Abstract
We introduce a stochastic sequence $\zeta(k)$ 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 the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. 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 estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the 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.
Highlights
A variety of non-stationary and long memory time series models are introduced and investigated by researchers in the past decade
Since the book by Box and Jenkins (1970), autoregressive moving average (ARMA) models integrated of order d are standard models used for time series analysis
We present results of investigation of stochastic sequences with periodically stationary long memory multiple seasonal increments motivated by articles by Dudek [8], Gould et al [14] and Reisen et al [44], who considered models with multiple seasonal patterns for inference and forecasting, and Hurd and Piparas [23], who introduced two models of periodic autoregressive time series with multiple periodic coefficients
Summary
A variety of non-stationary and long memory time series models are introduced and investigated by researchers in the past decade (see, for example, papers by Dudek and Hurd [9], Johansen and Nielsen [24], Reisen et al.[44]). We present results of investigation of stochastic sequences with periodically stationary long memory multiple seasonal increments motivated by articles by Dudek [8], Gould et al [14] and Reisen et al [44], who considered models with multiple seasonal patterns for inference and forecasting, and Hurd and Piparas [23], who introduced two models of periodic autoregressive time series with multiple periodic coefficients. We present definition, justification and a brief review of the spectral theory of stochastic sequences with periodically stationary multiple seasonal increments. This type of stochastic sequences will allow us to deal with a wide range of non-stationarity in time series analysis. Theorem 2.2 The structural function Ds(d)(m; μ1, μ2) of the vector-valued stochastic stationary GM increment sequence χμ(d,s) (ξ⃗(m)) can be represented in the form.
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