (m) + (m) at points m = 0, −1, −2, …, where (m) is a stationary sequence uncorrelated with (m), is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are derived in the case where spectral densities of stochastic sequences are exactly known and admit the canonical factorizations. In the case of spectral uncertainty, where spectral densities are not known exactly, but sets of admissible spectral densities are specified, the minimax-robust method is applied. Formulas and relations that determine the least favourable spectral densities and the minimax-robust spectral characteristics are proposed for the given sets of admissible spectral densities. The filtering problem for a class of cointegrated sequences is investigated.The crucial assumption of application of traditional methods of finding solution to the filtering problem for random processes is that spectral densities of the processes are exactly known. However, in practical situations complete information on spectral densities is impossible and the established results cannot be directly applied to practical filtering problems. This is a reason to apply the minimax-robust method of filtering and derive the minimax estimates since they minimize the maximum value of the mean-square errors for all spectral densities from a given set of admissible densities simultaneously. In this article, we deal with the problem of optimal estimation of functionals depending on the unknown values of a random process with stationary increments based on observations of the process and a noise. In the case where spectral densities of the processes are not exactly known, relations for determining least favourable spectral densities and minimax-robust spectral characteristics are proposed.

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We introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$  based on its  observations at points $ k<0$. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.