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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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