Title: Non-parametric cure models through extreme-value tail estimation
Abstract: In survival analysis, the estimation of the proportion of subjects who will never experience the event of interest, termed the cure rate, has received considerable attention recently. Its estimation can be a particularly difficult task when follow-up is not sufficient, that is when the censoring mechanism has a smaller support than the distribution of the target data. In the latter case, non-parametric estimators were recently proposed using extreme value methodology, assuming that the distribution of the susceptible population is in the Fréchet or Gumbel max-domains of attraction. In this paper, we take the extreme value techniques one step further, to jointly estimate the cure rate and the extreme value index, using probability plotting methodology, and in particular using the full information contained in the top order statistics. In other words, under sufficient or insufficient follow-up, we reconstruct the immune proportion. To this end, a Peaks-over-Threshold approach is proposed under the Gumbel max domain assumption. Next, the approach is also transferred to more specific models such as Pareto, log-normal and Weibull tail models, allowing to recognize the most important tail characteristics of the susceptible population. We establish the asymptotic behavior of our estimators under regularization. Though simulation studies, our estimators are show to rival and often outperform established models, even when purely considering cure rate estimation. Finally, we provide an application of our method to Norwegian birth registry data.
This is a joint work with Jan Berlant (KU Leuven) and Martin Bladt (University of Copenhagen)
Title: Trends in tail dependence of heteroscedastic extremes
Abstract: We consider multivariate extreme value statistics for independent but nonidentically distributed random vectors. In particular, the data may have varying tail copulas and also heteroscedastic marginal distributions. Assuming smoothly changing tail copulas, we propose a nonparametric estimator for the integrated tail copula, as well as one for the local tail copula. We establish the asymptotic behavior of both estimators. Notably, the heteroscedastic marginals do not affect the limiting processes. Finally we use the main result for the integrated tail copula to test for a constant tail copula across all observations.
This is a joint work with John Einmahl (Tilburg University).