Summary We propose a new class of survival models which naturally links a family of proper and improper population survival functions. The models resulting in improper survival functions are often referred to as cure rate models. This class of regression models is formulated through the Box–Cox transformation on the population hazard function and a proper density function. By adding an extra transformation parameter into the cure rate model, we are able to generate models with a zero cure rate, thus leading to a proper population survival function. A graphical illustration of the behavior and the influence of the transformation parameter on the regression model is provided. We consider a Bayesian approach which is motivated by the complexity of the model. Prior specification needs to accommodate parameter constraints due to the nonnegativity of the survival function. Moreover, the likelihood function involves a complicated integral on the survival function, which may not have an analytical closed form, and thus makes the implementation of Gibbs sampling more difficult. We propose an efficient Markov chain Monte Carlo computational scheme based on Gaussian quadrature. The proposed method is illustrated with an example involving a melanoma clinical trial.