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Weakly adaptive estimators in explosive autoregression. (English) Zbl 0709.62083
Let $$X(t)=\alpha \cdot X(t-1)+\epsilon (t)$$, $$t\geq 1$$, be an AR[1]- process with $$| \alpha | >1$$. The $$\epsilon$$ (t) are independent and have a common unknown density $$\psi$$. The paper establishes a set of sufficient conditions on $$\psi$$ under which the log-likelihood process $\Lambda_ n(t| \psi)=\log (dP^ n_{\alpha +\delta (n)\cdot t,\psi}/dP^ n_{\alpha,\psi})$ has a nondegenerate limit $$\Lambda$$ (t$$| \psi)$$ when the shrinking factor $$\delta$$ (n) is suitably choosen. This limit is neither LAN nor LAMN in general. Under some additional moment conditions the sequence of weakly adaptive Pitman-type estimators ${\hat \alpha}_ n=\int u\prod^{n}_{i=k+1}{\hat \psi}_ n(X_ i-uX_{i-1})du/\int \prod^{n}_{i=k+1}{\hat \psi}_ n(X_ i-uX_{i-1})du,$ where $${\hat \psi}$$ is a kernel type density estimate, is shown to be asymptotically minimax for the limit $$\Lambda$$ (t$$| \psi)$$ under a weighted squared error loss function.
Reviewer: R.Schlittgen

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62F12 Asymptotic properties of parametric estimators
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