Multiplier bootstrap of tail copulas with applications

TL;DR

This paper introduces two bootstrap methods for estimating the distribution of tail copulas, proving their consistency and relaxing previous smoothness assumptions, with applications in statistical inference.

Contribution

It proposes two novel multiplier bootstrap procedures for tail copula estimation, addressing the estimation of partial derivatives and broadening applicability under weaker assumptions.

Findings

Both bootstrap methods are proven consistent.

The common assumption of continuous partial derivatives is shown to be overly restrictive.

The methods are applicable to various statistical problems like goodness-of-fit testing.

Abstract

For the problem of estimating lower tail and upper tail copulas, 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 the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail 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 that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.