Abstract: This paper introduces a flexible skewed link function for modeling binary as well as ordinal data with covariates based on the generalized extreme value (GEV) distribution. Extreme value techniques have been widely used in many disciplines relating to risk analysis, but, applications to binary and ordinal data in a Bayesian context are sparse. There are a number of non-regular situations with the likelihood method for GEV models in which the usual asymptotic properties of MLE do not hold, suggesting Bayesian methodology for analyzing GEV models. We introduce the GEV distribution in reliability and survival models, and show that our proposed model leads to an extremely flexible hazard function. We investigate the properties of posterior distributions for binary and ordinal response models under the generalized extreme value link using a uniform prior distribution on the regression parameters. Necessary and sufficient conditions for the propriety of the posterior distribution are established. We consider similar issues for survival data models, where log survival time has a GEV distribution, and the propriety of the posterior distribution under a uniform prior on the regression coefficients is established. The flexibility of the proposed survival model is illustrated through a dataset involving a lung cancer clinical trial.

Key words and phrases: Complementary log-log link, generalized extreme value distribution, hazard function, improper prior, posterior propriety, skewness.