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Bootstrapping unstable first order autoregressive process with errors in the domain of attraction of stable law. (English) Zbl 0919.62110

Summary: We consider a first order autoregressive process \(X_t= \varphi X_{t-1} +\varepsilon_t\) where \(\{\varepsilon_t\}\) are independent and identically distributed random errors which belong to the domain of attraction of a stable law with index \(\alpha\in (0,2]\). We show that when \(\varphi =1\), Efron’s bootstrap of the \(M\)-estimator of \(\varphi\) is asymptotically valid if and only if \(E(\varepsilon^2_1) =\infty\). Also we show that for \(0< \alpha<2\) the usual bootstrap is asymptotically invalid for the least-squares estimate. A modification of the resampling sample size to \(m_n= o(n)\) makes the bootstrap asymptotically valid for all \(\alpha\in (0,2]\).

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
60E07 Infinitely divisible distributions; stable distributions
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