Extremal Dependence Structure Research Articles

Vecchia Likelihood Approximation for Accurate and Fast Inference with Intractable Spatial Max-Stable Models

Max-stable processes are the most popular models for high-impact spatial extreme events, as they arise as the only possible limits of spatially-indexed block maxima. However, likelihood inference for such models suffers severely from the curse of dimensionality, since the likelihood function involves a combinatorially exploding number of terms. In this article, we propose using the Vecchia approximation, which conveniently decomposes the full joint density into a linear number of low-dimensional conditional density terms based on well-chosen conditioning sets designed to improve and accelerate inference in high dimensions. Theoretical asymptotic relative efficiencies in the Gaussian setting and simulation experiments in the max-stable setting show significant efficiency gains and computational savings using the Vecchia likelihood approximation method compared to traditional composite likelihoods. Our application to extreme sea surface temperature data at more than a thousand sites across the entire Red Sea further demonstrates the superiority of the Vecchia likelihood approximation for fitting complex models with intractable likelihoods, delivering significantly better results than traditional composite likelihoods, and accurately capturing the extremal dependence structure at lower computational cost. Supplementary materials for this article are available online.

ENVIRONMENTAL CONTOUR ESTABLISHMENT BASED ON OPENLY AVAILABLE EMODNET WAVE AND WIND DATA FOLLOWING HEFFERNAN AND TAWN METHODOLOGIES

In this paper, I present methodologies and routines for inference of environmental contours based on openly available data from the European EMODnet service, specifically wind and wave timeseries from oil platform 6200146. The conditional marginal distribution models, known as environmental contours, is established based on measured wind and wave heights, at platform 6200146 located in the North Sea. The data is provided through the openly available data platform, EMODnet. The environmental contour model is established by fitting univariate distributions to each data-series independently, whereafter an extremal dependence structure is established following methodologies outlined in (Janet E. heffernan, 2004).

Pricing Multi-Event-Triggered Catastrophe Bonds Based on a Copula–POT Model

The constantly expanding losses caused by frequent natural disasters pose many challenges to the traditional catastrophe insurance market. The purpose of this paper is to develop an innovative and systemic trigger mechanism for pricing catastrophic bonds triggered by multiple events with an extreme dependence structure. Due to the bond’s low cashflow contingencies and the CAT bond’s high return, the multiple-event CAT bond may successfully transfer the catastrophe risk to the huge financial markets to meet the diversification of capital allocations for most potential investors. The designed hybrid trigger mechanism helps reduce the moral hazard and increase the bond’s attractiveness with a lower trigger likelihood, displaying the determinants of the wiped-off coupon and principal by both the magnitude and intensity of the natural disaster events involved. As the trigger indicators resulting from the potential catastrophic disaster might be associated with heavy-tailed margins, nested Archimedean copulas are introduced with marginal distributions modeled by a POT-GP distribution for excess data and common parametric models for moderate risks. To illustrate our theoretical pricing framework, we conduct an empirical analysis of pricing a three-event rainstorm CAT bond based on the resulting losses due to rainstorms in China during 2006–2020. Monte Carlo simulations are carried out to analyze the sensitivity of the rainstorm CAT bond price in trigger attachment levels, maturity date, catastrophe intensity, and numbers of trigger indicators.

Estimating POT second-order parameter for bias correction

The stable tail dependence function provides a full characterisation of the extremal dependence structures. Unfortunately, the estimation of the stable tail dependence function often suffers from significant bias, whose scale relates to the Peaks-Over-Threshold (POT) second-order parameter. For this second-order parameter, this paper introduces a penalised estimator that discourages it from being too close to zero. This paper then establishes this estimator's asymptotic consistency, uses it to correct the bias in the estimation of the stable tail dependence function, demonstrates its desirable empirical properties in the estimation of the extremal dependence structures, and illustrates it with an application to liability claim data.

Pairwise counter-monotonicity

We systematically study pairwise counter-monotonicity, an extremal notion of negative dependence. A stochastic representation and an invariance property are established for this dependence structure. We show that pairwise counter-monotonicity implies negative association, and it is equivalent to joint mix dependence if both are possible for the same marginal distributions. We find an intimate connection between pairwise counter-monotonicity and risk sharing problems for quantile agents. This result highlights the importance of this extremal negative dependence structure in optimal allocations for agents who are not risk averse in the classic sense.

