Stochastic Comparisons of General Proportional Mean Past Lifetime Frailty Model
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Stochastic Comparisons of General Proportional Mean Past Lifetime Frailty Model Fatemeh Hooti and Jafar Ahmadi Ferdowsi University of Mashhad, Mashhad, Iran
N. Balakrishnan McMaster University, Hamilton, Canada Abstract In this paper, the general proportional mean past lifetime frailty model is considered, from which the unconditional cumulative distribution and density functions of the lifetime variable are derived. Dependency between the two variables are studied. Stochastic comparisons are made through which it is shown that some well-known stochastic orderings between two frailty variables carry over to the corresponding lifetime variables. The effects of baseline variable and the frailty variable on the proposed frailty model are studied. The relative mean past lifetime ordering is introduced and some relative ordering between two lifetime random variables with different frailty variables are studied. Also a simulation study is given to illustrate some results. AMS (2000) subject classification. 62N05, 60E15. Keywords and phrases. Frailty model, PMPL model, Reversed hazard rate ordering, Likelihood ratio ordering, Stochastic ordering, Relative ordering, Totally positive of order 2, Reverse regular of order 2
1 Introduction In the literature, several models have been introduced for modeling and analyzing failure time data. The proportional hazard (PH) rate model, the proportional reversed hazard (PRH) rate model, the additive hazard model, the additive reversed hazard rate model, proportional mean residual life model and frailty model are all good examples. Finkelstein (1986) developed a method for fitting the proportional hazards regression model when the data set contains left, right, or interval censored observations. Sankaran et al. (2014) investigated additive reversed hazard rate models and estimated the population hazard function. Vaupel et al. (1979) introduced the term
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frailty, it can be attributed to a person who is more vulnerable than others. They used it in order to capture the difference between individuals at risks, even when they appear to have height, weight, age, etc., to be the same. Frailty models are extensively used in survival analysis to account for unobserved heterogeneity in individual risks to disease and death. This model is a random effect model for time to event data and takes into account that the population is not homogeneous. Heterogeneity is usually explained by covariates, but when important covariates have not been observed, this leads to unobserved heterogeneity. For more details, we refer the readers to the books by, Duchateau and Janssen (2008), Hanagal (2011) and Wienke (2011) and the references therein. In survival analysis, Cox proportional hazard (PH) model is one of the most popular statistical models which is used to investigate the relation between the survival time and covariates. But, as Zucker and Yang (2006) have mentioned, sometimes the covariates on the hazard can be reduced over time, while in PH model covariates have a fixed multiplicative eff
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