Abstract
The problem of optimal estimation of the linear functionals Aξ = ∫ ∞ 0 a(t)ξ(t)dt and AT ξ = ∫ T 0 a(t)ξ(t)dt depending on the unknown values of stochastic process ξ(t), t ∈ R, with stationary nth increments from observations of the process at points t < 0 is considered. Formulas for calculating the mean square error and the spectral characteristic of optimal linear estimates of the functionals are derived in the case where the spectral density of the process is exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the process is not exactly known, but a set of admissible spectral densities is given.
Highlights
Estimation of unknown values of stochastic processes is an important part of the theory of stochastic processes
Effective methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes were developed by Kolmogorov [1], Wiener[2], Yaglom [3, 4]
In papers by Moklyachuk [18] - [21] the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences
Summary
Estimation of unknown values of stochastic processes is an important part of the theory of stochastic processes. The spectral representation for stationary increments and canonical factorization for spectral densities were received, the problem of linear extrapolation of unknown value of stationary stochastic increment from observation of the process was solved Further results for such stochastic processes were presented by Pinsker [8], Yaglom and Pinsker [9]. In papers by Moklyachuk [18] - [21] the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences. Dubovets’ka and Moklyachuk [22] – [28] investigated the minimax-robust estimation problems (extrapolation, interpolation and filtering) for the linear functionals which depend on unknown values of periodically correlated stochastic processes. Theorem 3 A stochastic stationary increment process ξ(n)(t, τ ) is regular if and only if there exists an innovatti≥on0p}r,o∫c0e∞ss|φ{(ηn()t()t,:τt)|∈2dtR
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