Interpolation of functionals of stochastic sequences with stationary increments

Abstract

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.

Filtering Problem for Functionals of Stationary Sequences

“…Suppose that canonical factorizations (8), (19) hold true and let the linear operators Φ and Υ be defined in the same way as in lemma 2. Define the linear operator T in the space ℓ 2 by the matrix with elements (T) l,k = s(l − k), l, k ≥ 0, where the coefficients s(k), k ≥ 0, are defined in (20).…”

“…Such approach to filtering problems for stochastic sequence with stationary nth increments was applied in the papers by Luz and Moklyachuk [20], [21]. The method of projection in the Hilbert space is also applied to extrapolation and interpolation problems for stochastic sequences with stationary nth increments in papers by Luz and Moklyachuk [19], [22].…”

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

“…Using these Fourier coefficients we can represent equation (20) in terms of the system of linear equations…”

“…. is calculated by formula (19). The spectral characteristic h µ (λ) of the optimal estimate Aξ is calculated by formula (22).…”

“…The minimax interpolation problem for the linear functional A N ξ = ∑ N k=0 a(k)ξ(k) which depends on the unknown values of the sequence ξ(k) based on observations with and without noise was investigated in papers [19,20], and for the linear functional Aξ = ∫ ∞ 0 a(t)ξ(t)dt which depends on the unknown values of a random process ξ(t) in the paper [24].…”