## Efficient estimation of seasonal long-range dependent processes.(English)Zbl 1097.62092

A class of Gaussian seasonal long-memory processes is considered with the spectral density $f(\omega)=H(\omega)| \omega| ^{-\alpha}\prod_{i=1}^r \prod_{j=1}^m| \omega-\omega_{ij}| ^{-\alpha_i},$ where $$H$$ is symmetric, strictly positive, continuous and bounded, and $$\omega_{ij}$$ are known poles. $$f$$ is assumed to be known up to a parameter $$\vartheta$$ from a finite-dimensional compact set. Generalized ARMA and seasonal fractionally integrated ARMA models satisfy this specification. Consistency, asymptotic normality and Fisher’s efficiency of the maximum likelihood estimate for $$\vartheta$$ are demonstrated. Results of simulations and application to Internet traffic data are presented.

### MSC:

 62M15 Inference from stochastic processes and spectral analysis 62F12 Asymptotic properties of parametric estimators 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F10 Point estimation
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### References:

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