Abstract
The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations’ lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.
Highlights
Introduction and PreliminariesIn survival analysis, various semiparametric models have been introduced
Cox [1] introduced the proportional hazards (PH) model as one of the semiparametric models that has contributed greatly to the literature. He used censored failure times and assumed that multiple explanatory variables are available for each value. e hazard rate function (HRF) was assumed to be a function of the explanatory variables and the unknown regression coefficients were multiplied by an arbitrary and unknown function of time
He obtained a conditional likelihood leading to inferences about the unknown regression coefficients and presented some generalizations of his method. e conclusion was that the PH model can be applied in many fields, the motivation is medical
Summary
Cox [1] introduced the proportional hazards (PH) model as one of the semiparametric models that has contributed greatly to the literature He used censored failure times and assumed that multiple explanatory variables are available for each value. Gupta and Kirmani [10] made some preliminary stochastic comparisons in the multiplicative hazards frailty model. E aim of the present study was to consider a general MRL frailty model and to add a parameter to the family of distributions generated by (3) in order to consider the effects of a multiplicative parameter together with the effects of the baseline population and the effects of the weight function on the life span of the population. Considering the parameter as a random variable, mixture models of the MRL frailty model and the specific weighted PMRL model are generated. We have moved the proofs of the results to the appendix to reduce the complexity of the paper and smooth the content
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