The largest order statistics for the inradius in an isotropic STIT tessellation Abstract A planar stationary and isotropic STIT tessellation at time t > 0 is observed in the window \(W_{\rho }={t^{-1}}\sqrt {\pi \ \rho }\cdot [-\frac {1}{2},\frac {1}{2}]^{2}\), for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to infinity.

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails Abstract Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

Estimation of extremes for Weibull-tail distributions in the presence of random censoring Abstract The Weibull-tail class of distributions is a sub-class of the Gumbel extreme domain of attraction, and it has caught the attention of a number of researchers in the last decade, particularly concerning the estimation of the so-called Weibull-tail coefficient. In this paper, we propose an estimator of this Weibull-tail coefficient when the Weibull-tail distribution of interest is censored from the right by another Weibull-tail distribution: to the best of our knowledge, this is the first one proposed in this context. A corresponding estimator of extreme quantiles is also proposed. In both mild censoring and heavy censoring (in the tail) settings, asymptotic normality of these estimators is proved, and their finite sample behavior is presented via some simulations.

Modeling extreme negative returns using marked renewal Hawkes processes Abstract Extreme return financial time series are often challenging to model due to the presence of heavy temporal clustering of extremes and strong bursts of return volatility. One approach to model both these phenomena in extreme financial returns is the marked Hawkes self-exciting process. However, the Hawkes process restricts the arrival times of exogenously driven returns to follow a Poisson process and may fail to provide an adequate fit to data. In this work, we introduce a model for extreme financial returns, which provides added flexibility in the specification of the background arrival rate. Our model is a marked version of the recently proposed renewal Hawkes process, in which exogenously driven extreme returns arrive according to a renewal process rather than a Poisson process. We develop a procedure to evaluate the likelihood of the model, which can be optimized to obtain estimates of model parameters and their standard errors. We provide a method to assess the goodness-of-fit of the model based on the Rosenblatt residuals, as well as a procedure to simulate the model. We apply the proposed model to extreme negative returns for five stocks traded on the Australian Stock Exchange. The models identified for the stocks using in-sample data were found to be able to successfully forecast the out-of-sample risk measures such as the value at risk and provide a better quality of fit than the competing Hawkes model.

Improving precipitation forecasts using extreme quantile regression Abstract Aiming to estimate extreme precipitation forecast quantiles, we propose a nonparametric regression model that features a constant extreme value index. Using local linear quantile regression and an extrapolation technique from extreme value theory, we develop an estimator for conditional quantiles corresponding to extreme high probability levels. We establish uniform consistency and asymptotic normality of the estimators. In a simulation study, we examine the performance of our estimator on finite samples in comparison with a method assuming linear quantiles. On a precipitation data set in the Netherlands, these estimators have greater predictive skill compared to the upper member of ensemble forecasts provided by a numerical weather prediction model.

On the accuracy of Poisson approximation Abstract The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. From a practical point of view, it has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, the topic provides insights into Kolmogorov’s problem. The task of establishing an estimate with the best possible constant at the leading term remained open for decades. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We generalise and sharpen the corresponding inequalities of Prokhorov, LeCam, Barbour, Hall, Deheuvels, Pfeifer, and Roos. A new result is established for the intensively studied topic of Poisson approximation to the binomial distribution.

Extremal dependence of random scale constructions Abstract A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2) = R(W1,W2), with non-degenerate R > 0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

Improved estimation of the extreme value index using related variables Abstract Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.

Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d $\mathbb {R}^{d}$ Abstract Let X(t), \(t\in \mathbb {R}\) , be a d-dimensional vector-valued Brownian motion, d ≥ 1. For all \(\boldsymbol {b}\in \mathbb {R}^{d}\setminus (-\infty ,0]^{d}\) we derive exact asymptotics of $$ \mathbb{P}\{\boldsymbol{X}(t+s)-\boldsymbol{X}(t) >u\boldsymbol{b}\text{ for some } t\in[0,T],\ s\in[0,1]\} \quad\text{as } u\to\infty, $$ that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for X; we cover not only the case of a fixed time-horizon T > 0 but also cases where T → 0 or \(T\to \infty \) . Results for high level excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the ‘supremum’ of vector-valued Brownian motions.

Robust quantile estimation under bivariate extreme value models Abstract 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.