We are interested in the tail behavior of the randomly weighted sum $ \sum _{i=1}^{n}\theta _{i}X_{i}$ , in which the primary random variables X 1, …, X n are real valued, independent and subexponentially distributed, while the random weights 𝜃 1, …, 𝜃 n are nonnegative and arbitrarily dependent, but independent of X 1, …, X n . For various important cases, we prove that the tail probability of $\sum _{i=1}^{n}\theta _{i}X_{i}$ is asymptotically equivalent to the sum of the tail probabilities of 𝜃 1 X 1, …, 𝜃 n X n , which complies with the principle of a single big jump. An application to capital allocation is proposed.

Micro-data of European Union (EU) countries show that capital incomes account for a large part of disparity in populations and follow heavy-tailed distributions in many EU countries. Measuring and comparing the disparity requires incorporating the relative nature of ‘small’ and ‘large,’ and for this reason we employ the newly developed Zenga index of economic inequality. Its non-parametric estimator does not fall into any well known class of statistics. This makes the development of statistical inference a challenge even for light-tailed populations, let alone heavy-tailed ones, as is the case with capital incomes. In this paper we construct a heavy-tailed Zenga estimator, establish its asymptotic distribution, and derive confidence intervals. We explore the performance of the confidence intervals in a simulation study and draw conclusions about capital incomes in EU countries, based on the 2001 wave of the European Community Household Panel (ECHP) survey.

A sub-model of multivariate regular variation called hidden regular variation facilitates more accurate estimation of joint tail probabilities in the presence of asymptotic independence. A related concept called hidden domain of attraction can sometimes offer similar estimation assistance in circumstances where hidden regular variation is absent. Examples and discussion illustrate strengths and limitations of this concept. We outline estimation techniques where applicable.

A new approach to extreme value theory is presented for vector data with heavy tails. The tail index is allowed to vary with direction, where the directions are not necessarily along the coordinate axes. Basic asymptotic theory is developed, using operator regular variation and extremal integrals. A test is proposed to judge whether the tail index varies with direction in any given data set.

Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating “once per 10,000 years” still water level at Hoek van Holland, The Netherlands.

Under mild conditions on the covariance function of a stationary Gaussian process, the maxima behaves asymptotically the same as the maxima of independent, identically distributed Gaussian random variables. In order to achieve extremal clustering, Hsing et al. (Ann Appl Probab 6:671–686, 1996) considered a triangular array of Gaussian sequences in which the correlation between “neighboring” observations approaches 1 at a certain rate. Using analogues of the conditions of Hsing et al., which allows for strong local dependence among variables but asymptotic independence, it is possible to show that two-dimensional Gaussian random fields also exhibit extremal clustering in the limit. A closed form expression for the extremal index governing the clustering will be provided. The results apply to Gaussian random fields in which the spatial domain is rescaled.

Asymptotic properties of a shot noise sequence, {Xj = ∑ i ≤ jh(τj − τi)Ai}, are considered, where the marginal distribution of the Ai’s is regularly varying at infinity with negative index − α. The tail distribution of the Xj’s is studied with regard to higher order tail area expansions, validity of the Von Mises condition and the inheritability of the second order regular variation property. The Hill’s estimator for the tail index is shown to be asymptotically normally distributed provided the impulse response function h satisfies a mild integrability condition and the distribution of the Xj’s satisfies some regularity conditions.

Let X1,X2, ⋯ be a sequence of independent and identically distributed random variables and Mn = max {X1,X2, ⋯ , Xn}. Suppose that some of the random variables X1,X2, ⋯ , Xn can be observed and denote by \(\widetilde{M}_{n}\) the maximum of the observed random variables from the set {X1,X2, ⋯ , Xn}. The limiting distribution of random vector \((M_{n}-\widetilde{M}_{n}, \widetilde{M}_{n})\) is derived. The result is also extended to the case of stationary Gaussian sequences. In the end, the almost sure limit theorem on \(M_{n}-\widetilde{M}_{n}\) for a sequence of independent and identically distributed random variables is proved.

We provide a necessary and sufficient condition for the ratio of two jointly α-Frechet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).

The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution.