Framework Of Extreme Value Theory Research Articles

SPECIAL ISSUE LinStat: a new proposal for robust estimation of the extremal index

The extremal index (EI) is a parameter defined in the framework of extreme value theory, which measures the degree of dependence among exceedances above high fixed thresholds. When the EI exists and those exceedances occur in clusters, the EI is related to the dimension of the clusters and, in the limit distribution, it coincides with the reciprocal of the mean clusters dimension. There are different EI estimators according to the method used in the identification of those clusters. Some of them use the sample mean in the estimation of the cluster dimension. The present proposal is based on the ‘Runs’ estimator. It considers a negative binomial as the limit distribution of the number of exceedances that occur before a non-exceedance observation. Then its mean is estimated through the constant term of a negative binomial regression. The reciprocal of that value gives the EI estimate. The procedure makes use of robust estimators for counting processes, with known properties. Thus the negative binomial distribution is integrated in the estimation of the mean, while small deviations from the assumptions are controlled, including the occurrence of atypical cluster size values. A simulation study explores and compares the present robust proposal with other estimators.

Real-time crash risk forecasting using Artificial-Intelligence based video analytics: A unified framework of generalised extreme value theory and autoregressive integrated moving average model

With the recent advancements in computer vision and artificial intelligence, traffic conflicts occurring at an intersection and associated traffic characteristics can be obtained at the granular level of a signal cycle in real-time. This capability enables the estimation of the real-time crash risk using sophisticated modelling techniques, e.g., extreme value theory. However, these models are inherently incapable of forecasting the crash risk of future time periods based on the temporal dependency of crash risks. This study proposes a unified framework of extreme value theory and autoregressive integrated moving average models for forecasting crash risks at signalised intersections. At the first level of this framework, a non-stationary generalised extreme value model has been developed to estimate the real-time rear-end crash risk at the signal cycle level using the video data collected from three signalised intersections in Queensland, Australia. To capture the time-varying effect of different traffic conditions on conflict extremes, traffic flow, speed, shockwave area, and platoon ratio covariates are incorporated into the generalised extreme value model. The signal cycle-level crash risks obtained from the first level form a univariate time series, which is modelled using two variants of autoregressive integrated moving average model to forecast the crash risk of future signal cycles. Results reveal that the autoregressive integrated moving average model with exogenous variables outperforms the model without exogenous variables and can forecast the crash risk for the next 30–35 min with reasonable accuracy. Similarly, results also demonstrate that different crash risk patterns within a typical day are accurately predicted. The proposed framework helps identify the spatiotemporal windows where safety gradually deteriorates over time, thus enabling proactive safety assessment.

Statistical performance of local attractor dimension estimators in non-Axiom A dynamical systems.

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system's global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth's climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.

Impacts of Max-Stable Process Areal Exceedance Calculations to Study Area Sampling Density, Surface Network Precipitation Gage Extent and Density, and Model Fitting Method

Max-stable process (MSP) models can be fit to data collected over a spatial domain to estimate areal-based exceedances while accounting for spatial dependence in extremes. They have theoretical grounding within the framework of extreme value theory (EVT). In this work, we fit MSP models to three-day duration cool season precipitation maxima in the Willamette River Basin (WRB) of Oregon and to 48 h mid-latitude cyclone precipitation annual maxima in the Upper Trinity River Basin (TRB) of Texas. In total, 14 MSP models were fit (seven based on the WRB data and seven based on the TRB data). These MSP model fits were developed and applied to explore how user choices of study area sampling density, gage extent, and model fitting method impact areal precipitation-frequency calculations. The impacts of gage density were also evaluated. The development of each MSP involved the application of a recently introduced trend surface modeling methodology. Significant reductions in computing times were achieved, with little loss in accuracy, applying random sample subsets rather than the entire grid when calculating areal exceedances for the Cougar dam study area in the WRB. Explorations of gage extent revealed poor consistency among the TRB MSPs with modeling the generalized extreme value (GEV) marginal distribution scale parameter. The gauge density study revealed the robustness of the trend surface modeling methodology. Regardless of the fitting method, the final GEV shape parameter estimates for all fourteen MSPs were greater than their prescribed initial values which were obtained from spatial GEV fits that assumed independence among the extremes. When two MSP models only differed by their selected fitting method, notable differences were observed with their dependence and trend surface parameter estimates and resulting areal exceedances calculations.

