Abstract
This survey provides an overview of optimal estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence. Based on observations of the sequence without noise as well as observations of the sequence with a stationary noise, estimates could be obtained. Formulas for calculating the spectral characteristics and the mean-square errors of the optimal estimates of functionals are derived in the case of spectral certainty, where spectral densities of the sequences are exactly known. In the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics of estimates are presented for some special classes of admissible spectral densities.
Highlights
Theory of estimation of the unknown values of stationary stochastic processes based on a set of observations of the processes plays an important role in many practical applications
The maximum error gives a moving average stationary sequence which is least favourable in the given class of stationary sequences
IAnNthξi=s s∑ectNji=o0n we propose a(j)ξ(j) and aAmξ e=th∑od∞ jo=f0 solution a(j)ξ(j) of the mean square optimal linear estimation of the functionals which depend on the unknown values of a stationary stochastic sequence ξ(j) from the class Ξ of stationary stochastic sequences satisfying the conditions Eξ(j) = 0, E|ξ(j)|2 ≤
Summary
Theory of estimation of the unknown values of stationary stochastic processes based on a set of observations of the processes plays an important role in many practical applications. Franke [9, 10], Franke and Poor [11] investigated the minimax extrapolation and filtering problems for stationary sequences with the help of convex optimization methods This approach makes it possible to find equations that determine the least favourable spectral densities for various classes of admissible densities. Are outside the unit disk D = {z : |z| < 1}, equation (10) has a stationary solution which can be represented as one-sided moving average sequence In this case the correlation function R(n) is of the form (12).
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