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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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