A completely nonparametric method for the estimation of mixture cure models is proposed in this paper. The nonparametric estimator of the incidence introduced by Xu and Peng ( 2014) is extensively studied and a nonparametric estimator of the latency is presented. These estimators, which are based on the Beran estimator of the conditional survival function, are proved to be the local maximum likelihood estimators. An iid representation is obtained for the nonparametric incidence estimator. As a consequence, an asymptotically optimal bandwidth is found. Moreover, a bootstrap bandwidth selection method for the nonparametric incidence estimator is proposed. The introduced nonparametric estimators are compared with existing semiparametric approaches in a simulation study, in which the performance of the bootstrap bandwidth selector is also assessed. Finally, the presented method is applied to a database of colorectal cancer from the University Hospital of A Coruña (CHUAC).

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A short introduction to survival analysis and censored data is included in this paper. A thorough literature review in the field of cure models has been done. An overview on the most important and recent approaches on parametric, semiparametric and nonparametric mixture cure models is also included. The main nonparametric and semiparametric approaches were applied to a real time dataset of COVID-19 patients from the first weeks of the epidemic in Galicia (NW Spain). The aim is to model the elapsed time from diagnosis to hospital admission. The main conclusions, as well as the limitations of both the cure models and the dataset, are presented, illustrating the usefulness of cure models in this kind of studies, where the influence of age and sex on the time to hospital admission is shown.

In cancer studies there may be long term survivors, which lead to heavy censoring at the end of the follow-up period. Since a standard survival model is not appropriate to handle these data, a cure model is needed. In the literature, covariate significance tests for cure models are limited to parametric and semiparametric methods. We fill this important gap by proposing a nonparametric covariate significance test for the probability of cure in mixture cure models. A bootstrap method is proposed to approximate the null distribution of the test statistic. The procedure can be applied to any type of covariate, and could be extended to the multivariate setting. Its efficiency is evaluated in a Monte Carlo simulation study. Finally, the method is applied to a colorectal cancer dataset.