Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition

Details Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition

2006-11-13

arxiv.org/abs/math/0611370v1

Annals of Statistics 2006, Vol. 34, No. 4, 1987-2014

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000434 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000434

Consider $n$ i.i.d. random vectors on $\mathbb{R}^2$, with unknown, common distribution function $F$. Under a sharpening of the extreme value condition on $F$, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting distribution of the test statistic from the aforementioned weighted approximation. This limiting distribution contains unknown functional parameters. Therefore, we show that a version with estimated parameters converges weakly to the true limiting distribution. Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study. A real data application is also presented.

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