A semiparametric approach to estimate two seasonal fractional parameters in the SARFIMA model

Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations

“…where ω ∈ (−π, π] and g ψ (ω) is a continuous function, bounded above and away from zero and ω ι j 0 are poles for j = 1, ..., ξ ι , ι = 1, ..., L. The autocovariance function of X t behaves like γ X (h) ∼ K j 2d ι −1 cos( jω) as h → ∞ and K is a constant that does not depend on h. See, for example, Giraitis and Leipus [17], Palma [36], Arteche [2], Arteche and Robinson [3], Reisen et al [48] and references therein. For suitable choices of the fractionally differencing parameters d ι , ι = 1, ..., L, X t may have a finite number of zeros or singularities of order d 1 ,...,d L on the unit circle which allows the modeling of long and short memory data containing seasonal periodicities.…”

“…Reisen et al [47] estimated the fractional and seasonal parameters of SARFIMA models by means of a semiparametric procedure, considering a nonconstant conditional error variance. Reisen et al [48] studied the properties of the SARFIMA model when the data exhibits one and two seasonal periods and short-memory components. Ye et al [58] proposed a new method to estimate the fractional difference parameter in the SARFIMA model using tapered periodogram.…”

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

“…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]). Such models are used when analyzing data which arise in different field of economics, finance, climatology, air pollution, signal processing.…”

“…In this article, 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.…”

“…The spectral characteristic ⃗ h µ,N (λ) = {h µ,N,p (λ)} Tp=1 and the value of the mean square error ∆(f ; A M ζ) are calculated by formulas (42) and (43) respectively. In the case where the spectral density matrix f (λ) admits the canonical factorization(38), then the spectral characteristic ⃗ h µ,N (λ) and the value of the mean square error of the estimate A M ζ can be calculated by formulas(44) and (45) respectively.As a corollary from Theorem 3.4, one can obtain the mean square optimal estimate of the unobserved value ζ(M ), M ≥ 0 of a stochastic sequence ζ(m) with periodically stationary GM increments based on observations of the sequence ζ(m) at points m = −1, −2, . .…”