Abstract: In this paper the log-exponentiated-Weibull regression model is modified to allow the possibility that long term survivors are present in the data. The modification leads to a log-exponentiated-Weibull regression model with cure rate, encompassing as special cases the log-exponencial regression and log-Weibull regression models with cure rate typically used to model such data. The models attempt to estimate simultaneously the effects of covariates on the acceleration/deceleration of the timing of a given event and the surviving fraction; that is, the proportion of the population for which the event never occurs. Assuming censored data, we consider a classic analysis and Bayesian analysis for the parameters of the proposed model. The normal curvatures of local influence are derived under various perturbation schemes and two deviance-type residuals are proposed to assess departures from the log-exponentiated-Weibull error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.