Minimax interpolation of sequences with stationary increments and cointegrated sequences

Abstract

We consider the problem of optimal estimation of the linear functional AN ξ = N k=0 a(k)ξ(k) depending on the unknown values of a stochastic sequence ξ(m) with stationary increments from observations of the sequence ξ(m) + η(m) at points of the set Z \ {0, 1, 2, . . . , N }, where η(m) is a stationary sequence uncorrelated with ξ(m). We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.Keywords Stochastic sequence with stationary increments, cointegrated sequences, minimax-robust estimate, mean square error, least favorable spectral density, minimax-robust spectral characteristic 2010 MSC 60G10, 60G25, 60G35, 62M20, 93E10, 93E11

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).…”

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

“…depending on unknown values of a stochastic process ξ(t), t ∈ R, with stationary increments from observations of the stochastic process ξ(t) + η(t) for t ∈ R \ [0; T ], where η(t), t ∈ R, is a stationary stochastic process uncorrelated with ξ(t). A similar problem is considered in the paper by Luz and Moklyachuk [15] for the discrete time.…”