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

Biosynthesis of Cucurbita maxima trypsin inhibitor I in the methylotropic yeast Pichia pastoris

Squash inhibitors are the smallest natural serine protease inhibitors. Their compact, rigid nature has enabled detailed examination of their 3D structure by NMR and X-ray crystallography. Being of a convenient size to synthesise chemically, the effects on activity of selective substitutions and deletions within the sequence have also been investigated. Thus, this family of inhibitors is considered useful as a model system for the study of protein-protein interactions. Cucurbita maxima trypsin inhibitor I (CMTI I) may be thought of as representative of the squash inhibitors, for which there is detailed structural and functional information available. It is a 29 amino acid protein, with the tri-disulphide bridging pattern common to all squash inhibitors. There are only a few examples of squash inhibitors being produced by recombinant DNA technology. As this technique offers a relatively cheap way of producing large amounts of these proteins, further investigation is required. Problems have been experienced with the expression of disulphide-rich proteins in E. coli, as the cytosol of this microorganism is not conducive to their folding. Furthermore extraction of these proteins from the peri plasmic space is often required, resulting in a reduction in yield. To overcome these shortcomings, the methylotrophic yeast Pichia pastoris was investigated as an alternative means of expression, although at the inception of this work, no disulphide-rich proteins of this size had been expressed in P. pastoris. It was a challenge to investigate the feasibility of producing squash inhibitors in this expression host and to compare the activity of the recombinant inhibitor to that of native CMTI I

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

Long Strange Segments, Ruin Probabilities and the Effect of Memory on Moving Average Processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.Comment: 29 pages, minor changes and a few typo correction from last versio

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

Limit theorem for maximum of the storage process with fractional Brownian motion as input

AbstractThe maximum MT of the storage process Y(t)=sups⩾t(X(s)-X(t)-c(s-t)) in the interval [0,T] is dealt with, in particular, for growing interval length T. Here X(s) is a fractional Brownian motion with Hurst parameter, 0<H<1. For fixed T the asymptotic behaviour of MT was analysed by Piterbarg (Extremes 4(2) (2001) 147) by determining an approximation for the probability P{MT>u} for u→∞. Using this expression the convergence P{MT<uT(x)}→G(x) as T→∞ is derived where uT(x)→∞ is a suitable normalization and G(x)=exp(-exp(-x)) the Gumbel distribution. Also the relation to the maximum of the process on a dense grid is analysed

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

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