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.

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

“…Having canonical factorizations (35) and (36) we can calculate the values of mean-square errors ∆(h µ (f, g); f, g) and spectral characteristics h µ (f, g) by formulas (42), (41) respectively. However, such situation does not appear in practice since we do not know exactly spectral densities of the observed sequences.…”

“…Statement 2) comes from the relation Ψ µ Θ µ = Θ µ Ψ µ = I, which we need to prove. From factorizations (34) and (35) one can obtain…”

“…Thus the operator C µ = C 1 is determined by the matrix with elements (35) are found using the equality Ψ 1 Θ 1 = I from the proof of Lemma 3 putting θ 1 (0) = 1, θ 1 (1) = −x, θ 1 (2) = −y and θ 1 (l) = 0 for l ≥ 3:…”

“…Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions δ(f, g|D f × D g ) of the set D f × D g of spectral densities we describe relations that determine least favourable spectral densities in some special classes of spectral densities (see books [15,33,35] for additional details). …”

Filtering Problem for Functionals of Stationary Sequences

“…In the papers by Moklyachuk [23] - [28], Moklyachuk and Sidei [31] extrapolation, interpolation and filtering problems for stationary stochastic processes and sequences are investigated. The corresponding problems for vector-valued stationary sequences and processes are investigated by Moklyachuk and Masyutka [29], [30].…”