STATISTICS | RESEARCH ARTICLE Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences

1 * Abstract: The problem of optimal estimation of the linear functional A� = ∑∞ a(k)�(−k) depending on the unknown values of a stochastic sequence �(m) with nth stationary increments from observations of the sequence �(m )+ �(m) at points m = 0, −1, −2, …, where �(m) is a stationary sequence uncorrelated with �(m), is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are derived in the case where spectral densities of stochastic sequences are exactly known and admit the canonical factorizations. In the case of spectral uncertainty, where spectral densities are not known exactly, but sets of admissi- ble spectral densities are specified, the minimax-robust method is applied. Formulas and relations that determine the least favourable spectral densities and the minimax-robust spectral characteristics are proposed for the given sets of admissible spectral densities. The filtering problem for a class of cointegrated sequences is investigated.