Generalized extreme shock models with a possibly increasing threshold

We propose a generalized extreme shock model with a possibly increasing failure threshold. While standard models assume that the crucial threshold for the system may only decrease over time, because of weakening shocks and obsolescence, we assume that, especially at the beginning of the system's life, some strengthening shocks may increase the system tolerance to large shock. This is for example the case of turbines' running-in in the field of engineering. On the basis of parametric assumptions, we provide theoretical results and derive some exact and asymptotic univariate and multivariate distributions for the model. In the last part of the paper we show how to link this new model to some nonparametric approaches proposed in the literature

AN URN MODEL FOR CASCADING FAILURES ON A LATTICE

A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial network

Iterative Estimation of the Extreme Value Index

Let {Xn, n ≥ 1} be a sequence of independent random variables with common continuous distribution function F having finite and unknown upper endpoint. A new iterative estimation procedure for the extreme value index γ is proposed and one implemented iterative estimator is investigated in detail, which is asymptotically as good as the uniform minimum varianced unbiased estimator in an ideal model. Moreover, the superiority of the iterative estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation stud

On testing extreme value conditions

Applications of univariate extreme value theory rely on certain as- sumptions. Recently, two methods for testing these extreme value conditions are derived by [Dietrich, D., de Haan, L., Hüsler, J., Extremes 5: 71-85, (2002)] and [Drees, H., de Haan, L., Li, D., J. Stat. Plan. Inference, 136: 3498-3538, (2006)]. In this paper we compare the two tests by simulations and investigate the effect of a possible weight function by choosing a parameter, the test error and the power of each test. The conclusions are useful for extreme value application

Weak Convergence of the Empirical Mean Excess Process with Application to Estimate the Negative Tail Index

Let Y i , 1 ≤ i ≤ n be i.i.d. random variables with the generalized Pareto distribution W γ,σ with γ < 0. We define the empirical mean excess process with respect to {Y i , 1 ≤ i ≤ n} as in Eq. 2.1 (see below) and investigate its weak convergence. As an application, two new estimators of the negative tail index γ are constructed based on the linear regression to the empirical mean excess function and their consistency and asymptotic normality are obtaine

Review of testing issues in extremes: in honor of Professor Laurens de Haan

As a leading statistician in extreme value theory, Professor Laurens de Haan has made significant contribution in both probability and statistics of extremes. In honor of his 70th birthday, we review testing issues in extremes, which include research done by Professor Laurens de Haan and many others. In comparison with statistical estimation in extremes, research on testing has received less attention. So we also point out some practical questions in this directio

Tail approximations to the density function in EVT

Let X 1, X 2, ...,X n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a n > 0 and b n ∈ ℝ such that the limit of P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x) exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x) under some second order conditio

GENERALIZED EXTREME SHOCK MODELS WITH A POSSIBLY INCREASING THRESHOLD

We propose a generalized extreme shock model with a possibly increasing failure threshold. Although standard models assume that the crucial threshold for the system might only decrease over time, because of weakening shocks and obsolescence, we assume that, especially at the beginning of the system's life, some strengthening shocks might increase the system tolerance to large shock. This is, for example, the case of turbines' running-in in the field of engineering. On the basis of parametric assumptions, we provide theoretical results and derive some exact and asymptotic univariate and multivariate distributions for the model. In the last part of the article we show how to link this new model to some nonparametric approaches proposed in the literatur