Conditional extreme value models: fallacies and pitfalls

Bi-free extreme values

We consider heavy tailed time series whose finite-dimensional distributions are extremally independent in the sense that extremely large values cannot be observed consecutively. This calls for methods beyond the classical multivariate extreme value theory which is convenient only for extremally dependent multivariate distributions. We use the Conditional Extreme Value approach to study the effect of an extreme value at time zero on the future of the time series. In formal terms, we study the limiting conditional distribution of future observations given an extreme value at time zero. To this purpose, we introduce conditional scaling functions and conditional scaling exponents. We compute these quantities for a variety of models, including Markov chains, exponential autoregressive models, stochastic volatility models with heavy tailed innovations or volatilities.

Abstract. For a wide range of applications in hydrology, the probability distribution of precipitation maxima represents a fundamental quantity to build dykes, propose flood planning policies, or more generally, to mitigate the impact of precipitation extremes. Classical Extreme Value Theory (EVT) has been applied in this context by usually assuming that precipitation maxima can be considered as Independent and Identically Distributed (IID) events, which approximately follow a Generalized Extreme Value distribution (GEV) at each recording site. In practice, weather stations records can not be considered as independent in space. Assessing the spatial dependences among precipitation maxima provided by two Belgium measurement networks is the main goal of this work. The pairwise dependences are estimated by a variogram of order one, also called madogram, that is specially tailored to be in compliance with spatial EVT and to capture EVT bivariate structures. Our analysis of Belgium precipitation maxima indicates that the degree of dependence varies greatly according to three factors: the distance between two stations, the season (summer or winter) and the precipitation accumulation duration (hourly, daily, monthly, etc.). Increasing the duration (from one hour to 20 days) strengthens the spatial dependence. The full independence is reached after about 50 km (100 km) for summer (winter) for a duration of one hour, while for long durations only after a few hundred kilometers. In addition this dependence is always larger in winter than in summer whatever is the duration. An explanation of these properties in terms of the dynamical processes dominating during the two seasons is advanced.

This paper presents a new model for characterizing temporal dependence in exceedances above a given threshold. Our model is based on a class of stationary, infinitely divisible stochastic processes known as trawl processes. For use with extreme values, our model is constructed by embedding a trawl process in a hierarchical framework. This ensures that the marginal distribution is a generalized Pareto, as expected from classical extreme value theory. We also consider a modified version of this model that works with a wider class of generalized Pareto distributions (GPDs) and has the advantage of separating marginal and temporal dependence properties. The model is illustrated via various applications to environmental time series; thus, we show that the model offers considerable flexibility in capturing the dependence structure of extreme value data.

For integers n ≥ r, we treat the rth largest of a sample of size n as an $\mathbb {R}^{\infty }$ -valued stochastic process in r which we denote as M(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M(r) and M(r) itself, after norming and centering. In continuous time, an analogous process Y(r) based on a two-dimensional Poisson process on $\mathbb {R}_{+}\times \mathbb {R}$ is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t > 0. This necessitates a different approach to the asymptotics in this case.

The problem of extreme wind prediction for determination of design wind speed is considered. Drawbacks of the presently used method based on the classical extreme-value theory are pointed out, and a step toward better statistical treatment of data is presented. The new procedure, which is based on estimating the tail of a probability distribution, forms a powerful and flexible class of alternatives to the traditional methods. Here, rather than the annual maxima, all the large values are included in the analysis, regardless of their occurrence time. For the parametric family of distributions involved (which take only three forms, like the classical extreme-value theory), the available estimating methods are described, and an illustrating example is presented using a set of published data. The general advantages of the procedure over the subsample method developed by Gumbel are examined.