Modelling Dependency Structures of Carbon Trading Markets between China and European Union: From Carbon Pilot to COVID-19 Pandemic

The exploration of the dependency structure of the Chinese and EU carbon trading markets is crucial to the construction of a globally harmonized carbon market. In this paper, we studied the characteristics of structural interdependency between China’s major carbon markets and the European Union (EU) carbon market before and after the launch of the national carbon emissions trading scheme (ETS) and the occurrence of the new coronavirus (COVID-19) by applying the C-vine copula method, with the carbon trading prices of the EU, Beijing, Shanghai, Guangdong, Shenzhen and Hubei as the research objects. The study shows that there exists a statistically significant dependence between the EU and the major carbon markets in China and their extremal dependences and dependence structures are different at different stages. After the launch of the national carbon ETS, China has become more independent in terms of interdependency with the EU carbon market, and is more relevant between domestic carbon markets. Most importantly, we found that the dependence between the EU and Chinese carbon markets has increased following the outbreak of COVID-19, and tail dependency structures existed before the launch of the national carbon ETS and during the outbreak of the COVID-19. The results of this study provide a basis for the understanding of the linkage characteristics of carbon trading prices between China and the EU at different stages, which in turn can help market regulators and investors to formulate investment decisions and policies.

Flexible modeling of multivariate spatial extremes

We develop a novel multi-factor copula model for multivariate spatial extremes, which is designed to capture the different combinations of within- and cross-field extremal dependence structures between different spatial processes. Our proposed model, which can be seen as a multi-factor copula model, can capture all possible distinct combinations of extremal dependence structures within each individual spatial process while allowing flexible cross-field extremal dependence structures for both upper and lower tails. We show how to perform Bayesian inference for the proposed model using a Markov chain Monte Carlo algorithm based on carefully designed block proposals with an adaptive step size. In our real data application, we apply our model to study the upper and lower extremal dependence structures of the daily maximum air temperature (TMAX) and daily minimum air temperature (TMIN) from the state of Alabama in the southeastern United States. The fitted multivariate spatial model is found to provide a good fit in the lower and upper joint tails, both in terms of the spatial dependence structure within each individual process, as well as in terms of the cross-field dependence structure. Our results suggest that the TMAX and TMIN processes are quite strongly spatially dependent over the state of Alabama, and moderately cross-dependent. From a practical perspective, this implies that it may be worthwhile to model them jointly when interest lies in computing spatial risk measures that involve both quantities.

Structure Learning for Extremal Tree Models

AbstractExtremal graphical models are sparse statistical models for multivariate extreme events. The underlying graph encodes conditional independencies and enables a visual interpretation of the complex extremal dependence structure. For the important case of tree models, we develop a data-driven methodology for learning the graphical structure. We show that sample versions of the extremal correlation and a new summary statistic, which we call the extremal variogram, can be used as weights for a minimum spanning tree to consistently recover the true underlying tree. Remarkably, this implies that extremal tree models can be learned in a completely non-parametric fashion by using simple summary statistics and without the need to assume discrete distributions, existence of densities or parametric models for bivariate distributions.

Asymptotic Dependence Modelling of the BRICS Stock Markets

With the use of empirical data, this paper focuses on solving financial and investment issues involving extremal dependence of 10 pairwise combinations of the 5 BRICS (Brazil, Russia, India, China, and South Africa) stock markets. Daily closing equity indices from 5 January 2010 to 6 August 2018 are used in the study. Unlike previous literature, we use bivariate point process and conditional multivariate extreme value models to investigate the extremal dependence of the stock market returns. However, it is observed that the point process was able to model many more extreme observations or exceedances that contribute to the likelihood estimation. It gives more information than the threshold excess method of the CMEV model. This study shows varying levels of low extremal dependence structure whose outcomes are highly beneficial to investors, portfolio managers and other market participants interested in maximising investment returns and financial gains.

Dependence bounds for the difference of stop-loss payoffs on the difference of two random variables

This paper considers the difference of stop-loss payoffs where the underlying is a difference of two random variables. The goal is to study whether the comonotonic and countermonotonic modifications of those two random variables can be used to construct upper and lower bounds for the expected payoff, despite the fact that the payoff function is neither convex nor concave. The answer to the central question of the paper requires studying the crossing points of the cumulative distribution function (cdf) of the original difference with the cdf's of its comonotonic and countermonotonic transforms. The analysis is supplemented with a numerical study of longevity trend bonds, using different mortality models and population data. The numerical study reveals that for these mortality-linked securities the three pairs of cdf's generally have unique pairwise crossing points. Under symmetric copulas, all crossing points can reasonably be approximated by the difference of the marginal medians, but this approximation is not necessarily valid for asymmetric copulas. Nevertheless, extreme dependence structures can give rise to bounds if the layers of the bond are selected to hedge tail risk. Further, the dependence uncertainty spread can be low if the layers are selected to hedge median risk, and, subject to a trade-off, to hedge tail risk as well.

