Bivariate Extreme Value Distribution Research Articles

On approximating dependence function and its derivatives

AbstractBivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.

A Generalized Framework for Risk-Based Extreme Load Analysis in Offshore System Design

Abstract The primary aim of this research is to consider the correlation among environmental factors in calculating 100 and 1000 years of extreme load design criteria. This is done by considering load as energy transferred from external environment to the offshore system. Also, incorporating spatial and temporal dependence of environmental variables in the context of offshore design. A bivariate extreme value distribution and a conditional joint return level function are developed using the Gumbel–Hougaard copula. The offshore design risk criteria are developed for the finer grid locations (0.1 deg × 0.1 deg latitude/longitude grid) considering joint extreme wind and wave energy. The developed approach is tested using data for the Flemish Pass basin off the east coast of Canada. Along with the primary aim, the impact of climate change is investigated (time and space variability) by implementing the proposed methodology in two cases: the periods from 1959 to 1988 and 1989 to 2018. This study observed that climate change has caused 30% less correlation between wind speed and wave height in recent years (1989–2018) compared to the period of 1959–1988. The proposed extreme design wind speed is 39.7 m/s, and significant wave height is 16.4 m; their joint exceeding probability is 5.80 × 10−5 over an annual basis for a scenario of 100-year.

Statistical inference for tail-based cumulative residual entropy

Tail risk analysis plays a central strategic role in risk management, and focuses on the problem of risk measurement in the tail regions of extreme risks. In this paper, by extending the entropy method, we propose a quantity called Tail-based Cumulative Residual Entropy (TCRE) in a bivariate set-up as an alternative measurement of qualifying tail risk variability, which is less addressed in the literature. The TCRE in a bivariate set-up can be viewed as the tail cumulative residual entropy for a random vector (X,Y) with some dependence structure, and the combination of it with the Marginal Expected Shortfall (MES) constitutes a new risk measure called Marginal CRE Shortfall (MCRES). For a wide class of bivariate extreme value distributions, we construct a nonparametric estimator of the proposed TCRE measure and establish an asymptotic distribution of the estimator with extreme high-level tail. We use extreme value techniques to calculate the limit of the proposed measure as the risk level approaches to the extreme status and present how the tail dependence structure and marginal risk severity have an influence on its limits. Furthermore, a simulation study and real data analysis of the proposed TCRE are presented to understand and verify the properties of TCRE and its estimators.

Probable maximum precipitation estimation over western Iran based on remote sensing observations: comparing deterministic and probabilistic approaches

ABSTRACT Reliable estimation of probable maximum precipitation (PMP) is critical to ensure the safety and resilience of communities. The aim of this study is to improve the estimation of 24-h PMP using ground-based and remotely sensed data, particularly over data-scarce regions. Gumbel copula, as a bivariate extreme value distribution based on a moisture maximization method, was applied to estimate PMP. The framework allows us to examine the simultaneous occurrence of extreme precipitable water vapour (PW) and precipitation efficiency (PE) and determines extreme PW values using a regional remote sensing algorithm. This novel framework was compared with conventional methods including the Hershfield and moisture maximization approaches, which do not consider the dependencies between extreme PW and PE. The results demonstrate the importance of considering the dependence structure between extreme PW and PE in the estimation of PMP and the applicability of remotely sensed data, especially for data-scarce regions.

Robust quantile estimation under bivariate extreme value models

In risk quantification of extreme events in multiple dimensions, a correct specification of the dependence structure among variables is difficult due to the limited size of effective data. This paper studies the problem of estimating quantiles for bivariate extreme value distributions, considering that an estimated Pickands dependence function may deviate from the truth within some fixed distance. Our method thus finds optimal upper and lower bounds for the true but unknown dependence function, based on which robust quantile bounds are obtained. A simulation study shows the usefulness of our robust estimates that can supplement traditional error estimation methods.

