Multivariate Extreme Value Distribution Research Articles

The maximum domain of attraction of multivariate extreme value distributions is small

Consider the set of Borel probability measures on Rk and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.

Conditional modelling approach to multivariate extreme value distributions: application to extreme rainfall events in South Africa

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. The results showed that the multivariate conditional modelling fitted to daily maximum rainfall events provided apparent benefits in terms of improved precision in the estimation of the marginal parameters of generalised Pareto distribution. The conditional modelling approach provided all forms of dependence using Laplace marginal transformations, for which all weather stations are not extreme equally. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors, and compared with the prediction obtained from the conditional model fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, and the extremal dependence structure described consistently for extreme daily maximum rainfall events between weather stations. The current study contributes towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with e.g., flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by responsible authorities.

An Application of Multivariate Extreme Value Distribution to d4PDF: Co-occurrence Probability of Extreme Floods in Two River Basins

An Application of Multivariate Extreme Value Distribution to d4PDF: Co-occurrence Probability of Extreme Floods in Two River Basins

PREDIKSI CURAH HUJAN EKSTREM DI KOTA SEMARANG MENGGUNAKAN SPATIAL EXTREME VALUE DENGAN PENDEKATAN MAX STABLE PROCESS (MSP)

This research covers Spatial Extreme Value method application with Max-Stable Process (MSP) approach that will be used to analysis Extreme Rainfall in Semarang city. Extreme value sample are selected by Block Maxima methods, it will be estimated into Spatial Extreme Value form by including location factors. Then it transform to Frechet distribution because it has a heavy tail pattern. Max Stable Process (MSP) is an extension of the multivariate extreme value distribution into infinite dimension of the Extreme Value Theory. After the best model of extreme rainfall data in Semarang is obtained, then calculated the prediction of extreme rainfall with a certain time period. Predictions are calculated using a return level, predictions of extreme rainfall using the return period of the next two years, at the Semarang City Climatology Station predicted to be a maximum of 100.7539 mm. At the Tanjung Mas Rain Monitoring Station it is predicted that a maximum of 100.1052 mm, Ahmad Yani Rain Monitoring Station is predicted to be a maximum of 109.9379 mm. Maximum prediction of extreme rainfall can also be calculated for future t (time) periods.

The infinite extendibility problem for exchangeable real-valued random vectors

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on $\mathbb{R}^d$, which is: Can a given random vector $\vec{X} = (X_1,\ldots,X_d)$ be represented in distribution as the first $d$ members of an infinite exchangeable sequence of random variables? This is the case if and only if $\vec{X}$ has a stochastic representation that is "conditionally iid" according to the seminal de Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model $\vec{X}$ is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when $\vec{X}$ takes values in $\{0,1\}^d$ has a spherical law, a law with $\ell_1$-norm symmetric survival function, or a law with $\ell_{\infty}$-norm symmetric density. The solutions in all These cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on $\mathbb{R}$, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when $\vec{X}$ has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a L\'evy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems.

Stochastic generation of spatially coherent river discharge peaks for continental event-based flood risk assessment

Abstract. We present a new method to generate spatially coherent river discharge peaks over multiple river basins, which can be used for continental event-based probabilistic flood risk assessment. We first extract extreme events from river discharge time series data over a large set of locations by applying new peak identification and peak-matching methods. Then we describe these events using the discharge peak at each location while accounting for the fact that the events do not affect all locations. Lastly we fit the state-of-the-art multivariate extreme value distribution to the discharge peaks and generate from the fitted model a large catalogue of spatially coherent synthetic event descriptors. We demonstrate the capability of this approach in capturing the statistical dependence over all considered locations. We also discuss the limitations of this approach and investigate the sensitivity of the outcome to various model parameters.

Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution

We revisit multivariate extreme-value theory modeling by emphasizing multivariate regular variation and a multivariate version of Breiman’s Lemma. This allows us to recover in a simple framework the most popular multivariate extreme-value distributions, such as the logistic, negative logistic, Dirichlet, extremal- t and Hüsler–Reiß models. We then focus on the Hüsler–Reiß Pareto model and its surprising exponential family property. After a thorough study of this exponential family structure, we focus on maximum likelihood estimation: we prove the existence of asymptotically normal maximum likelihood estimators and provide simulation experiments assessing their finite-sample properties. We also consider the generalized Hüsler–Reiß Pareto model with different tail indices and a likelihood ratio test for discriminating constant tail index versus varying tail indices.

