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A minimum distance estimator for first-order autoregressive processes. (English) Zbl 0607.62102
Consider the first-order stationary autoregressive model \(X_ k=\beta X_{k-1}+U_ k\), \(| \beta | <1\), where \(\{U_ k\}\) are i.i.d. \((0,\sigma^ 2)\). The paper proposes a minimum distance Cramér-von Mises-type estimator of \(\beta\). Consider the empirical process \[ S(t,\Delta)=\sum^{n}_{k=1}X_{k-1}I(X_ k\leq t+\Delta X_{k-1}) \] and let \(Q(\Delta)=\int^{\infty}_{-\infty}S^ 2(t,\Delta)dH(t)\), where H is a finite measure on (\({\mathbb{R}},{\mathcal B})\). Define an estimator \({\hat \beta}\) by Q(\({\hat \beta}\))\(=\inf_{\Delta} Q(\Delta)\). Then it is proved that \(\sqrt{n}({\hat \beta}-\beta)\) is asymptotically normal under appropriate assumptions. Some results of independent interest are also established in the course of the proofs.
Reviewer: P.A.Morettin
MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
60G10 Stationary stochastic processes
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