• Proposes the Multimodal Bivariate Extreme Value (MMBEV) distribution. • Extends bivariate extreme value theory to capture multimodal dependencies. • Applies MMBEV to model extreme climate data with complex dependencies. • Demonstrates MMBEV’s flexibility through simulations and real-world analysis. • Establishes links between MMBEV and bivariate Weibull distributions. Modeling multivariate extreme values in complex systems increasingly demands flexible distributions capable of capturing multimodal behavior. Classical bivariate extreme value (BEV) distributions, though well established, are inherently unimodal and may inadequately represent data with heterogeneity or regime-switching—features commonly observed in environmental and financial applications. This paper introduces the Multimodal Bivariate Extreme Value (MBEV) distribution, a new class that extends the traditional BEV framework to accommodate multimodal structures through additional shape parameters. Unlike previous generalizations restricted to specific cases such as the bivariate Gumbel model, the MBEV encompasses the entire BEV family. We explore theoretical properties of the MBEV model, including its stochastic representation and its connection to the bivariate Weibull distribution. Through Monte Carlo simulations, we assess the performance of maximum likelihood estimators and demonstrate their robustness. An application to climate data from Brasília, Brazil, illustrates the practical value of the MBEV model in capturing complex dependencies among extreme variables such as wind gust speed, relative humidity, and dew point temperature. Model selection criteria (AIC and BIC) confirm the superiority of the MBEV model over classical BEV distributions. Overall, the MBEV model offers a flexible and interpretable framework for modeling bivariate extremes, with potential applications in climatology, survival analysis, reliability, and finance. This work advances the frontier of extreme value modeling by addressing multimodal dependence in heterogeneous multivariate contexts.

PurposeIn this paper, the aim is to investigate the tail behavior of daily stock returns for three emerging stock in the Gulf region (Bahrain, Oman, and Saudi Arabia) over the period 1998‐2005. In addition, the aim is also to test whether the distributions are similar across these markets.Design/methodology/approachFollowing McNeil and Frey, Wanger and Marsh, and Bystrom, extreme value theory (EVT) methods are utilized to examine the asymptotic distribution of the tail for daily returns in the Gulf region. As a first step and to obtain independent and identically distributed residuals series, the returns are prefiltered with an ordinary time‐series model, taking into account the observed Gulf return dynamics. Then, the “Peaks‐Over‐Threshold” (POT) model is applied to estimate the tails of the innovational distribution.FindingsNot only is the heavy tail found to be a facial appearance in these markets, but also POT method of modelling extreme tail quantiles is more accurate than conventional methodologies (historical simulation and normal distribution models) in estimating the tail behavior of the Gulf markets returns. Across all return series, it is found that left and right tails behave very different across countries.Research limitations/implicationsThe results show that risk models that are able to exploit tail behavior could lead to more accurate risk estimates. Thus, participants in the Gulf equity markets can rely on EVT‐based risk model when assessing their risks.Originality/valueThe paper extends previous studies in two aspects. First, it extends the classical unconditional extreme value approach by first filtering the data by using AR‐FIAPARCH model to capture some of the dependencies in the stock returns, and thereafter applying ordinary extreme value techniques. Second, it provides a broad analysis of return dynamics of the Gulf markets.

The classical extreme value theory based on generalized extreme value (GEV) distribution and generalized Pareto distribution (GPD) is applied to the wave height estimate based on wave hindcast data covering a period of 31 years for a location in the eastern Arabian Sea. Practical concern such as the threshold selection and model validation is discussed. Estimates of wave height having different return periods are compared with estimates obtained from different distributions. On comparing the distributions fitted to the GEV with annual maximum approach and GPD with peaks over threshold approach have indicated that both GEV and GPD models gave similar or comparable wave height for the study area since there is no multiple storm event in a year. Influence of seasonality on wave height having different return period is also studied.

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

The largest possible earthquake magnitude based on geographical characteristics for a selected return period is required in earthquake engineering, disaster management, and insurance. Ground-based observations combined with statistical analyses may offer new insights into earthquake prediction. In this study, to investigate the seismic characteristics of different geographical regions in detail, clustering was used to provide earthquake zoning for Mainland China based on the geographical features of earthquake events. In combination with geospatial methods, statistical extreme value models and the right-truncated Gutenberg–Richter model were used to analyze the earthquake magnitudes of Mainland China under both clustering and non-clustering. The results demonstrate that the right-truncated peaks-over-threshold model is the relatively optimal statistical model compared with classical extreme value theory models, the estimated return level of which is very close to that of the geographical-based right-truncated Gutenberg–Richter model. Such statistical models can provide a quantitative analysis of the probability of future earthquake risks in China, and geographical information can be integrated to locate the earthquake risk accurately.

