Abstract
We introduce a new modelling for long-term survival models, assuming that the number of competing causes follows a mixture of Poisson and the Birnbaum-Saunders distribution. In this context, we present some statistical properties of our model and demonstrate that the promotion time model emerges as a limiting case. We delve into detailed discussions of specific models within this class. Notably, we examine the expected number of competing causes, which depends on covariates. This allows for direct modeling of the cure rate as a function of covariates. We present an Expectation-Maximization (EM) algorithm for parameter estimation, to discuss the estimation via maximum likelihood (ML) and provide insights into parameter inference for this model. Additionally, we outline sufficient conditions for ensuring the consistency and asymptotic normal distribution of ML estimators. To evaluate the performance of our estimation method, we conduct a Monte Carlo simulation to provide asymptotic properties and a power study of LR test by contrasting our methodology against the promotion time model. To demonstrate the practical applicability of our model, we apply it to a real medical dataset from a population-based study of incidence of breast cancer in São Paulo, Brazil. Our results illustrate that the proposed model can outperform traditional approaches in terms of model fitting, highlighting its potential utility in real-world scenarios.