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Nonparametric estimation for long-range dependent sequences. (English) Zbl 1029.62035

Doukhan, Paul (ed.) et al., Theory and applications of long-range dependence. Boston, MA: Birkhäuser. 303-311 (2003).
Summary: We consider density estimation and regression problems for long-range dependent processes. We recall here limit theorems for stationary Gaussian subordinated and infinite moving average processes. Let \(f_n\) be a kernel functional estimate defined from a sequence of \(n\) observations of the process. We observe strong differences in the convergence of the estimate compared to the case of weak dependent processes. Denoting \(\varphi_n(x)\) the recentered and rescaled estimate \(\text{(Var} f_n(x))^{-1/2} [f_n(x)-f(x)]\), we have:
\(\bullet\) \(\varphi_n(x) @> d >> \sigma(x){\mathcal Z}\), for long-range dependent processes, where \(Z\) is a random variable independent of \(x\) so that the limiting process is degenerated.
\(\bullet\) \(\varphi_n(x) @>fidi>> \dot W(x)\) for weak dependent processes, where \(\dot W\) is a Gaussian process.
This difference may be a first step to a test of the dependence type of a process.
For the entire collection see [Zbl 1005.00017].

MSC:

62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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