Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach

In this paper we investigate the impact of model mis-specification, in terms of the dependence structure in the extremes of a spatial process, on the estimation of key quantities that are of interest to hydrologists and engineers. For example, it is often the case that severe flooding occurs as a result of the observation of rainfall extremes at several locations in a region simultaneously. Thus, practitioners might be interested in estimates of the joint exceedance probability of some high levels across these locations. It is likely that there will be spatial dependence present between the extremes, and this should be properly accounted for when estimating such probabilities. We compare the use of standard models from the geostatistics literature with max-stables models from extreme value theory. We find that, in some situations, using an incorrect spatial model for our extremes results in a significant under-estimation of these probabilities which – in flood defence terms – could lead to substantial under-protection.     

Max-stable processes for modeling extremes observed in space and time

Max-stable processes have proved to be useful for the statistical modeling of spatial extremes. For statistical inference it is often assumed that there is no temporal dependence; i.e., that the observations at spatial locations are independent in time. In a first approach we construct max-stable space–time processes as limits of rescaled pointwise maxima of independent Gaussian processes, where the space–time covariance functions satisfy weak regularity conditions. This leads to so-called Brown–Resnick processes. In a second approach, we extend Smith’s storm profile model to a space–time setting. We provide explicit expressions for the bivariate distribution functions, which are equal under appropriate choice of the parameters. We also show how the space–time covariance function of the underlying Gaussian process can be interpreted in terms of the tail dependence function in the limiting max-stable space–time process.     

Local likelihood smoothing of sample extremes

Trends in sample extremes are of interest in many contexts, an example being environmental statistics. Parametric models are often used to model trends in such data, but they may not be suitable for exploratory data analysis. This paper outlines a semiparametric approach to smoothing sample extremes, based on local polynomial fitting of the generalized extreme value distribution and related models. The uncertainty of fits is assessed by using resampling methods. The methods are applied to data on extreme temperatures and on record times for the women's 3000 m race.     

Statistics for Extreme Sea Currents

Estimates of various characteristics of extreme sea currents, such as speeds and their directions, are required when designing offshore structures. This paper extends standard statistical methods for extreme values to handle the directionality, temporal dependence and tidal non-stationarity that are present in sea current extremes. The methods are applied to a short period of data from the Inner Dowsing Light Tower in the North Sea. Substantial benefits, over existing methods, are obtained from our analysis of the sea current by decomposing it into tide and surge currents. In particular, we find that at the Inner Dowsing the strong directionality in extreme sea current speeds is completely explained by the tidal current and directionality in the non-extreme surge currents. This finding aids model fitting and extrapolation.     

Extremal analysis of processes sampled at different frequencies

The observed extremes of a discrete time process depend on the process itself and the sampling frequency. We develop theoretical results which show how to account for the effect of sampling frequency on extreme values, thus enabling us to analyse systematically extremal data from series with different sampling rates. We present statistical methodology based on these results which we illustrate though simulations and by applications to sea-waves and rainfall data.     

Modelling extreme wind speeds in regions prone to hurricanes

Extreme wind speeds can arise as the result of a simple pressure differential, or a complex dynamic system such as a tropical storm. When sets of record values comprise a mixture of two or more different types of event, the standard models for extremes based on a single limiting distribution are not applicable. We develop a mixture model for extreme winds arising from two distinct processes. Working with sequences of annual maximum speeds obtained at hurricane prone locations in the USA, we take a Bayesian approach to inference, which allows the incorporation of prior information obtained from other sites. We model the extremal behaviour for the contrasting wind climates of Boston and Key West, and show that the standard models can give misleading results at such locations.     

Investigating precipitation extremes in South Carolina with focus on the state's October 2015 precipitation event

The October 2015 precipitation event in the Southeastern United States brought large amounts of rainfall to South Carolina, with particularly heavy amounts in Charleston and Columbia. The subsequent flooding resulted in numerous casualties and hundreds of millions of dollars in property damage. Precipitation levels were so severe that media outlets and government agencies labeled this storm as a 1 in 1000-year event in parts of the state. Two points of discussion emerged as a result of this event. The first was related to understanding the degree to which this event was anomalous; the second was related to understanding whether precipitation extremes in South Carolina have changed over recent time. In this work, 50 years of daily precipitation data at 28 locations are used to fit a spatiotemporal hierarchical model, with the ultimate goal of addressing these two points of discussion. Bayesian inference is used to estimate return levels and to perform a severity-area-frequency analysis, and it is determined that precipitation levels related to this event were atypical throughout much of the state, but were particularly unusual in the Columbia area. This analysis also finds marginal evidence in favor of the claim that precipitation extremes in the Carolinas have become more intense over the last 50 years.     

A hierarchical model for extreme wind speeds

Summary.  A typical extreme value analysis is often carried out on the basis of simplistic inferential procedures, though the data being analysed may be structurally complex. Here we develop a hierarchical model for hourly gust maximum wind speed data, which attempts to identify site and seasonal effects for the marginal densities of hourly maxima, as well as for the serial dependence at each location. A Gaussian model for the random effects exploits the meteorological structure in the data, enabling increased precision for inferences at individual sites and in individual seasons. The Bayesian framework that is adopted is also exploited to obtain predictive return level estimates at each site, which incorporate uncertainty due to model estimation, as well as the randomness that is inherent in the processes that are involved.     

