Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

TL;DR

This paper develops a minimax-robust method for forecasting sequences with complex, periodically stationary long-memory patterns, providing formulas for optimal estimation and robustness against spectral density uncertainties.

Contribution

It introduces a new class of stochastic sequences with generalized increments and derives minimax-robust estimation formulas for them.

Findings

Formulas for optimal linear estimates with known spectral densities.

Methods for calculating mean square errors and spectral characteristics.

Robust estimation formulas when spectral densities are uncertain.

Abstract

We introduce stochastic sequences $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequences $\zeta(k)$ based on their observations at points $ k<0$. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of sequences are not exactly known while some sets of admissible spectral densities are given.