Semiparametric Copula-Based Confidence Intervals on Level Curves for the Evaluation of the Risk Level Associated to Bivariate Events

Abstract

If $(X,Y)$ is a random pair with distribution function $F_{X,Y}(x,y)=\mathbb{P}(X\le x,Y\le y)$, one can define the level curve of probability $p$ as the values of $(x,y)\in {ℝ}^2$ such that $F_{X,Y}(x,y)=p$. This level curve is at the base of bivariate versions of return periods for the assessment of risk associated with extreme events. In most uses of bivariate return periods, the values taken by $(X,Y)$ on this level curve are accorded equal significance. This paper adopts an innovative point-of-view by showing how to build confidence sets for the values of a pair of continuous random variables on a level curve. To this end, it is shown that the conditional distribution of $(X,Y)$ given that the pair belongs to the level curve can be written in terms of the copula that characterizes its dependence structure. This allows for the definition of confidence sets on the level curve. It is suggested that the latter be estimated semi-parametrically, where the copula is assumed to belong to a given parametric family, and the marginals are replaced by their empirical counterparts. Formulas are derived for the Farlie–Gumbel–Morgenstern, Archimedean, Normal, and Student copulas. The methodology is illustrated on the risk level associated with the daily concentration of atmospheric pollutants.