Minimax-Robust Estimation Problems for Stationary Stochastic Sequences

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.

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

“…This method allows us to find estimates that minimize the maximum values of the mean-square errors of the estimates for all spectral density matrices from a given class of admissible spectral density matrices. For description of the minimax method we introduce the following definitions (see Moklyachuk [23,24], and Moklyachuk and Masytka [28]).…”

“…Note, that the form of the functional ∆(h(F 0 , G 0 ); F, G) is convenient for application of the Lagrange method of indefinite multipliers for finding solution to the problem (18). Making use of the method of Lagrange multipliers and the form of subdifferentials of the indicator functions we describe relations that determine the least favourable spectral densities in some special classes of spectral densities (see books by Moklyachuk [22,23], Moklyachuk and Masyutka [28] for additional details).…”

“…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).…”

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

“…This approach makes it possible to find equations that determine the least favorable spectral densities for some classes of admissible densities. In the papers by Moklyachuk [22,23] results of investigation of the extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences are described. The problem of estimation of functionals which depend on the unknown values of multivariate stationary stochastic processes is the aim of the papers by Moklyachuk and Masyutka [25] - [27].…”

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

“…In papers by Moklyachuk [20] - [22] the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences. For more results and details see, for example, book by Moklyachuk [29], articles and book by Moklyachuk and Masyutka [26] − [31]. Dubovets'ka and Moklyachuk [3] − [8] investigated the minimax-robust estimation problems (extrapolation, interpolation and filtering) for periodically correlated stochastic processes.…”