Testing for lower tail dependence in extreme value models

ABSTRACTThis paper proposes some tests for lower tail dependence in extreme value models. Let be a random vector which follows in its lower tail a bivariate extreme value distribution with unit Frechet margins. We show that the conditional distribution function (df) of X+Y, given that X+Y <c, has a limiting df uniform on , i.e. , as if and only if X, Y have a lower tail dependence. We recommend using Fisher's κ, Chi-square goodness-of-fit, Kolmogorov–Smirnov, Cramer–von Mises and Anderson–Darling tests for lower tail dependence. Simulations show that, except the Fisher's κ test, all tests have good performance in terms of the size and power. Finally, by using two real datasets, we illustrate the application of the proposed statistics in testing for lower tail dependence.

Bayesian Inference with M-splines on Spectral Measure of Bivariate Extremes

We consider a Bayesian methodology with M-splines for the spectral measure of a bivariate extreme-value distribution. The tail of a bivariate distribution function F in the max-domain of attraction of an extreme-value distribution function G may be approximated by that of its extreme value attractor. The function G is characterized by a probability measure with expectation equal to 1/2, called the spectral measure, and two extreme-value indices. This spectral measure determines the tail dependence structure of F. The approximation of the spectral measure is proposed thanks to a non-parametric Bayesian estimator that guarantees to fulfill a moment and a shape constraint. The problem of routine calculation of posterior distributions for both coefficients and knots of M-splines is addressed using the Markov chain Monte Carlo (MCMC) simulation technique of reversible jumps.

Probable Maximum Precipitation: Its Estimation and Uncertainty Quantification Using Bivariate Extreme Value Analysis

AbstractProbable maximum precipitation (PMP) is the key parameter used to estimate the probable maximum flood (PMF), both of which are important for dam safety and civil engineering purposes. The usual operational procedure for obtaining PMP values, which is based on a moisture maximization approach, produces a single PMP value without an estimate of its uncertainty. We therefore propose a probabilistic framework based on a bivariate extreme value distribution to aid in the interpretation of these PMP values. This 1) allows us to evaluate estimates from the operational procedure relative to an estimate of a plausible distribution of PMP values, 2) enables an evaluation of the uncertainty of these values, and 3) provides clarification of the impact of the assumption that a PMP event occurs under conditions of maximum moisture availability. Results based on a 50-yr Canadian Centre for Climate Modelling and Analysis Regional Climate Model (CanRCM4) simulation over North America reveal that operational PMP estimates are highly uncertain and suggest that the assumption that PMP events have maximum moisture availability may not be valid. Specifically, in the climate simulated by CanRCM4, the operational approach applied to 50-yr data records produces a value that is similar to the value that is obtained in our approach when assuming complete dependence between extreme precipitation efficiency and extreme precipitable water. In contrast, our results suggest weaker than complete dependence. Estimates from the operational approach are 15% larger on average over North America than those obtained when accounting for the dependence between precipitation efficiency and precipitable water extremes realistically. A difference of this magnitude may have serious implications in engineering design.

Time-varying extreme value dependence with application to leading European stock markets

Extremal dependence between international stock markets is of particular interest in today’s global financial landscape. However, previous studies have shown this dependence is not necessarily stationary over time. We concern ourselves with modeling extreme value dependence when that dependence is changing over time, or other suitable covariate. Working within a framework of asymptotic dependence, we introduce a regression model for the angular density of a bivariate extreme value distribution that allows us to assess how extremal dependence evolves over a covariate. We apply the proposed model to assess the dynamics governing extremal dependence of some leading European stock markets over the last three decades, and find evidence of an increase in extremal dependence over recent years.

Some results on joint record events

Let X1,X2,… be independent and identically distributed random variables on the real line with a joint continuous distribution function F. The stochastic behavior of the sequence of subsequent records is well known. Alternatively to that, we investigate the stochastic behavior of arbitrary Xj,Xk,j<k, under the condition that they are records, without knowing their orders in the sequence of records. The results are completely different. In particular it turns out that the distribution of Xk, being a record, is not affected by the additional knowledge that Xj is a record as well. On the contrary, the distribution of Xj, being a record, is affected by the additional knowledge that Xk is a record as well. If F has a density, then the gain of this additional information, measured by the corresponding Kullback–Leibler distance, is j∕k, independent of F. We derive the limiting joint distribution of two records, which is not a bivariate extreme value distribution. We extend this result to the case of three records. In a special case we also derive the limiting joint distribution of increments among records.

