In this paper the generalized log-gamma model is modified for possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the acceleration/deceleration of the timing of a given event and surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. We consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model.