Workshop on Time Series Analysis

WORKSHOP ON TIME SERIES ANALYSIS, Monday, February 13.

LOCATION:  Store auditorium, Nørre Alle 53, NEXS (Idraet).  (Next to HCØ. 

SCHEDULE:

13:15-13:45  Jeffrey Collamore, Large deviations and conditioned limit laws for matrix recursive sequences

13:45-14:15  Johannes Heiny, Eigenvalues of large heavy-tailed random matrices.

14:15-14:30  Break

14:30-15:00  Claudia Klüppelberg, Risk in a large claims insurance market with bipartite graph structure.

15:00-15:30  Richard Davis, Extreme Value Analysis Without the Largest Values: What Can Be Done?

ABSTRACTS:

Speaker:  Jeffrey Collamore (Univ. Copenhagen)

Title:  Large deviations and conditioned limit laws for matrix recursive sequences

Abstract:   Motivated by branching processes in random environments, matrix recursions were originally introduced in the seminal paper of Kesten (1973), who studied the sequence V(n) = A(n) V(n-1) + B(n), where {(A(n)} is an i.i.d. sequence of random matrices and {B(n)} an i.i.d. sequence of random vectors. Under a stationarity condition, Kesten showed that V(n) converges to a random variable V, whose tail decays at a specified polynomial rate.  Extensions of this work have been the subject of extensive work in pure and applied probability, and the estimate arises in a variety of applications (e.g. GARCH financial time series models).  In this talk, we focus on the path behavior particularly in the case where {V(n)} is "explosive," deriving a large deviation limit law and conditioned limit theorem.  (Based on joint work with Anand Vidyashankar.)

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Speaker:  Johannes Heiny (Univ. Copenhagen)

Title:  Eigenvalues of large heavy-tailed random matrices

Abstract:  This talk is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries.

   In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index alpha<4, we employ a large deviations approach to show that the extreme eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables.  The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension of the problem at hand.

   We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals of the eigenvalues.

   In the second part, we show that the largest and smallest eigenvalues of a high-dimensional sample correlation matrix possess almost sure non-random limits if the truncated variance of the entry distribution is ‘’almost slowly varying'', a condition we describe via moment properties of self-normalized sums. We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment.

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Speaker:  Claudia Klüppelberg (TU Munich)

Title:  Risk in a large claims insurance market with bipartite graph structure

Abstract:  We model the influence of sharing large exogeneous losses to the reinsurance market by a bipartite graph. Using Pareto-tailed claims and multivariate regular variation we obtain asymptotic results for the Value-at-Risk and the Conditional Tail Expectation. We address the following problems in our setting of networks of agents.

  (1) We explain the influence of the network structure on diversification effects in different network scenarios. As is well-known in a non-network setting, if the Pareto exponent is larger than 1, then for the individual agent (reinsurance company) diversification is beneficial, whereas when it is less than 1, concentration on a few objects is the better strategy.

  (2) An additional aspect is the amount of uninsured losses which have to be covered by society. In our setting, diversification is never detrimental concerning the amount of uninsured losses. If the Pareto-tailed claims have finite mean, diversification turns out to be never detrimental, both for society and for individual agents. In contrast, if the Pareto-tailed claims have infinite mean, a conflicting situation may arise between the incentives of individual agents and the interest of some regulator to keep the risk for society small.

  (3) We also obtain asymptotic results for conditional risk measures based on the Value-at-Risk and the Conditional Tail Expectation. These results allow us to assess the influence of an individual institution on the systemic or market risk and vice versa through a collection of conditional risk measures.

  (4) For large markets Poisson approximations of the relevant constants are provided. Differences of the conditional risk measures for an underlying homogeneous and inhomogeneous random graph are illustrated by simulations.

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Speaker:  Richard Davis (Columbia University)

Title:  Extreme Value Analysis Without the Largest Values: What Can Be Done?

Abstract: In recent years, there has been growing interest in inference related problems in the traditional extreme value theory setup in which the data has been truncated above some large value.  The principal objectives have been to estimate the parameters of the model, usually in a Pareto or a generalized Pareto distribution (GPD) formulation, together with the truncated value. Ultimately, the Hill estimator plays a starring role in this work. In this paper we take a different perspective. Motivated by data coming from a large network, the Hill estimator appeared to exhibit smooth “sample path” behavior as a function of the number of upper order statistics used in constructing the estimator. This became more apparent as we artificially truncated more of the upper order statistics. Building on this observation, we introduce a new parameterization into the Hill estimator that is a function of delta and theta, that correspond, respectively, to the proportion of extreme values that have been truncated and the path behavior of the “Hill estimator”.  As a function of (delta,theta), we establish functional convergence of the renormalized Hill estimator to a Gaussian process. Based on this limit theory, an estimation procedure is developed to estimate the number of censored observations and other extreme value parameters including alpha, the index of regular variation and the bias of Hill’s estimate. We illustrate this approach in both simulations and with real data.  (This is joint work with Gennady Samorodnitsky and Jingjing Zou.)