Multiplier bootstrap of tail copulas - with applications

Abstract

In the problem of estimating the lower and upper tail copula we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures we prove consistency. For these investigations we demonstrate that the common assumption of the existence of continuous partial derivatives in the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to asymptotic independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions which are satisfied for many commonly used models. These results are applied in several statistical problems including minimum distance estimation and goodness-of-fit testing. AMS Subject Classification: Primary 62G32 ; secondary 62G20