Transportation cost-information and concentration inequalities for bifurcating Markov chains

We investigate the transportation cost-information inequalities for bifurcating Markov chains which are a class of processes indexed by binary tree. These processes provide models for cell growth when each individual in one generation gives birth to two offsprings in the next one. Transportation cost inequalities provide useful concentra-tion inequalities. We also study deviation inequalities for the empiri-cal means under relaxed assumptions on the Wasserstein contraction of the Markov kernels. Applications to bifurcating non linear autore-gressive processes are considered: deviation inequalities for pointwise estimates of the non linear leading functions

On the study of the Beran estimator for generalized censoring indicators

Along with the analysis of time-to-event data, it is common to assume that only partial information is given at hand. In the presence of right-censored data with covariates, the conditional Kaplan-Meier estimator (also referred as the Beran estimator) is known to propose a consistent estimate for the lifetimes conditional survival function. However, a necessary condition is the clear knowledge of whether each individual is censored or not, although, this information might be incomplete or even totally absent in practice. We thus propose a study on the Beran estimator when the censoring indicator is not clearly specified. From this, we provide a new estimator for the conditional survival function and establish its asymptotic normality under mild conditions. We further study the supervised learning problem where the conditional survival function is to be predicted with no censorship indicators. To this aim, we investigate various approaches estimating the conditional expectation for the censoring indicator. Along with the theoretical results, we illustrate how the estimators work for small samples by means of a simulation study and show their practical applicability with the analysis of synthetic data and the study of real data for the prognosis of monoclonal gammopathy

Testing for sufficient follow-up in censored survival data by using extremes

In survival analysis, it often happens that some individuals, referred to as cured individuals, never experience the event of interest. When analyzing time-to-event data with a cure fraction, it is crucial to check the assumption of `sufficient follow-up', which means that the right extreme of the censoring time distribution is larger than that of the survival time distribution for the non-cured individuals. However, the available methods to test this assumption are limited in the literature. In this article, we study the problem of testing whether follow-up is sufficient for light-tailed distributions and develop a simple novel test. The proposed test statistic compares an estimator of the non-cure proportion under sufficient follow-up to one without the assumption of sufficient follow-up. A bootstrap procedure is employed to approximate the critical values of the test. We also carry out extensive simulations to evaluate the finite sample performance of the test and illustrate the practical use with applications to leukemia and breast cancer datasets.Comment: 16 pages, 2 figures and 4 tables are adde

Bias-corrected and robust estimation of the bivariate stable tail dependence function

We consider the estimation of the bivariate stable tail dependence function and propose a bias-corrected and robust estimator. We establish its asymptotic behavior under suitable assumptions. The finite sample performance of the proposed estimator is examined on a simulation study involving both uncontaminated and contaminated samples

Non-parametric cure rate estimation under insufficient follow-up by using extremes

An important research topic in survival analysis is related to the modelling and estimation of the cure rate, i.e. the proportion of subjects who will never experience the event of interest. However, most estimation methods proposed so far in the literature do not handle the case of insufficient follow‐up, i.e. when the right end point of the support of the censoring time is strictly less than that of the survival time of the susceptible subjects, and consequently these estimators overestimate the cure rate in that case. We fill this gap by proposing a new estimator of the cure rate that makes use of extrapolation techniques from the area of extreme value theory. We establish the asymptotic normality of the estimator proposed and show how the estimator works for small samples by means of a simulation study. We also illustrate its practical applicability through the analysis of data on the survival of breast cancer patients

Non-parametric cure rate estimation under insufficient follow-up using extremes

status: publishe

Local estimation of the conditional stable tail dependence function

International audienceWe consider the local estimation of the stable tail dependence function when a random covariate is observed together with the variables of main interest. Our estimator is a weighted version of the empirical estimator adapted to the covariate framework. We provide the main asymptotic properties of our estimator, when properly normalized, in particular the convergence of the empirical process towards a tight centered Gaussian process. The finite sample performance of our estimator is illustrated on a small simulation study and on a dataset of air pollution measurements

Bias correction in conditional multivariate extremes

International audienceWe consider bias-corrected estimation of the stable tail dependence function in the regression context. To this aim, we first estimate the bias of a smoothed estimator of the stable tail dependence function, and then we subtract it from the estimator. The weak convergence, as a stochastic process, of the resulting asymptotically unbiased estimator of the conditional stable tail dependence function, correctly normalized, is established under mild assumptions, the covariate argument being fixed. The finite sample behaviour of our asymptotically unbiased estimator is then illustrated on a simulation study and compared to two alternatives, which are not bias corrected. Finally, our methodology is applied to a dataset of air pollution measurements