Multivariate Regional Frequency Analysis Using Conditional Extreme Values Approach

AbstractFloods can be characterized by various correlated variables such as peak flow, flood volume, duration, and time to peak. Hence, flood risk can be better assessed by performing frequency analysis in a multivariate (rather than a univariate) framework. However, multivariate approaches involve sophisticated analyses that require considerably more data than univariate approaches. In a univariate framework, flood risk assessment at an ungauged or data‐sparse location/site is generally performed through regional frequency analysis (RFA), based on information pooled from a group of sites resembling the target site. However, RFA has received little attention in the multivariate framework. The available literature recommends the use of index‐flood approach (IFA) for multivariate RFA (MRFA), even though IFA has theoretical shortcomings. Another issue is that marginals of all the flood‐related variables may not exhibit extreme behavior simultaneously. Conventionally used at‐site multivariate models are not suitable for describing the dependence structure of extremes in the regions of the support of the joint distribution where only some variables exhibit extreme behavior. This article proposes a conditional extreme values approach (CEA) to address the aforementioned issues in MRFA. Its effectiveness is demonstrated through Monte Carlo simulation experiments and a case study on watersheds from a flood‐prone region in India. Results indicate that the proposed approach can reliably predict the joint distribution of multiple flood‐related variables (and thus flood risk) at ungauged/sparsely gauged sites by effectively capturing the regional dependence structure between those variables.

Asymptotic theory for the inference of the latent trawl model for extreme values

AbstractThis article develops statistical inference methods and their asymptotic theory for the latent trawl model for extremes, which captures serial dependence in the time series of exceedances above a threshold. We review two methods based on pairwise likelihood and show that they underestimate the serial dependence in the extremes. We propose two generalized method of moments procedures based on auto‐covariance matching to overcome this shortcoming. Out of those four inference approaches, two are single‐stage strategies while the others have two stages, and we provide central limit theorems in the sense of weakly approaching sequences of distributions for all of them. This additional flexibility ensures good behavior between the estimators and estimates of the limiting distribution. In an empirical illustration using London air pollution data, we find that the two‐stage auto‐covariance matching scheme yields a high‐quality inference. It comprises two interpretable steps and correctly captures the serial dependence structure of extremes while performing on par with other methods in terms of marginal fit.

Modeling Concurrent Hydroclimatic Extremes With Parametric Multivariate Extreme Value Models

AbstractEstimating the dependence structure of concurrent extremes is a fundamental issue for accurate assessment of their occurrence probabilities. Identifying the extremal dependence behavior is also crucial for scientific understanding of interactions between the variables of a multidimensional environmental process. This study investigates the suitability of parametric multivariate extreme value models to correctly represent and estimate the dependence structure of concurrent extremes. Probabilistic aspects of multivariate extreme value theory with point process representation are discussed and illustrated with application to the concurrence of rainfall deficits, soil moisture deficits, and high temperatures. Application is concerned with the investigation of extremal behavior and risk assessment in Marathwada, a drought‐prone region of Maharashtra state, India. To characterize the multivariate extremes, marginal distributions are specified first and transformed into unit Fréchet margins. Standardized distributions are represented by a Poisson point process and coordinates of data points are transformed to pseudo‐polar coordinates to make the dependence form more explicit. The extremal dependence structure is described through angular densities on the unit simplex. Strong dependence is observed between soil moisture deficits and high temperatures, whereas rainfall deficits are mildly dependent on these two variables. Overall, a weak dependence is observed between the variables considered. Estimated extremal dependence is further used to compute probabilities of a few critical extreme combinations. Results demonstrate the ability of parametric multivariate models to characterize the complex dependence structure of concurrent extremes. These models can provide a powerful new perspective for appropriate statistical analysis of dependent hydroclimatic extremes in higher dimensions.

Recognizing a spatial extreme dependence structure: A deep learning approach

AbstractUnderstanding the behavior of extreme environmental events is crucial for evaluating economic losses, assessing risks, and providing health care, among many other related aspects. In a spatial context, relevant for environmental events, the dependence structure is extremely important, influencing joint extreme events and extrapolating on them. Thus, recognizing or at least having preliminary information on the patterns of these dependence structures is a valuable knowledge for understanding extreme events. In this study, we address the question of automatic recognition of spatial asymptotic dependence versus asymptotic independence, using a convolutional neural network (CNN). We designed a CNN architecture as an efficient classifier of a dependence structure. Extremal dependence measures are used to train the CNN. We tested our methodology on simulated and real datasets: air temperature data at 2 m over Iraq and rainfall data along the east coast of Australia.