Diagnostics for dependence within time series extremes

Summary. The analysis of extreme values within a stationary time series entails various assumptions concerning its long- and short-range dependence. We present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events. These tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables. Our diagnostic aids are illustrated through theoretical examples, simulation studies and by application to rainfall and exchange rate data. On the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed.     

多元极值的参数建模方法及其金融应用：最新进展述评

 由于现实中的极值事件往往倾向于同时或相继发生,因此多元极值研究正成为极值统计学的理论前沿和研究热点。本文对该领域中参数建模方法的最新进展做了系统性述评,包括经典多元极值理论、Ledford-Tawn-Ramos方法和Heffernan和Tawn条件法等,并指出了这些建模方法的优缺点以及未来可能的理论突破点。本文还全面分析了近年来多元极值分析方法在金融领域的国内外应用现状,并探讨其未来的应用前景,可能是在金融传染、组合问题和系统性风险管理等方面。     

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

Modeling and predicting extreme cyber attack rates via marked point processes

Cyber attacks have become a problem that is threatening the economy, human privacy, and even national security. Before we can adequately address the problem, we need to have a crystal clear understanding about cyber attacks from various perspectives. This is a challenge because the Internet is a large-scale complex system with humans in the loop. In this paper, we investigate a particular perspective of the problem, namely the extreme value phenomenon that is exhibited by cyber attack rates, which are the numbers of attacks against a system of interest per time unit. It is important to explore this perspective because understanding the statistical properties of extreme cyber attack rates will pave the way for cost-effective, if not optimal, allocation of resources in real-life cyber defense operations. Specifically, we propose modeling and predicting extreme cyber attack rates via marked point processes, while using the Value-at-Risk as a natural measure of intense cyber attacks. The point processes are then applied to analyze some real data sets. Our analysis shows that the point processes can describe and predict extreme cyber attack rates at a very satisfactory accuracy.     

Estimation of extremes in corrosion engineering

SUMMARY This paper reviews a number of extreme value models which have been applied to corrosion problems. The techniques considered are used to model and predict the statistical behaviour of corrosion extremes, such as the largest pit, thinnest wall, maximum penetration or similar assessment of corrosion phenomenon. These techniques can be applied to measurements over a regular grid or to measurements of selected extremes, and can be adapted to accommodate all values over a selected threshold, or a selected number of the largest values-or only the single largest value. Data can come from one coupon or several coupons, and can be modelled to allow for dependence on environmental conditions, surface area examined, and duration of exposure or of experimentation. The techniquesare demonstrated on data from laboratory experiments and also on data collected in an industrial context.     

Anticipating catastrophes through extreme value modelling

Summary. When catastrophes strike it is easy to be wise after the event. It is also often argued that such catastrophic events are unforeseeable, or at least so implausible as to be negligible for planning purposes. We consider these issues in the context of daily rainfall measurements recorded in Venezuela. Before 1999 simple extreme value techniques were used to assess likely future levels of extreme rainfall, and these gave no particular cause for concern. In December 1999 a daily precipitation event of more than 410 mm, almost three times the magnitude of the previously recorded maximum, caused devastation and an estimated 30000 deaths. We look carefully at the previous history of the process and offer an extreme value analysis of the data—with some methodological novelty—that suggests that the 1999 event was much more plausible than the previous analyses had claimed. Deriving design parameters from the results of such an analysis may have had some mitigating effects on the consequences of the subsequent disaster. The themes of the new analysis are simple: the full exploitation of available data, proper accounting of uncertainty, careful interpretation of asymptotic limit laws and allowance for non-stationarity. The effect on the Venezuelan data analysis is dramatic. The broader implications are equally dramatic; that a naïve use of extreme value techniques is likely to lead to a false sense of security that might have devastating consequences in practice.     

Use of tail areas for testing goodness-of-fit

Characterizing a set of data as a random sample from a specified distribution is often a precursor to statistical inference or hypothesis testing involving the extremes of the distribution -precisely the regions of greatest uncertainty. It seems reasonable then to exploit as best we can our limited knowledge of this region. Toward this end we investigate here the areas in the tails of the distribution as determined by the extreme order statistics as a criterion for testing goodness-of-fit.     

Outliers and order statistics

Outliers are to be found among the extremes of a data set. Extremes are examples of order statistics. It is thus relevant to ask to what extent the statistical methods (and probabilistic properties) of outliers and of order statistics coincide and depend on each other. Whilst clear overlap is identifiable, aims and procedures are often quite distinct and each topic plays its own important role in the panoply of statistical principles and methodology.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

Exact maximum likelihood for incomplete data from a correlated gaussian process

In many situations in which a variable is measured at locations in time or space the observed data can be regarded as incomplete, the missing data sites perhaps completing a regular pattern such as a rectangular grid. In this paper general methods not dependent on the sequential nature of time are considered for estimating the parameters of Gaussian processes. An example is given.