Characterizations of Extreme Value Extended Marshall-Olkin Models with Exponential Marginals

We construct and characterize bivariate extreme value distributions with exponential marginals generated by the stochastic representation (X1,X2) = (min(T1,T3), min(T2,T3)) where the random variable T3 is independent of random variables T1 and T2 which are assumed to be dependent. A building procedure is suggested when the joint distribution of (T1,T2) is absolutely continuous and Ti's are not necessarily exponentially distributed, i=1,2,3. The Pickands representation of the vector (X1,X2) is computed. We illustrate the general relations by examples.

Spectral density regression for bivariate extremes

We introduce a density regression model for the spectral density of a bivariate extreme value distribution, that allows us to assess how extremal dependence can change over a covariate. Inference is performed through a double kernel estimator, which can be seen as an extension of the Nadaraya–Watson estimator where the usual scalar responses are replaced by mean constrained densities on the unit interval. Numerical experiments with the methods illustrate their resilience in a variety of contexts of practical interest. An extreme temperature dataset is used to illustrate our methods.

Bayesian framework for parametric bivariate accelerated lifetime modeling and its application to hospital acquired infections.

Infectious diseases that can be spread directly or indirectly from one person to another are caused by pathogenic microorganisms such as bacteria, viruses, parasites, or fungi. Infectious diseases remain one of the greatest threats to human health and the analysis of infectious disease data is among the most important application of statistics. In this article, we develop Bayesian methodology using parametric bivariate accelerated lifetime model to study dependency between the colonization and infection times for Acinetobacter baumannii bacteria which is leading cause of infection among the hospital infection agents. We also study their associations with covariates such as age, gender, apache score, antibiotics use 3 months before admission and invasive mechanical ventilation use. To account for singularity, we use Singular Bivariate Extreme Value distribution to model residuals in Bivariate Accelerated lifetime model under the fully Bayesian framework. We analyze a censored data related to the colonization and infection collected in five major hospitals in Turkey using our methodology. The data analysis done in this article is for illustration of our proposed method and can be applied to any situation that our model can be used.

Flood Double Frequency Analysis: 2D-Archimedean Copulas vs Bivariate Probability Distributions

Floods are multidimensional phenomena which are conventionally studied considering the pairwise dependence between peak flow and flood volume or peak flow and duration. In this paper, the flood phenomenon is analysed based on the peak flow-flood volume dependence. The paper presents a comparison between two methodologies for the double frequency analysis using a bivariate probability distribution and the copulas approach. The comparison is performed with data from a case study example of the Ashuapmushuan river basin in Quebec, Canada. In the presented example the bivariate Extreme Value distribution type I (Gumbel) is used for comparison with the bivariate Gumbel-Hougaard Archimedean copula. For the parameters estimation of the the latter, the Kendall method and the maximum likelihood method are employed. Based on the derived results of the analysed example, it can be concluded that for engineering purposes, the copulas approach, regardless of the method of its parameter estimation, provides a simple and accurate approach for the frequency analysis of floods and the estimation of the design variables thereof. The Kendall estimation parameter method, as simpler method, is easier to apply.

On the distribution of some ordered variables

Motivated by a recent article published by Adam and Tawn, we characterize the distribution of two random variables X, Y ordered linearly like X < Y. We suppose that the random variables follow a bivariate extreme value distribution.