Modeling the dependence structure and systemic risk of all listed insurance companies in the Chinese insurance market

AbstractAs a developing country, China's insurance market is already the second largest insurance market in the world, and its development is inseparable from the comprehensive development of insurance companies' business. At present, insurance companies are engaged in both traditional and nontraditional insurance businesses. Thus, to avoid extreme financial events, the insurance market requires more risk management. Hence, examining systemic risk in the insurance market in China is an interesting research topic. This paper studies the comovement between all listed insurance companies and the insurance market in China by modeling the average and extreme dependence structure using different copula specifications. Furthermore, it analyzes both downside and upside risk spillover effects from insurance companies to the insurance market by quantifying the following three market risk measures: the value‐at‐risk (VaR), the conditional VaR (CoVaR) and the delta CoVaR (ΔCoVaR). Finally, we test the significance of risk spillover effects and provide a formal ranking of the listed insurance companies with respect to their contribution to both the downside and upside systemic risk of the insurance market.

Bootstrapping Hill estimator and tail array sums for regularly varying time series

In the extreme value analysis of stationary regularly varying time series, tail array sums form a broad class of statistics suitable to analyze their extremal behavior. This class includes for example, the Hill estimator or estimators of the extremogram and the tail dependence coefficient. General asymptotic theory for tail array sums has been developed by Rootzén et al. (Ann. Appl. Probab. 8 (1998) 868–885) under mixing conditions and in Kulik et al. (Stochastic Process. Appl. 129 (2019) 4209–4238) for functions of geometrically ergodic Markov chains. A more general framework of cluster functionals is presented in Drees and Rootzén (Ann. Statist. 38 (2010) 2145–2186). However, the resulting limiting distributions turn out to be very complex and cumbersome to estimate as they usually depend on the whole extremal dependence structure of the time series. Hence, a suitable bootstrap procedure is desired, but available bootstrap consistency results for tail array sums are scarce. In this paper, following Drees (Drees (2015)), we consider a multiplier block bootstrap to estimate the limiting distribution of tail array sums. We prove that, conditionally on the data, an appropriately constructed multiplier block bootstrap statistic converges to the correct limiting distribution. Interestingly, in contrast, it turns out that an apparently natural, but naïve application of the multiplier block bootstrap scheme does not yield the correct limit. In simulations, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed multiplier block bootstrap over some obvious competitors. The proposed bootstrap scheme proves to be computationally efficient in comparison to other approaches.

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.

Statistical Modelling of Extreme Rainfall Indices using Multivariate Extreme Value Distributions

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. Moreover, 1-day to 5-day indices of rainfall events were constructed to investigate the frequencies and intensities of rainfall events for selected weather stations. Then, the conditional multivariate modelling was fitted to investigate dependence between series of extreme rainfall events. The conditional multivariate modelling has provided all forms of dependence, using Laplace marginal transformations, for which all weather stations are not equally extreme. 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 modelling that was fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, as well as the extremal dependence structure described consistently for indices of rainfall events between weather stations. The modelling framework and results of this study contribute 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 communities.

Spatial deformation for nonstationary extremal dependence

AbstractModeling the extremal dependence structure of spatial data is considerably easier if that structure is stationary. However, for data observed over large or complicated domains, nonstationarity will often prevail. Current methods for modeling nonstationarity in extremal dependence rely on models that are either computationally difficult to fit or require prior knowledge of covariates. Sampson and Guttorp (1992) proposed a simple technique for handling nonstationarity in spatial dependence by smoothly mapping the sampling locations of the process from the original geographical space to a latent space where stationarity can be reasonably assumed. We present an extension of this method to a spatial extremes framework by considering least squares minimization of pairwise theoretical and empirical extremal dependence measures. Along with some practical advice on applying these deformations, we provide a detailed simulation study in which we propose three spatial processes with varying degrees of nonstationarity in their extremal and central dependence structures. The methodology is applied to Australian summer temperature extremes and UK precipitation to illustrate its efficacy compared with a naive modeling approach.

Modeling Time-Varying Tail Dependence, with Application to Systemic Risk Forecasting

Abstract Empirical evidence for multivariate stock suggests that there are changes from asymptotic independence to asymptotic dependence and vice versa. Under asymptotic independence, the probability of joint extremes vanishes, whereas under asymptotic dependence, this probability remains positive. In this paper, we propose a dynamic model for bivariate extremes that allows for smooth transitions between regimes of asymptotic independence and asymptotic dependence. In doing so, we ignore the bulk of the distribution and only model the joint tail of interest. We propose a maximum-likelihood estimator for the model parameters and demonstrate its accuracy in simulations. An empirical application to losses on the CAC 40 and DAX 30 illustrates that our model provides a detailed description of changes in the extremal dependence structure. Furthermore, we show that our model issues adequate forecasts of systemic risk, as measured by CoVaR. Finally, we find some evidence that our CoVaR forecasts outperform those of a benchmark dynamic t-copula model.returns