An Extreme Value Bayesian Lasso for the Conditional Left and Right Tails

We introduce a novel regression model for the conditional left and right tail of a possibly heavy-tailed response. The proposed model can be used to learn the effect of covariates on an extreme value setting via a Lasso-type specification based on a Lagrangian restriction. Our model can be used to track if some covariates are significant for the lower values, but not for the (right) tail—and vice versa; in addition to this, the proposed model bypasses the need for conditional threshold selection in an extreme value theory framework. We assess the finite-sample performance of the proposed methods through a simulation study that reveals that our method recovers the true conditional distribution over a variety of simulation scenarios, along with being accurate on variable selection. Rainfall data are used to showcase how the proposed method can learn to distinguish between key drivers of moderate rainfall, against those of extreme rainfall. Supplementary materials accompanying this paper appear online.

Practical issues with modeling extreme Brazilian rainfall

Accurately quantifying extreme rainfall is important for the design of hydraulic structures, for flood mapping and zoning and for disaster management. In order to produce maps of estimates of 25-year rainfall return levels in Brazil, we selected 893 shorter and 104 longer rainfall time series from the Agência Nacional de Águas (ANA), and applied the framework of extreme value theory. Care was needed to reduce the impact of poor data. Estimates of the shape parameter of the extreme-value model fitted to rainfall data are typically biased, so we discuss an empirical correction that takes into account not only the sample-size bias, but also a so-called penultimate approximation that we use to inform a Bayesian spatial latent variable model for the annual rainfall maxima. This model accounts for subtle patterns of spatial variation in the data and provides plausible return level estimates.

International Capital Flows and Extreme Exchange Market Pressure: Evidence from Emerging Market Economies

In response to the currency crises in the emerging market economies (EMEs) during the 1990s, earlier studies tended to put emphasis on identifying and explaining currency crash, which is an extreme event mostly associated with massive capital reversals. After the 2008 global financial crisis, the focus shifted towards enormous capital inflows which have put a sharp appreciation pressure on domestic currency and inflated a large housing and construction bubble. In this paper, we examine the foreign exchange instabilities of a group of EMEs between 1995Q1 and 2019Q4 using the exchange market pressure (EMP) index by taking into considerations both extreme positive and negative episodes. The identification of tail observations is carried out under the framework of Extreme Value Theory (EVT) to handle asymmetric and heavy-tailed data. A panel multinomial logit model is used to explore whether the predictors differ between extreme positive and negative EMP events. Our findings show that (1) there is asymmetry in the EMP distributions, where the occurrence of currency crises is more frequent than excessive appreciations in most EMEs, (2) portfolio and credit flows are significant predictors to both extreme events, and (3) by distinguishing the residency of capital flows, foreign credit flow is the key factor that contributes to the devaluation pressure in the EMEs.

Assessing the Role of Ordering in Sequential English Auctions – Evidence from the Online Western Video Market Auction

In a sequential English auction, where multiple homogenous objects are offered, it is often found that the price tends to decrease over the course of the auction, which is referred to as the “declining price anomaly”. This article studies the price trend in a sequential online live cattle auction. Motivated by the fact that in an English first‐price auction, the winning price is the maximum among all the bids offered, we examined the existence of the declining price anomaly by introducing the Generalized Extreme Value theory framework, and nesting it within a hedonic regression, where the price of cattle relates to the market values for cattle attributes. Empirical results indicate that the unit price difference between the first and the last lot sold ranges from $7.82 to $2.31, depending on the average animal weight in that lot. On average, the unit price of a stocker cattle lot sold in the last position of a typical sale is $5/cwt (or 3%) less than if it was sold in the first place, regardless of the scale of the sale and keeping all other factors constant. Based on the theoretical framework developed by Brendstrup and Paarsch (2006), we find that the buyer's maximum valuation on the subsequent object actually declines as he wins more objects, which provides a plausible explanation for the declining prices anomaly in sequential auctions.

Forecasting Value-at-Risk of Financial Markets under the Global Pandemic of COVID-19 Using Conditional Extreme Value Theory

The recent global pandemic of coronavirus (COVID-19) has had an enormous impact on the financial markets across the world. It has created an unprecedented level of risk uncertainty, prompting investors to impetuously dispose of their assets leading to significant losses over a very short period. In this paper, the conditional heteroscedastic models and extreme value theory are combined to examine the extreme tail behaviour of stock indices from major economies over the period before and during the COVID-19 pandemic outbreak. Daily returns data of stock market indices from twelve different countries are used in this study. The paper implements a dynamic method for forecasting a one-day ahead Value at Risk. As a first step, a comprehensive in-sample volatility modelling is implemented with skewed Student’s-t distribution assumption and their goodness of fit is determined using information selection criteria. In the second step, the VaR quantiles are estimated with the help of conditional Extreme Value Theory framework and then used to estimate the out-of-sample VaR forecasts. Backtesting results suggest that the conditional EVT based models consistently produce a better 1-day VaR performance compared with conditional models with asymmetric probability distributions for return innovations and maybe a better option in the estimation of VaR. This emphasizes the importance of modelling extreme events in stock markets using conditional extreme value theory and shows that the ability of the model to capture volatility clustering accurately is not sufficient for a correct assessment of risk in these markets.

