There is an abundant literature in statistics, biostatistics and econometrics on the modelling, estimation and inference of regression models for survival data subject to censoring. However, only a few of them consider a potential multimodality of the time-to-event. To the best of our knowledge, there is no model that takes into account both multimodality and the possible presence of a cure fraction, i.e. the presence of a fraction of subjects who do not experience the event of interest. Our aim is to develop a modelling approach that takes both these aspects into account. This is particularly useful in contexts such as modelling cancer recurrence, where recurrences may occur in several waves, but with a proportion of patients never relapsing.

In this work, we have built a model that considers both multimodality of the time-to-event and a cure fraction. To achieve this aim, we developed an accelerated failure time model in which the error term is assumed to follow a mixture of Sinh-Cauchy distributions. This approach offers greater robustness by combining the flexibility of mixture models with that of the Sinh-Cauchy distribution. We studied the properties of this distribution and implemented an estimation method using the EM algorithm. A simulation study was carried out to illustrate the performance of the proposed approach.

Simulations have demonstrated the relevance and effectiveness of our approach for modelling multimodal time-to-event data with a proportion of cure. The methodology implemented to estimate the various parameters of the model provides reliable results in terms of bias, variance and MSE. Further investigations are ongoing on the selection of the number of components in the mixture, but preliminary results indicate that large flexibility is already achieved with a limited number of mixture components.

The results obtained show that the proposed model is an interesting alternative to traditional cure models in the presence of multimodality and cure, while also providing good results for unimodal data. It therefore constitutes a more flexible and robust approach. In the context of multimodal survival data with a proportion of cure. In the following, we intend to apply our methodology to real data.