Extreme rainfall prediction using spatial extreme value by Max Stable Process (MSP) Smith model approach

This research covers Spatial Extreme Value method application with Max-Stable Process (MSP) approach that will be used to analysis Extreme Rainfall in Semarang city. Extreme value sample are selected by Block Maxima methods, it will be estimated into Spatial Extreme Value form by including location factors. Then it transform to Frechet distribution because it has a Heavy tail pattern. Max Stable Process (MSP) is an extension of the multivariate extreme value distribution into infinite dimension of the Extreme Value Theory. MSP using Generalized Extreme Value (GEV) approach, it has three forms distribution that is Gumbel, Frechet and Weibull. Extreme Value Samples that are selected by Block Maxima will follow GEV distribution. The best model that obtained based on Max-Stable with Smith model can be used to predict Extreme Rainfall by count Return Level. Then can be further analysis of various aspects, so it can provide a suggestion about the potential of natural disasters which are caused by extreme rainfall. Extreme rainfall prediction using the Smith model in the return period of the next two years, at the Semarang City Climatology, Tanjung Mas and Ahmad Yani Station is predicted to be a maximum of 100.7539 mm, 100.1052 mm, and 109.9379 mm.

Extremal properties of the multivariate extended skew-normal distribution, Part B

The skew-normal and related families are flexible and asymmetric parametric models suitable for modelling a diverse range of systems. We show that the multivariate maximum of a high-dimensional extended skew-normal random sample has asymptotically independent components and derive the speed of convergence of the joint tail. To describe the possible dependence among the components of the multivariate maximum, we show that under appropriate conditions an approximate multivariate extreme-value distribution that leads to a rich dependence structure can be derived.

Estimation of intensity–duration–frequency curves using max-stable processes

We present an approach to estimate intensity–duration–frequency (IDF) curves based on max-stable processes. The proposed method has been inspired by the seminal study of Nadarajah et al. (J R Stat Soc B 60(2):473–496, 1998), who used a multivariate extreme value distribution (MEVD) to estimate (IDF) curves from rainfall records. Max-stable processes are continuous extensions of MEVD (i.e. the marginal distributions of rainfall maxima at different durations are generalized extreme valued), which are more flexible, allow for extreme rainfall estimation at any arbitrary duration d (i.e. not just a discrete set, as is the case of MEVD), while preserving asymptotic dependencies. The latter characteristic of IDF estimates results from the combined effect of the statistical structure of rainfall (i.e. temporal dependencies), as well as the IDF construction process, which involves averaging of the original series to obtain rainfall maxima at different temporal resolutions. We apply the method to hourly precipitation data, and compare it to empirical estimates and the results produced by a semiparametric approach. Our findings indicate that max-stable processes fit well the statistical structure and inter-dependencies of annual rainfall maxima at different durations, produce slightly more conservative estimates relative to semiparametric methods, while allowing for extrapolations to durations and return periods beyond the range of the available data. The proposed statistical model is fully parametric and likelihood based, providing a theoretically consistent basis in solving the problem at hand.

A general approach to generate random variates for multivariate copulae

SummaryWe suggest two methods for simulating from a multivariate copula in an arbitrary dimension. Although our main emphasis in this paper is on multivariate extreme value distributions, the proposed methods can be applied to any copula. The basic idea is to approximate the (unknown) density of the copula by a distribution that has a piece‐wise constant (histogram) density. This is achieved by partitioning the support of a given copula C into a large number of hyper‐rectangles and using them to generate random variates from an approximation of the copula. We suggest two methods for finding this approximation which correspond to either finding hyper‐rectangles which have equal probability mass with respect to C, or determining a partition using hyper‐squares of equal volume and finding the corresponding probability mass of each hyper‐square. We also discuss how the generated random variates can be used as proposals in a Metropolis–Hastings algorithm, when C is an absolutely continuous distribution function, to generate a sequence of random variates from C. An implementation of the proposed methodologies is provided for the statistical computing and graphics environment R in our package called SimCop.

