Parametric rates of nonparametric estimators and predictors for continuous time processes. (English) Zbl 0885.62041

Bosq, Denis

Ann. Stat. 25, No. 3, 982-1000 (1997).

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)

Keywords:

prediction; density; large deviation inequality; Borel-Cantelli lemma; parametric rate