Bivariate extreme value theory was used to estimate a rare event (see de Haan and de Ronde [1998. Sea and wind: multivariate extremes at work. Extremes 1, 7-45]). This procedure involves estimating a tail dependence function. There are several estimators for the tail dependence function in the literature, but their limiting distributions depend on partial derivatives of the tail dependence function. In this paper smooth estimators are proposed for estimating partial derivatives of bivariate tail dependence functions and their asymptotic distributions are derived as well. A simulation study is conducted to compare different estimators of partial derivatives in terms of both mean squared errors and coverage accuracy of confidence intervals of the bivariate tail dependence function based on these different estimators of partial derivatives.
Bibliographical noteFunding Information:
We thank two reviewers, an associate editor and executive editor for helpful comments. Peng's research was supported by NSF grant DMS-04-03443 and a Humboldt research fellowship, and Qi's research was supported in part by NSF grant DMS 06-04176.
- Bivariate extremes
- Confidence interval
- Smooth estimation
- Spectral measure
- Tail dependence function