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Parametric rates of nonparametric estimators and predictors for continuous time processes. (English) Zbl 0885.62041
Summary: We show that local irregularity of observed sample paths provides additional information which allows nonparametric estimators and predictors for continuous time processes to reach parametric rates in mean square as well as in a.s. uniform convergence. For example, we prove that under suitable conditions the kernel density estimator \(f_T\) associated with the observed sample path \((X_t,\;0\leq t\leq T)\) satisfies \[ \sup_{x\in\mathbb{R}} \bigl|f_T(x)- f(x)\bigr |= o\bigl( \ln_kT(\ln T/T)^{1/2} \bigr) \quad \text{a.s., } k\geq 1, \] where \(f\) denotes the unknown marginal density of the stationary process \((X_t)\) and where \(\ln_k\) denotes the \(k\) th iterated logarithm.
The proof uses a special Borel-Cantelli lemma for continuous time processes together with a sharp large deviation inequality. Furthermore the parametric rate obtained above is preserved by using a suitable sampling scheme.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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