Generalized dimensions, large deviations and the distribution of rare events

Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a chaotic dynamical system, generalized dimensions take on a dynamical meaning, as they provide the rate function for the large deviations of the first hitting time, which is the (average) time required to connect any two different regions in phase space. We prove this result rigorously under a set of stringent assumptions. As a consequence, the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions. Numerical examples, presented along with the theory, suggest that the validity of this technique reaches far beyond the range covered by the theorem. We state our result within the framework of extreme value theory. This approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure. This suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension. We show an application of this technique to experimental data from climate dynamics.

Statistics of the separation between sliding rigid rough surfaces: Simulations and extreme value theory approach.

When a rigid rough solid slides on a rigid rough surface, it experiences a random motion in the direction normal to the average contact plane. Here, through simulations of the separation at single-point contact between self-affine topographies, we characterize the statistical and spectral properties of this normal motion. In particular, its rms amplitude is much smaller than that of the equivalent roughness of the two topographies and depends on the ratio of the slider's lateral size over a characteristic wavelength of the topography. In addition, due to the nonlinearity of the sliding contact process, the normal motion's spectrum contains wavelengths smaller than the smallest wavelength present in the underlying topographies. We show that the statistical properties of the normal motion's amplitude are well captured by a simple analytic model based on the extreme value theory framework, extending its applicability to sliding-contact-related topics.

Estimation of Tail Index and Value-at-Risk for the TA25 and the USD-ILS Exchange Rate Under Assumption of Pareto Distribution

While it is clear that returns of financial assets are not well described by the normal distribution, it is unclear how best to describe them. One distribution suggested is the Pareto distribution. I apply an extreme value theory framework to estimate the tails of the distributions of returns of the TA25, the Tel-Aviv Stock Exchange's leading stock index and the USD-ILS exchange rate, under the assumption returns follow the Pareto distribution. I find that the left tail of the TA25 is lighter than that of the S&P500, suggesting less extreme events in the TA25 and the right tail of the USD-ILS exchange rate is heavier than the left tail, indicating that there are more extreme events when the ILS weakens against the USD than vice versa. This may be due to central bank intervention. I use the estimated tail indexes to assess the value-at-risk of the TA25 and USD-ILS and find the estimations fit historical values well for relatively high percentiles, which are most problematic to estimate.

Bivariate extreme value modeling for road safety estimation

Surrogate safety measures have been advocated as a complementary approach to study safety from a broader perspective than relying on crash data alone. This study proposes an approach to incorporate different surrogate safety measures in a unified framework for road safety estimation within the bivariate extreme value theory framework. The model structure, model specification, threshold selection method, and parameter estimation method of the bivariate threshold excess model are introduced. Two surrogate safety measures, post encroachment time (PET) and length proportion of merging (LPM), are chosen to characterize the severity of merging events on freeway entrance merging areas. Based on the field data collected along Highway 417 in the City of Ottawa, Ontario, Canada, the bivariate modelling methods with seven distribution functions are applied and compared, and the model with logistic distribution function is selected as the best model. The best bivariate models’ estimation results are then evaluated by comparing them to their two marginal (univariate Generalized Pareto distribution) models. The results show that the bivariate models tend to generate crash estimates that are much closer to observed crashes than univariate models. A more important finding is that incorporating two surrogate safety measures into the bivariate models can significantly reduce the uncertainty of crash estimates. The efficiency of a bivariate model is not evidently better than either of its marginal models, but it is expected to be improved with data of a prolonged observation period. This study is also a step forward in the direction of developing multivariate safety hierarchy models, since models of the safety hierarchy have been predominantly univariate.

Endpoint estimation for observations with normal measurement errors

This paper investigates the estimation of the finite endpoint of a distribution function when the observations are contaminated by normally distributed measurement errors. Under the framework of Extreme Value Theory, we propose a class of estimators for the standard deviation of the measurement errors as well as for the endpoint. Asymptotic theories for the proposed estimators are established while their finite sample performance is demonstrated by simulations. In addition, we apply the proposed methods to the outdoor long jump data to estimate the ultimate limit for human beings in the long jump.

