Estimation of reinforced urn processes under left-truncation and right-censoring

We propose a nonparametric estimator for bivariate left-truncated and right-censored (LTRC) observations that combines the Expectation-Maximization (EM) algorithm and the Reinforced Urn Process (RUP). The resulting Expectation-Reinforcement (ER) algorithm allows for the inclusion of experts' knowledge in the form of a prior distribution, thus belonging to the class of Bayesian models. This can be relevant in applications where the data is incomplete, due to biases in the sampling process, as in the case of left-truncation and right-censoring. With this new approach, the distribution of the truncation variables is also recovered, granting further insight into those biases, and playing an important role in applications like prevalent cohort studies. The estimators are tested numerically using artificial and empirical datasets and compared with other methodologies such as copula models and the Kaplan-Meier estimator.