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Semiparametric estimation for seasonal long-memory time series using generalized exponential models. (English) Zbl 1247.62224

Summary: We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [J. Beran, Biometrika 80, No. 4, 817–822 (1993; Zbl 0800.62521)] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the P. Bloomfield model [Biometrika 60, 217–226 (1973; Zbl 0261.62074)], which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [H.L. Gray, N.-F. Zhang and W.A. Woodward, J. Time Ser. Anal. 10, No. 3, 233–257 (1989; Zbl 0685.62075)] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
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