Adaptive estimation of the conditional intensity of marker-dependent counting processes

Abstract

International audienceWe propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a non asymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation

Nonparametric estimation for survival data with censoring indicators missing at random

“…The proof follows the lines of Corollary 1 (p.1178) in Comte et al (2011). At this point, to state our final result for the estimation of λ (stated in Corollary 2), we have to construct and study an estimator of ζ.…”

“…which is a natural extension to the MAR case of the contrast introduced in Comte et al (2011). We note that, with assumption (A0-4) and the definition of ζ, we have…”

“…Note that these models are drawn from Wang and Shen (2008) or Zhou and Sun (2003) for the Exponential model, and Comte et al (2011) for the non-linear regression, the Pareto distribution can be viewed as a generalization of the unconditional setting of the previous paragraph.…”

“…Many steps of the proof below will refer to the one given in Comte et al (2011), the main difference lies in the additional term R n .…”

“…In addition, C is assumed to be independent of T conditionally to X, see e.g. Comte et al (2011) for comments on this assumption.…”