Copula Structure Analysis Based on Robust and Extreme Dependence Measures

In this paper we extend the standard approach of correlation structure analysis in order to reduce the dimension of highdimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptical copulae a correlation-like structure remains but different margins and non-existence of moments are possible. Moreover, elliptical copulae allow also for a copula structure analysis of dependence in extremes. After introducing the new concepts and deriving some theoretical results we observe in a simulation study the performance of the estimators: the theoretical asymptotic behavior of the statistics can be observed even for a sample of only 100 observations. Finally, we test our method on real financial data and explain differences between our copula based approach and the classical approach. Our new method yields a considerable dimension reduction also in non-linear models

Empirical Likelihodd Methods for an AR(1) process with ARCH(1) errors

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data

Max-linear models on infinte graphs generated by Bernoulli bond percolation

We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs, and investigate their relations to classical percolation theory. We formulate results for the oriented square lattice graph $\mathbb{Z}^2$ and nearest neighbor bond percolation. Focus is on the dependence introduced by this graph into the max-linear model. As a natural application we consider communication networks, in particular, the distribution of extreme opinions in social networks.Comment: 18 page

Extreme value theory for moving average processes with light-tailed innovations

We consider stationary infinite moving average processes of the form $Y_n = \sum c_i Z_{n+i}$, where the sum ranges over the integers, (Z_i) is a sequence of iid random variables with ``light tails'' and (c_i) is a sequence of positive and summable coefficients. By light tails we mean that Z_0 has a bounded density $f(t)$ behaving asymptotically like $v(t) \exp (-g(t) )$, where v(t) behaves roughly like a constant as t goes to infinity, and g(t) is strictly convex satisfying certain asymptotic regularity conditions. We show that the iid sequence associated with Y_0 is in the maximum domain of attraction of the Gumbel distribution. Under additional regular variation conditions on g, it is shown that the stationary sequence (Y_n) has the same extremal behaviour as its associated iid sequence. This generalizes results of Rootz\'en (1986, 1987), where $g(t) = t^p$ and $v(t)=c t^d$ for p > 1, positive c and a real constant d