Giné, Evarist; Guillou, Armelle 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 Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 6, 907-921 (2002). 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. Cited in 105 Documents MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 60F15 Strong limit theorems 62G30 Order statistics; empirical distribution functions Keywords:uniformly almost sure rates; kernel density estimators; general upper bound; exact bound Citations:Zbl 0376.61024; Zbl 0376.62024 PDFBibTeX XMLCite \textit{E. Giné} and \textit{A. Guillou}, Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 6, 907--921 (2002; Zbl 1011.62034) Full Text: DOI Numdam EuDML