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Empirical process of the squared residuals of an ARCH sequence. (English) Zbl 1012.62053
Summary: We derive the asymptotic distribution of the sequential empirical process of the squared residuals of an ARCH(p) sequence. Unlike the residuals of an ARMA process, these residuals do not behave in this context like asymptotically independent random variables, and the asymptotic distribution involves a term depending on the parameters of the model. We show that in certain applications, including the detection of changes in the distribution of the unobservable innovations, our result leads to asymptotically distribution free statistics.

MSC:
62G30 Order statistics; empirical distribution functions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20 Asymptotic properties of nonparametric inference
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
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