This paper develops a modification of the shared frailty model that allows for the distribution of the response and the random effects to be determined by the data (from within the exponential family of distributions). A simulated maximum likelihood and a computationally simpler simulated linear least squares approach to estimating this model are also presented. These generalizations are important for modelling high-temperature deformation because they allow life time predictions to be made with levels of confidence attached to them, for the real world situation where repeated experimental observations are not normally distributed and not independent of each other. An application to a commonly used material in power generation revealed that ignoring these characteristics lead to an underestimate of the minimum safe life when operating at 773 K and stresses above 235 MPa.