Nonparametric estimation for survival data with censoring indicators missing at random

Abstract

To cite this version:Elodie Brunel, Fabienne Comte, Agathe Guilloux. Nonparametric estimation for survival data with censoring indicators missing at random. Abstract. In this paper, we consider the problem of hazard rate estimation in presence of covariates, for survival data with censoring indicators missing at random. We propose in the context usually denoted by MAR (missing at random, in opposition to MCAR, missing completely at random, which requires an additional independence assumption), nonparametric adaptive strategies based on model selection methods for estimators admitting finite dimensional developments in functional orthonormal bases. Theoretical risks bounds are provided, they prove that the estimators behave well in term of Mean Square Integrated Error (MISE). Simulation experiments illustrate the statistical procedure.

Estimation/Imputation Strategies for Missing Data in Survival Analysis

“…which is a particular case of the contrast introduced in Brunel et al Brunel et al (2013). Indeed, if we compute the expectation of this theoretical contrast, we obtain, under the MAR assumption and using (4) and (5),…”

“…We also notice that the logit estimator of ζ versus the nonparametric one is nearly impossible to beat at least for sample sizes n = 200, 1000. Of course, when the logit assumption is violated, for large sample (n = 5000) and important missing rate 45%, the results are deteriorating, see Brunel et al (2013), see also Comte et al ?. Below, we give steps for obtaining the second inequality.…”

“…Brunel et al (2013) in a more general setting, and a sketch of the proof of (14) is given in appendix.…”

“…These quantities can have different orders, depending on the second step strategy, • the last terms C ′′ 1 /n or C ′′ 2 /n are negligible. We can obtain adaptive nonparametric rate with the same model selection principle for ζ and π, if we apply the nonparametric strategy (see Theorem 2 for ζ in Brunel et al (2013) Brunel et al (2013)). …”

“…Below, we give steps for obtaining the second inequality. The reference centered empirical process is It is easy to see that (see Brunel et al Brunel et al (2013)):…”