We show how to obtain theoretical and numerical estimates of correlation dimension and phase space contraction by using the extreme value theory. The maxima of suitable observables sampled along the trajectory of a chaotic dynamical system converge asymptotically to classical extreme value laws where: (i) the inverse of the scale parameter gives the correlation dimension and (ii) the extremal index is associated with the rate of phase space contraction for backward iteration, which in dimension 1 and 2, is closely related to the positive Lyapunov exponent and in higher dimensions is related to the metric entropy. We call it the Dynamical Extremal Index. Numerical estimates are straightforward to obtain as they imply just a simple fit to a univariate distribution. Numerical tests range from low dimensional maps, to generalized Henon maps and climate data. The estimates of the indicators are particularly robust even with relatively short time series.

Spinodal decomposition describes the growth of domains in a system that is quenched from a state where a single homogenous phase is stable to a state where two (or more) phases coexist in equilibrium. The classical theories (Lifshitz, 1962; Lifshitz and Slyozov, 1961), predict that during the growth the typical size R of the domains as a function of time t can be described by a power law, R ∝ t a . The exponent a is independent of microscopic details but does depend on the conservation laws. There have been numerous studies confirming this prediction, both experimentally and numerically (Gunton et al., 1983; Furukawa, 1985). Contrary to the case of a non-conserved order parameter where there is excellent agreement with theory, experiments and numerical calculations for the conserved order parameter case yield exponents that are systematically smaller than the classical value a = 1/3. This difference is attributed to finite size corrections which mask the asymptotic behaviour in the accessible range of domain sizes (Huse, 1986). Recent experimental studies were performed in alloys (Gaulin and Spooner, 1987), polymer blends (Wiltzius et al., 1988), and microemulsions (Roux, 1986), and theoretical calculations include lattice gas simulations (Amer et al., 1988), solutions of the Cahn-Hillard equation (Oono and Puri, 1987) and a cell dynamics approach (Toral et al., 1988).

Current estimates of the European windstorm climate and their associated losses are often hampered by either relatively short, coarse resolution or inhomogeneous datasets. This study tries to overcome some of these shortcomings by estimating the European windstorm climate using dynamical seasonal-to-decadal (s2d) climate forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF). The current s2d models have limited predictive skill of European storminess, making the ensemble forecasts ergodic samples on which to build pseudoclimates of 310–396 yr in length. Extended winter (October–April) windstorm climatologies are created using scalar extreme wind indices considering only data above a high threshold. The method identifies up to 2363 windstorms in s2d data and up to 380 windstorms in the 40-yr ECMWF Re-Analysis (ERA-40). Classical extreme value analysis (EVA) techniques are used to determine the windstorm climatologies. Differences between the ERA-40 and s2d windstorm climatologies require the application of calibration techniques to result in meaningful comparisons. Using a combined dynamical–statistical sampling technique, the largest influence on ERA-40 return period (RP) uncertainties is the sampling variability associated with only 45 seasons of storms. However, both maximum likelihood (ML) and L-moments (LM) methods of fitting a generalized Pareto distribution result in biased parameters and biased RP at sample sizes typically obtained from 45 seasons of reanalysis data. The authors correct the bias in the ML and LM methods and find that the ML-based ERA-40 climatology overestimates the RP of windstorms with RPs between 10 and 300 yr and underestimates the RP of windstorms with RPs greater than 300 yr. A 50-yr event in ERA-40 is approximately a 40-yr event after bias correction. Biases in the LM method result in higher RPs after bias correction although they are small when compared with those of the ML method. The climatologies are linked to the Swiss Reinsurance Company (Swiss Re) European windstorm loss model. New estimates of the risk of loss are compared with those from historical and stochastically generated windstorm fields used by Swiss Re. The resulting loss-frequency relationship matches well with the two independently modeled estimates and clearly demonstrates the added value by using alternative data and methods, as proposed in this study, to estimate the RP of high RP losses.

In this chapter, we discuss an algorithm for the adaptive estimation of a positive extreme value index, γ, the primary parameter in Statistics of Extremes. Apart from classical extreme value index estimators, we suggest the consideration of associated second-order corrected-bias estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds.