Minimax interpolation of stochastic processes with stationary increments from observations with noise

Abstract

We deal with the problem of optimal estimation of the linear functionals constructed from the missed values of a continuous time stochastic process ξ(t) with periodically stationary increments at points t ∈ [0; (N + 1)T ] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions {ξj , j ∈ Z} or corresponding to it an (infinite dimensional) vector stationary increment sequence { ξIn the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

“…The least favorable spectral densities F 0 (λ), G 0 (λ) in the classes D 0 × D ε for the optimal linear interpolation of the functional A s ⃗ ξ are determined by relations (19), (20) for the first pair D 1 0 × D 1 ε of sets of admissible spectral densities; (21), (22) for the second pair D 2 0 × D 2 ε of sets of admissible spectral densities; (23), (24) for the third pair D 3 0 × D 3 ε of sets of admissible spectral densities; (25), (26) for the fourth pair D 4 0 × D 4 ε of sets of admissible spectral densities; the minimality condition (1); the constrained optimization problem (15) and restrictions on densities from the corresponding classes D 0 × D ε . The minimax-robust spectral characteristic of the optimal estimate of the functional A s ⃗ ξ is determined by the formula (8).…”