When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Abstract

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epiand hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies L p convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.

Modelling Extremes Using Approximate Bayesian Computation

“…Under minimal assumptions, the estimator is consistent and asymptotically normal with a convergence rate of √ k (Einmahl et al, 2012;Bücher et al, 2014). Alternatively, the marginal distributions might be estimated by F j,2 (X ij ) = R ij,n /n or F j,3 (X ij ) = (R ij,n − 1/2)/n, resulting in estimators that are asymptotically equivalent to ℓ.…”

Beyond Dimension two: A Test for Higher-Order Tail Risk

“…converges to a sum of a centered Gaussian field and univariate centered Gaussian processes with given covariance structure (Einmahl et al (2012), Bücher et al (2014)). If X is asymptotically independent,ˆ (x) is still asymptotically normal but with degenerate variance (Hüsler & Li 2009).…”

“…The second assumption restricts the speed with which k grows to infinity, and in combination with (A1) guarantees that an asymptotic bias term for the left hand side of equation (11) vanishes (see Resnick & de Haan (1996), Einmahl et al (2008) for details). According to Bücher et al (2014) a smoothness assumption for the STDF is not required. In particular, we do not need to impose that partial derivatives of exist for the asymptotic result to hold.…”

“…It is a general and flexible concept of tail dependence and allows for straightforward non-parametric estimation, bearing a smaller risk of model misspecification than alternative parametric approaches. Furthermore, its statistical properties are well understood for X of dimension beyond two (Einmahl et al (2012), Bücher et al (2014)). …”

“…Huang (1992), Dietrich et al (2003), Einmahl et al (2006), Drees et al (2006), Einmahl et al (2012), Bücher et al (2014)). Importantly, the STDF is a convex function and homogeneous of degree one, i.e.…”