Filtering problem for sequences with periodically stationary multi-seasonal increments with spectral densities allowing canonical factorizations

Abstract

We consider a stochastic sequence $\xi(m)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional $A{\xi}=\sum_{k=0}^{\infty}{a}(k) {\xi}(-k)$. Formulas that determine the least favourable spectral densities and the minimax (robust) spectralcharacteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.