Copula-based conditional tail indices

Abstract

Consider a multivariate distribution of $(X,Y)$, where $X$ is a vector of predictor variables and $Y$ is a response variable. Results are obtained for comparing the conditional and marginal tail indices, ${\xi }_{Y|X}\left(x\right)$ and ${\xi }_Y$, based on conditional distributions $\left\{F_{Y|X}\left(\cdot \right|x)\right\}$ and marginal distribution $F_Y$, respectively. For a multivariate distribution based on a copula, the conditional tail index can be decomposed into a product of copula-based conditional tail indices and the marginal tail index. In some applications, one may want ${\xi }_{Y|X}\left(x\right)$ to be non-constant, and some new copula families are derived to facilitate this.