A bivariate power generalized Weibull distribution: A flexible parametric model for survival analysis

We are concerned with the flexible parametric analysis of bivariate survival data. Elsewhere, we argued in favour of an adapted form of the 'power generalized Weibull' distribution as an attractive vehicle for univariate parametric survival analysis. Here, we additionally observe a frailty relationship between a power generalized Weibull distribution with one value of the parameter which controls distributional choice within the family and a power generalized Weibull distribution with a smaller value of that parameter. We exploit this relationship to propose a bivariate shared frailty model with power generalized Weibull marginal distributions linked by the BB9 or 'power variance function' copula, then change it to have adapted power generalized Weibull marginals in the obvious way. The particular choice of copula is, therefore, natural in the current context, and the corresponding bivariate adapted power generalized Weibull model a novel combination of pre-existing components. We provide a number of theoretical properties of the models. We also show the potential of the bivariate adapted power generalized Weibull model for practical work via an illustrative example involving a well-known retinopathy dataset, for which the analysis proves to be straightforward to implement and informative in its outcomes.