Clustering rates and Chung type functional laws of the iterated logarithm for empirical and quantile processes

Following the works of Berthet (1997), we first obtain exact clustering rates in the functional law of the iterated logarithm for the uniform empirical and quantile processes and for their increments. In a second time, we obtain functional Chung-type limit laws for the local empirical process for a class of target functions on the border of the Strassen set

Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process

International audienceWe establish a Glivenko-Cantelli and a Donsker theorem for a class of random discrete measures which generalize the empirical measure, under conditions on uniform entropy numbers that are common in empirical processes theory. Some illustrative applications in nonparametric Bayesian theory and randomly sized sampling are provided

Empirical likelihood confidence bands for mean functions of recurrent events with competing risks and a terminal event

In this paper, we study recurrent events with competing risks in the presence of a terminal event and a censorship. We focus our attention on the mean functions which give the mean number of events of a specific type that have occured up to a time $t$. Using empirical likelihood ratio techniques, we are able to build confidence bands for these functions. We have a data set of nosocomial infections in an intensive care unit of a french hospital. For each patient, we know if and when he caught an infection, what infection it was (septicemia, urinary tract infection...), if and when he died and when he left the hospital. Our model fits this context and will be used to build confidence bands for one type of nosocomial infection and even a confidence tube for two types

A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed $h>0$ and $z\in \mathbb{R}^d$: G_n(K,h,z):=\sum_{i=1}^n K \Bigl(\frac{Z_i-z}{h^{1/d}}\Big) - \mathbbE \Bigl(K \Bigl(\frac{Z_i-z}{h^{1/d}}\Big)\Big), where the $Z_i$ are i.i.d. on $\mathbb{R}^d$. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\mathcal{G}$ and on the law of $Z_1$, we establish a uniform functional limit law for the collections of processes $\bigl\{G_n(\cdot,h_n,z), z\in H, h\in [h_n,\mathfrak{h}_n]\big\}$, where $H\subset \mathbb{R}^d$ is a compact set with nonempty interior and where $h_n$ and $\mathfrak{h}_n$ satisfy the Cs\"{o}rg\H{o}-R\'{e}v\'{e}sz-Stute conditions.Comment: Published in at http://dx.doi.org/10.1214/08-EJS193 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org