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Whittle estimator for finite-variance non-Gaussian time series with long memory. (English) Zbl 0945.62085
Let \(\{X_t\}\) be a zero-mean Gaussian time series with long memory. Then the series has spectral density \[ f(x)= |x|^{-\alpha} L(1/|x|),\quad x\in [-\pi,\pi],\;0<\alpha<1, \] where \(L\) is a slowly varying function at infinity. Define \(Y_t =G(X_t)\) where \(G\) is a polynomial. Assume that \(EY_t=0\) and that \(Y_t\) has spectral density \[ s_{\theta}(x)= \sigma^2 |x|^{-\alpha_G(\theta)} L_{G,\theta}(1/|x|),\quad \sigma>0,\;0\leq\alpha_G(\theta)<1, \] \(L_{G,\theta}\) is a slowly varying function and \(\theta\) belongs to a compact subset of \(R^p\). Let \(\hat{\theta}_N\) be the Whittle estimator of the true value of the parameter \(\theta=\theta_0\) based on \(Y_1,\dots, Y_N\).
The authors show that under some general assumptions \(\hat{\theta}_N\) is a consistent estimator of \(\theta_0\). If \(Y_t\) is Gaussian, then \(\sqrt N(\hat{\theta}_N-\theta_0)\) is asymptotically Gaussian. If \(Y_t\) is not Gaussian, then \(\sqrt N\) consistency of the Whittle estimator may not hold and the limit may not be Gaussian, even if \(Y_t\) has short memory.
Reviewer: J.Anděl (Praha)

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
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