An Extreme Value Model for U.S. Hail Size

The spatial distribution of return intervals for U.S. hail size is explored within the framework of extreme value theory using observations from the period 1979–2013. The center of the continent has experienced hail in excess of 5 in. (127 mm) during the past 30 yr, whereas hail in excess of 1 in. (25 mm) is more common in other regions, including the West Coast. Observed hail sizes show heavy quantization toward fixed-diameter reference objects and are influenced by spatial and temporal biases similar to those noted for hail occurrence. Recorded hail diameters have been growing in recent decades because of improved reporting. These data limitations motivate exploration of extreme value distributions to represent the return periods for various hail diameters. The parameters of a Gumbel distribution are fit to dithered observed annual maxima on a national 1° × 1° grid at locations with sufficient records. Gridded and kernel-smoothed return sizes and quantiles up to the 200-yr return period are determined for the fitted Gumbel distribution. These results are used to illustrate return levels for hail greater than a given size for at least one location within each 1° × 1° grid box for the United States.

The Role of the Skewed Distributions in the Framework of Extreme Value Theory (EVT)

In this paper, we analyze the role of the heavy tail and skewed distribution in market risk estimation (Value at Risk (VaR)). In particular, we are interested in knowing if in the framework of the conditional extreme value theory, the estimation of the volatility model below heavy tail and skewed distribution contributes to improve the VaR estimation respect to these obtained from a symmetric distribution. The study has been carried out for six individual assets belonging to the digital sector: ADP, Amazon, Cerner, Apple, Microsoft and Telefonica. The analysis period runs from January 1st, 2008 to the end of December 2013. Although the evidence found is a little bit weak, the results obtained seem to indicate that the heavy tail and skewed distribution outperforms the symmetric distribution both in terms of accuracy VaR estimations as in terms of firm’s loss function. Furthermore, the market risk capital requirements fixed on the base of the VaR estimations are also lowest below a skewed distribution.

Tail asymptotics of generalized deflated risks with insurance applications

Let X and S∈(0,1) be two independent risk variables. This paper investigates approximations of generalized deflated risks E{XκI{SX>x}} with a flexible constant κ≥0 under extreme value theory framework. Our findings are illustrated by three applications concerning higher-order tail approximations of deflated risks as well as approximations of the Haezendonck–Goovaerts and expectile risk measures. Numerical analyses show that higher-order approximations obtained in this paper significantly improve lower-order approximations.

Conditional independence and conditioned limit laws

Conditioned limit laws constitute an important and well developed framework of extreme value theory that describe a broad range of extremal dependence forms including asymptotic independence. We explore the assumption of conditional independence of X1 and X2 given X0 and study its implication in the limiting distribution of (X1,X2) conditionally on X0 being large. We show that under random norming, conditional independence is always preserved in the conditioned limit law but might fail to do so when the normalisation does not include the precise value of the random variable in the conditioning event.

Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization

The probabilistic framework of extreme value theory is well-known: the dependence structure of large events is characterized by an angular measure on the positive orthant of the unit sphere. The family of these angular measures is non-parametric by nature. Nonetheless, any angular measure may be approached arbitrarily well by a mixture of Dirichlet distributions. The semi-parametric Dirichlet mixture model for angular measures is theoretically valid in arbitrary dimension, but the original parametrization is subject to a moment constraint making Bayesian inference very challenging in dimension greater than three. A new unconstrained parametrization is proposed. This allows for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity of the Markov chain are verified and the algorithm is tested up to dimension five. In this non identifiable setting, convergence monitoring is performed by integrating the sampled angular densities against Dirichlet test functions.

Study of the Tail Dependence Structure in Global Financial Markets Using Extreme Value Theory

Abstract: The presence of tail dependencies invalidates the multivariate normality assumptions in portfolio risk management. The identification of tail (in)dependencies has drawn major attention in empirical financial studies. Yet it is still a challenging issue both theoretically and practically. Previous studies based on either a restrictive model or the null hypothesis of tail (perfect) dependence does not well describe or interpret extreme co-movements in financial markets. This paper examines tail dependence structures underlying a broad range of financial asset classes employing the newly developed tail quotient correlation coefficients. In theory, the original tail quotient correlation coefficient proposed in (Zhang 2008) is adapted to incorporate cases with varying data driven random thresholds. Our empirical results demonstrate different tail dependence structures underlying various global financial markets. Either omission or unanimous treatment of the tail dependence structures for different financial markets will lead to erroneous conclusions or suboptimal investment choices. The multivariate extreme value theory framework in this study has the potential to serve as an useful tool in exploiting arbitrage opportunities, optimizing asset allocations, and building robust risk management strategies.