Propriety of posterior distributions arising in categorical and survival models under generalized extreme value distribution

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. However, its application in the binary and ordinal data from a Bayesian context is sparse and its strength as a link function has recently been explored. There are a number of non-regular situations with likelihood method for GEV models where usual asymptotic properties of MLE do not hold. That is why Bayesian methodology is preferred for analyzing GEV models. We also introduce the GEV distribution in reliability and survival models. We show that our proposed model leads to an extremely flexible hazard function. In this paper, first we investigate the property 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 with the generalized extreme value link function are established. Next we consider similar issues for survival data models, where log survival time has a GEV distribution and propriety of the posterior distribution under 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. 1 Statistica Sinica: Preprint doi:10.5705/ss.2011.011