RESUMO
Frailty models are important for survival data because they allow for the possibility of unobserved heterogeneity problem. The problem of heterogeneity can be existed due to a variety of factors, such as genetic predisposition, environmental factors, or lifestyle choices. Frailty models can help to identify these factors and to better understand their impact on survival. In this study, we suggest a novel quasi xgamma frailty (QXg-F) model for the survival analysis. In this work, the test of Rao-Robson and Nikulin is employed to test the validity and suitability of the probabilistic model, we examine the distribution's properties and evaluate its performance in comparison with many relevant cox-frailty models. To show how well the QXg-F model captures heterogeneity and enhances model fit, we use simulation studies and real data applications, including a fresh dataset gathered from an emergency hospital in Algeria. According to our research, the QXg-F model is a viable replacement for the current frailty modeling distributions and has the potential to improve the precision of survival analyses in a number of different sectors, including emergency care. Moreover, testing the ability and the importance of the new QXg-F model in insurance is investigated using simulations via different methods and application to insurance data.