Weighted estimation of the dependence function for an extreme-value distribution

Bivariate extreme-value distributions have been used in modeling extremes in environmental sciences and risk management. An important issue is estimating the dependence function, such as the Pickands dependence function. Some estimators for the Pickands dependence function have been studied by assuming that the marginals are known. Recently, Genest and Segers [Ann. Statist. 37 (2009) 2990-3022] derived the asymptotic distributions of those proposed estimators with marginal distributions replaced by the empirical distributions. In this article, we propose a class of weighted estimators including those of Genest and Segers (2009) as special cases. We propose a jackknife empirical likelihood method for constructing confidence intervals for the Pickands dependence function, which avoids estimating the complicated asymptotic variance. A simulation study demonstrates the effectiveness of our proposed jackknife empirical likelihood method.

Optimal designs for contingent response models with application to toxicity–efficacy studies

We describe a general family of contingent response models. These models have ternary outcomes constructed from two Bernoulli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. A bivariate extreme value distribution illustrates the model and optimal design results. To provide a motivating context, we call the two binary events that compose the contingent responses toxicity and efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response (toxicity, efficacy without toxicity, neither efficacy nor toxicity). The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. While optimal designs for contingent response models are numerically found, limiting optimal designs can be expressed in closed forms. In particular, in the family of four parameter bivariate location-scale models we study, as the marginal probability functions of toxicity and no efficacy diverge, limiting D optimal designs are shown to consist of a mixture of the D optimal designs for each failure (toxicity and no efficacy) univariately. Limiting designs are also obtained for the case of equal scale parameters.

Expansions for bivariate extreme value distributions

Expressions for moment properties have not been known even for the simplest of the bivariate extreme value distributions. Here, simple expansions are derived for various properties of any given bivariate extreme value distribution. Each expansion is a single infinite sum. The properties considered include product moments, conditional moments, joint moment generating function and others. Computational efficiencies of these expansions are established with respect to standard approaches for computing moment properties.

Risk Assessment of Droughts in Gujarat Using Bivariate Copulas

This study presents risk assessment of hydrologic extreme events droughts in Saurashtra and Kutch region of Gujarat state, India. Drought is a recurrent phenomenon and risk assessment of droughts can play an important role in proper planning and management of water resources in the study region. In the study, drought events are characterized by severity and duration, and drought occurrences are modeled by Standardized Precipitation Index (SPI) computed on mean areal precipitation, aggregated at a time scale of 6 months for the period 1900–2008. After evaluating several distribution functions, drought variable—severity is best described by non-parametric kernel density, whereas duration is best fitted by exponential distribution. Considering the extreme nature of drought variables, the upper tail dependence copula families including two Archimedean—Gumbel-Hougaard, BB1 and one elliptical—Student’s t copulas are evaluated for modeling joint distribution of drought variables. On evaluating their performance using various goodness-of-fit measures, Gumbel-Hougaard copula is found to be the best performing copula in modeling the joint dependence structure of drought variables. Also, while comparing with traditional bivariate distributions, the copula based distributions are resulted in better performance as compared to bivariate log-normal and the logistic model for bivariate extreme value distributions. Then joint and conditional return periods of drought characteristics are derived, which can be helpful for risk based planning and management of water resources systems in the study region.

A test when the Fisher information may be infinite, exemplified by a test for marginal independence in extreme value distributions

For testing independence between the components of the multivariate logistic extreme value distribution, the Fisher Information is infinite and the score statistic converges only at the rate of O(n−1/2logn). Consequently, the asymptotic optimality properties of the usual likelihood based methods do not necessarily carry over to this case. Motivated by this specific problem, this paper develops a general method based on the Cramér–von Mises discrepancy. This method does not require finite Fisher information. It is shown that this test statistic converges at the regular rate of O(n−1/2). The test statistic has a closed form and it can be computed easily without using an iterative method or numerical integration. The asymptotic critical values can be obtained from a table for chi-square distribution. In a simulation study involving the test of marginal independence in bivariate extreme value distributions, the test proposed in this paper performed better than its main competing ones. While the method was motivated by, and developed for, a particular topic in inference for multivariate extreme value distributions, it is applicable more broadly than just for inference in multivariate extreme value distributions.