Robust Forecasting of Sequences with Periodically Stationary Long Memory Multiplicative Seasonal Increments Observed with Noise and Cointegrated Sequences

TL;DR

This paper develops robust methods for optimal estimation of linear functionals from sequences with periodically stationary long memory and seasonal increments, observed with noise, providing formulas for mean square errors and spectral characteristics.

Contribution

It introduces formulas for calculating optimal estimates and their errors, and proposes minimax spectral characteristics for cases with uncertain spectral densities.

Findings

Derived formulas for mean square errors of estimates.

Provided spectral characteristics of optimal estimators.

Developed minimax spectral strategies for uncertain spectral densities.

Abstract

The problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequence with periodically stationary increments based on observations of the sequence with a periodically stationary noise is considered. 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 the 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 the sequence are not exactly known while some sets of admissible spectral densities are given.