Robust interpolation of sequences with periodically stationary multiplicative seasonal increments

TL;DR

This paper develops methods for robust interpolation of complex stochastic sequences with seasonal and fractional patterns, providing formulas for optimal estimation and robustness under spectral uncertainty.

Contribution

It introduces a new approach for interpolating sequences with periodically stationary fractional increments, including formulas for mean square errors and robust spectral characteristics.

Findings

Derived formulas for optimal linear interpolation of sequences with seasonal fractional increments.

Provided methods for calculating mean square errors and spectral characteristics.

Developed minimax spectral characteristics for uncertain spectral densities.

Abstract

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.