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Rates of strong uniform consistency for multivariate kernel density estimators. (Vitesse de convergence uniforme presque sûre pour des estimateurs à noyaux de densités multivariées). (English) Zbl 1011.62034

Summary: Let \(f_n\) denote the usual kernel density estimator in several dimensions. It is shown that if \(\{a_n\}\) is a regular band sequence, \(K\) is a bounded square integrable kernel of several variables, satisfying some additional mild conditions, and if the data consist of an i.i.d. sample from a distribution possessing a bounded density \(f\) with respect to the Lebesgue measure on \(\mathbb{R}^d\), then \[ \limsup_{n\to\infty} (na^d_n/ \log a_n^{-1})^{1/2} \sup_{t\in\mathbb{R}^d} \bigl|f_n(t)- Ef_n(t)\bigr|\leq C\sqrt{ \|f\|_\infty \int K^2(x)dx}\quad \text{a.s.} \] for some absolute constant \(C\) that depends only on \(d\). With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to \(\sqrt{2d \|f\|_\infty \int K^2(x)dx}\). Convergence of the moment generating functions is also proved. Neither of these results requires \(f\) to be strictly positive. These results improve upon, and extend to several dimensions, results by B.W. Silverman [Ann. Stat. 6, 177-184 (1978; Zbl 0376.62024)] for univariate densities.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions
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