Minimax-robust prediction 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) 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.

Extrapolation Problem for Stationary Sequences with Missing Observations

“…Let ξ(j) be a stationary stochastic sequence with the spectral density f (λ) which satisfy the minimality condition (20). The spectral characteristic h N (e iλ ) and the mean-square error ∆(h N , f ) of the optimal linear estimate N ξ of the functional A N ξ can be calculated by the formulas (23), (24) …”

“…Let the density f 0 (λ) ∈ D W and satisfy the minimality condition (20). The spectral density f 0 (λ) is the least favorable in the class D W for the optimal extrapolation of the functional Aξ if…”

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

“…In the book by Moklyachuk and Golichenko [24] results of investigation of the interpolation, extrapolation and filtering problems for periodically correlated stochastic sequences are proposed. In their papers Luz and Moklyachuk [18] - [20] deal with the problems of estimation of functionals which depend on the unknown values of stochastic sequences with stationary increments. Prediction problem for stationary sequences with missing observations is investigated in papers by Bondon [1,2], Cheng, Miamee and Pourahmadi [5], Cheng and Pourahmadi [6], Kasahara, Pourahmadi and Inoue [15], Pourahmadi, Inoue and Kasahara [33], Pelagatti [32].…”

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

“…See the book by Golichenko and Moklyachuk (2014) for more relative results and references. The minimax-robust extrapolation, interpolation and filtering problems for stochastic sequences and processes with nth stationary increments were solved by Luz and Moklyachuk (2012, 2015a, 2015b, 2015c, 2016a, 2016bMoklyachuk & Luz, 2013). The obtained results are applied to find solution of the extrapolation and filtering problems for cointegrated sequences Moklyachuk, 2014b, 2015c).…”