The efficacy of life insurance company general account equity asset allocations: a safety-first perspective using vine copulas

AbstractWe study the asset allocation decision of a life insurance company’s general account with respect to the possibility of large negative economic shocks and examine how this account is affected by policyholder investment decisions in the company’s separate account. This is accomplished using a performance metric that incorporates downside risk measured using univariate and multivariate extreme value distributions. Because of its well-known price volatility, diversification attributes, and significant weight in the combined general and separate accounts, our primary focus is the company’s equity investments. Although industry asset allocations have varied over the past two decades, we find that the actual allocations to equity in the general account are close to the allocation percentages suggested by our extreme value metrics and both are far below the maximum values indicated by the relevant regulatory bodies.

Multidimensional extremal dependence coefficients

Extreme value modeling has been attracting the attention of researchers in diverse areas such as the environment, engineering, and finance. Multivariate extreme value distributions are particularly suitable to model the tails of multidimensional phenomena. The analysis of the dependence among multivariate maxima is useful to evaluate risk. Here we present new multivariate extreme value models, as well as, coefficients to assess multivariate extremal dependence.

Bernstein polynomial angular densities of multivariate extreme value distributions

To model the angular measure of a multivariate extreme value distribution, we develop a mean-constrained Bernstein polynomial over the (p−1)-dimensional simplex, along with a generalization that places mass on the simplex boundaries.

Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions

AbstractMultivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component‐wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max‐stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.

Detecting breaks in the dependence of multivariate extreme-value distributions

In environmental sciences, it is often of interest to assess whether the dependence between extreme measurements has changed during the observation period. The aim of this work is to propose a statistical test that is particularly sensitive to such changes. The resulting procedure is also extended to allow the detection of changes in the extreme-value dependence under the presence of known breaks in the marginal distributions. Simulations are carried out to study the finite-sample behavior of both versions of the proposed test. Illustrations on hydrological data sets conclude the work.

Exact simulation of max-stable processes.

Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

A saddlepoint approach to estimating joint extreme value distributions for vector non-stationary Gaussian processes

The focus of this study is on estimating the multivariate extreme value distributions associated with a vector of mutually correlated non-stationary Gaussian processes. This involves computing the joint crossing statistics of the vector processes by assuming the crossings to be Poisson counting processes. A mathematical artifice is adopted to take into account the dependencies that exist between the crossings of the processes. The crux in the formulation lies in the evaluation of a high-dimensional integral, which can be computationally expensive. This difficulty is bypassed by using saddlepoint approximation. The developments are illustrated through two numerical examples and are validated using Monte Carlo simulations. In the second example, reliability analysis is carried out for a multiply supported pipeline on an offshore jacket structure subjected to wave loading using the proposed formulation.

Multivariate Extreme Value Distributions for Vector of Non-stationary Gaussian Processes

The focus of this study is on estimating the multivariate extreme value distributions associated with a vector of mutually correlated non-stationary Gaussian processes. This involves computing the joint crossing statistics of the vector processes by assuming the crossings to be Poisson counting processes. A mathematical artifice is adopted to take into account the dependencies that exist between the crossings of the processes. The crux in the formulation lies in the evaluation of a four-dimensional integral, which can be computationally expensive. This difficulty is bypassed by using saddlepoint approximation to reduce the dimension of the integral to be numerically computed to just two. The developments are illustrated through a numerical example and are validated using Monte Carlo simulations.

Systemic Tail Risk Distribution

We introduce the systemic tail risk distribution of a financial market to characterize the asset return linkages during financial crisis. This distribution provides the probabilities that several assets of a market lose a large part of their nominal value given that the price of at least one of them collapses. It introduces a new way of assessing the stability of a financial market during potential systemic risk events. We propose a new type of multivariate extreme value distribution for high-dimensional vectors to model the extremal dependence between asset prices, and we use efficient likelihood inference methods to estimate the parameters of the systemic tail risk distribution. Our real data show that the empirical static systemic tail risk distribution is U-shaped, while the empirical conditional distribution is L-shaped.