Extremes最新文献

In this paper, we present several semiparametric approaches for the inference of univariate and multivariate extremes to resolve the tasks from the EVA (2023) Conference Data Challenge. We implement generalized additive models to capture the flexible relationship for point and interval estimations of the conditional quantiles. We also adopt (L^{p})-quantile to estimate the marginal quantiles of extreme levels. To predict probabilities of multivariate extreme events, we implement conditional methods by Heffernan and Tawn (Royal J. Stat. Soc.: Ser. B (Statistical Methodology) 66(3), 497–546, 2004) and Keef et al. (J. Multivar. Anal. 115, 396–404, 2013). We further validate predicted models, evaluating their performance scores constructed based on the notion of an equally extreme level of quantiles and cross-validation to select the best estimates to achieve high accuracy. When estimating the excess probability of 50-dimensional data, we cluster variables with high correlation after simple data exploration and combine the results obtained from each cluster. Finally, we also provide post-mortem analysis based on the ground truth.

Capturing the extremal behaviour of data often requires bespoke marginal and dependence models which are grounded in rigorous asymptotic theory, and hence provide reliable extrapolation into the upper tails of the data-generating distribution. We present a modern toolbox of four methodological frameworks, motivated by classical extreme value theory, that can be used to accurately estimate extreme exceedance probabilities or the corresponding level in either a univariate or multivariate setting. Our frameworks were used to facilitate the winning contribution of Team Yalla to the EVA (2023) Conference Data Challenge, which was organised for the 13(^text {th}) International Conference on Extreme Value Analysis. This competition comprised seven teams competing across four separate sub-challenges, with each requiring the modelling of data simulated from known, yet highly complex, statistical distributions, and extrapolation far beyond the range of the available samples in order to predict probabilities of extreme events. Data were constructed to be representative of real environmental data, sampled from the fantasy country of “Utopia”.

This paper presents the contribution of Team Yahabe to the EVA (2023) Conference Data Challenge. We tackle the four problems posed by the organizers by revisiting the current and existing literature on conditional univariate and multivariate extremes. We highlight overarching themes linking the four tasks, ranging from model validation at extremely high quantile levels to building customized estimation strategies that leverage model assumptions.

We conduct a non-asymptotic study of the Cross-Validation (CV) estimate of the generalization risk for learning algorithms dedicated to extreme regions of the covariates space. In this context which has recently been analysed from an Extreme Value Analysis perspective, the risk function measures the algorithm’s error given that the norm of the input exceeds a high quantile. The main challenge within this framework is the negligible size of the extreme training sample with respect to the full sample size and the necessity to re-scale the risk function by a probability tending to zero. We open the road to a finite sample understanding of CV for extreme values by establishing two new results: an exponential probability bound on the K-fold CV error and a polynomial probability bound on the leave-p-out CV. Our bounds are sharp in the sense that they match state-of-the-art guarantees for standard CV estimates while extending them to encompass a conditioning event of small probability. We illustrate the significance of our results regarding high dimensional classification in extreme regions via a Lasso-type logistic regression algorithm. The tightness of our bounds is investigated in numerical experiments.

We investigate the behavior of extreme values in Gaussian triangular arrays under strong dependence conditions. By extending previous results, we establish conditions for convergence to a mixture of Gaussian and Gumbel distributions without requiring stationarity. Our findings offer insights into the application of these models, particularly for analyzing air ozone concentrations.

We consider the estimation of the marginal excess moment (MEM), which is defined for a random vector (X, Y) and a parameter (beta >0) as (mathbb {E}[(X-Q_{X}(1-p))_{+}^{beta }|Y> Q_{Y}(1-p)]) provided (mathbb {E}|X|^{beta }< infty ), and where (y_{+}:=max (0,y)), (Q_{X}) and (Q_{Y}) are the quantile functions of X and Y respectively, and (pin (0,1)). Our interest is in the situation where the random variable X is of Weibull-type while the distribution of Y is kept general, the extreme dependence structure of (X, Y) converges to that of a bivariate extreme value distribution, and we let (p downarrow 0) as the sample size (n rightarrow infty ). By using extreme value arguments we introduce an estimator for the marginal excess moment and we derive its limiting distribution. The finite sample properties of the proposed estimator are evaluated with a simulation study and the practical applicability is illustrated on a dataset of wave heights and wind speeds.

For ({varvec{B}_{H}(t)= (B_{H,1}(t) ,ldots ,B_{H,d}(t))^{{top }},tge 0}), where ({B_{H,i}(t),tge 0}, 1le ile d) are mutually independent fractional Brownian motions, we obtain the exact asymptotics of

where A is a non-singular (dtimes d) matrix and (varvec{mu }=(mu _1,ldots , mu _d)^{{top }}in mathbb {R}^d), (varvec{nu }=(nu _1, ldots , nu _d)^{{top }} in mathbb {R}^d) are such that there exists some (1le ile d) such that (mu _i>0, nu _i>0.)

Modelling the extremal dependence of bivariate variables is important in a wide variety of practical applications, including environmental planning, catastrophe modelling and hydrology. The majority of these approaches are based on the framework of bivariate regular variation, and a wide range of literature is available for estimating the dependence structure in this setting. However, such procedures are only applicable to variables exhibiting asymptotic dependence, even though asymptotic independence is often observed in practice. In this paper, we consider the so-called ‘angular dependence function’; this quantity summarises the extremal dependence structure for asymptotically independent variables. Until recently, only pointwise estimators of the angular dependence function have been available. We introduce a range of global estimators and compare them to another recently introduced technique for global estimation through a systematic simulation study, and a case study on river flow data from the north of England, UK.

We consider the random connection model in which an edge between two Poisson points at distance r is present with probability g(r). We conduct an extreme value analysis on this model, namely by investigating the longest edge with at least one endpoint within some finite observation window, as the volume of this window tends to infinity. We show that the length of the latter, after normalizing by some appropriate centering and scaling sequences, asymptotically behaves like one of each of the three extreme value distributions, depending on choices of the probability g(r). We prove our results by giving a formal construction of the model by means of a marked Poisson point process and a Poisson coupling argument adapted to this construction. In addition, we study a discrete variant of the model. We obtain parameter regimes with varying behavior in our findings and an unexpected singularity.

Hüsler–Reiss vectors and Brown–Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components of the bivariate Hüsler–Reiss vector, extend these to the case of the Brown–Resnick field, and thoroughly study the properties of the resulting dependence measure. The use of correlation is justified by spatial risk theory, while power transforms are insightful when taking correlation as dependence measure, and are moreover very suited damage functions for weather events such as wind extremes or floods. This makes our theoretical results worthwhile for, e.g., actuarial applications. We finally perform a case study involving insured losses from extreme wind speeds in Germany, and obtain valuable conclusions for the insurance industry.