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Logistic regression model is widely used in many studies to investigate therelationship between a binary response variable $Y$ and a set of potentialpredictors $\mathbf X$. The binary response may represent, for example, theoccurrence of some outcome of interest ($Y=1$ if the outcome occurred and $Y=0$otherwise). When the dependent variable $Y$ represents a rare event, thelogistic regression model shows relevant drawbacks. In order to overcome thesedrawbacks we propose the Generalized Extreme Value (GEV) regression model. Inparticular, we suggest the quantile function of the GEV distribution as linkfunction, so our attention is focused on the tail of the response curve forvalues close to one. A sample of observations is said to contain a curefraction when a proportion of the study subjects (the so-called curedindividuals, as opposed to the susceptibles) cannot experience the outcome ofinterest. One problem arising then is that it is usually unknown who are thecured and the susceptible subjects, unless the outcome of interest has beenobserved. In these settings, a logistic regression analysis of the relationshipbetween $\mathbf X$ and $Y$ among the susceptibles is no more straightforward.We develop a maximum likelihood estimation procedure for this problem, based onthe joint modeling of the binary response of interest and the cure status. Weinvestigate the identifiability of the resulting model. Then, we conduct asimulation study to investigate its finite-sample behavior, and application toreal data.

Zero-inflated generalized extreme value regression model for binary data and application in health study