Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences

Abstract

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

Filtering of multidimensional stationary sequences with missing observations

“…Let the function F 0 (λ) + G(λ) satisfy the minimality condition (1). The spectral density F 0 (λ) is the least favorable in the classes D k 0 , k = 1, 4, for the optimal linear filtering of the functional A ξ if it satisfies relations (22), (25), (28), (31), respectively, and the pair (F 0 (λ), G(λ)) is a solution of the optimization problem (19). The minimax-robust spectral characteristic of the optimal estimate of the functional A ξ is determined by formula (11).…”

Interpolation Problem for Stationary Sequences with Missing Observations

“…If the sequence a(j), j ∈ S, is strictly positive and coefficients r k = r −k = P a(k)a −1 (0), k ∈ S, satisfy the inequality (22), then the least favourable in the class D u v spectral density for the optimal linear estimate of the functional A s ξ is density (17) of the autoregression stochastic sequence of order M s−1 + N s . The minimax characteristic h(f 0 ) of the estimate can be calculated by the formula (18).…”

“…The minimax characteristic h(f 0 ) of the estimate can be calculated by the formula (18). If the inequality (22) is not satisfied, then the least favourable spectral density in D u v is determined by relation (23) and the extremum condition (14). The minimax characteristic of the estimate is calculated by formula (11).…”

Filtering Problem for Random Processes with Stationary Increments

“…The minimax-robust extrapolation, interpolation and filtering problems for stochastic sequences and random processes with n th stationary increments are investigated by Luz and Moklyachuk Moklyachuk, 2012 -2015b;Moklyachuk and Luz, 2013). In particular, the minimax-robust filtering problem for such sequences is investigated in papers by Luz and Moklyachuk (2013b;2014b). Random processes with stationary increments with continuous time are considered in the articles by Luz and Moklyachuk (2014a;2015a), where the authors investigated the extrapolation and the interpolation problems.…”