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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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