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The canonical dependence function θ (z), z ∈ [0,1], is introduced and studied in detail for distributions, which belong to the δ-neighborhood of a bivariate generalized Pareto distribution. We establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of n i.i.d. observations and derive efficient estimators of θ (z) from the Hájek-LeCam Convolution Theorem. These...
Methods are given for simulating from symmetric and asymmetric versions of the multivariate logistic distribution, and from other multivariate extreme value distributions based on the well known logistic model. We consider two general approaches. The first approach uses transformations to derive random variables with a joint distribution function from which it is easy to simulate. The second approach...
In the stereological unfolding problem of size and shape factor of spheroidal particles the extremal shape factor is investigated. The transformation of particle parameters is shown to be stable with respect to the domain of attraction in three situations: (a) Conditional distribution of shape factor given particle size. (b) Conditional distribution of shape factor given particle section size. (c)...
Modeling of extreme values in the presence of heterogeneity is still a relatively unexplored area. We consider losses pertaining to several related categories. For each category, we view exceedances over a given threshold as generated by a Poisson process whose intensity is regulated by a specific location, shape and scale parameter. Using a Bayesian approach, we develop a hierarchical mixture prior,...
An asymptotically normally distributed estimate for the expected value of a positive random variable with infinite variance is introduced. Its behavior relative to estimation using the sample mean is investigated by simulations. An example of how to apply the estimate to file-size measurements on Internet traffic is also shown.
Arnold and Villaseñor (1999) raised several questions for upper records, including characterizing all limit distributions of normalized partial sums of upper records. We provide some answers in the case when the distribution from which the samples are drawn is bounded above. When the distribution is not bounded above, we give sufficient conditions on the distribution for the properly normalized partial...
The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample...
The sea elevation at a fixed point is modeled as a sum of a Gaussian process plus a quadratic random correction term. It is shown that the process can also be written as a quadratic form of a vector valued Gaussian process with arbitrary mean. The saddlepoint method is used to approximate the intensity μ (u), say, the sea level crosses the level u. The accuracy of the proposed method is studied. In...
New methods for identifying clusters of extreme values are proposed that exploit additional knowledge of the trajectory of the process around extreme events. These methods lead directly to new estimators of cluster characteristics, such as the extremal index, which are shown to have both substantially reduced bias and greater insensitivity to cluster identification parameters than existing methods...
We evaluate sharp upper bounds for the consecutive spacings of order statistics from an i.i.d. sample, measured in scale units generated by various central absolute moments of the parent distribution. The bounds are based on the projection method combined with the Hölder inequalities. We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained...
The dependence structure in the tails of bivariate random variables is studied by means of appropriate copulae. Weak convergence results show that these copulae are natural dependence structures for joint tail events. The results obtained apply to particular types of copulae such as archimedean copulae and the Gaussian copula. Further, connections to multivariate extreme value theory are investigated...
Let {Xk, 1 ≤ k ≤ n} be n independent and real-valued random variables with common subexponential distribution function, and let {θk, 1 ≤ k ≤ n} be other n random variables independent of {Xk, 1 ≤ k ≤ n} and satisfying a ≤ θk ≤ b for some 0 < a ≤ b < ∞ for all 1 ≤ k ≤ n. This paper proves that the asymptotic relations P (max1 ≤ m ≤ n ∑k=1m θkXk > x) ∼ P (sumk=1n θk...
Let {W(s)}s ≥ 0 be a standard Wiener process. The supremum of the squared Euclidian norm ⊬Y (t)⊬2, of the R2-valued process Y(t)=(√1/tW(t), √ {12/t3 int0ts dW (s)−√ {3/t} W(t)), t ∈ [α, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {supt ∈ [α, 1] ⊬Y(t)√2 >...
General theory on the extremes of stationary processes leads only to a limited representation for extreme-state behaviour, usually summarised by the extremal index. In practice this means that other quantities such as the duration of extreme episodes or aggregate of threshold exceedances within a cluster require stronger model assumptions. In this paper we propose a model based on a Markov assumption...
We present a method for deriving the limiting distribution of the maximum of a normed empirical moment generating function process indexed by one parameter. We first extend slightly the results of Csörgő et al. (1986b) to provide the rate of convergence for a Gaussian approximation to a non-Donsker empirical process. In cases we consider, the maximum tends to infinity in probability, but when appropriately...
Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define Ru(t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (Ru)u>0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence...
We consider the convergence of the maxima of a triangular array of random variables. Sufficient and necessary conditions are discussed, assuming that the underlying distributions are twice differentiable. Our results extend those known for the iid. case.
Let F be a distribution function in the domain of attraction of an extreme value distribution $$\mathcal{H}_{\gamma } $$ . In case γ ≥ 0 and F has an infinite end-point, we study the asymptotic behaviour of the relative approximation error $$\varepsilon _{\alpha } $$ of a high quantile $$q_{\alpha } $$ such that $$1 - F{\left( {q_{\alpha } } \right)} = \alpha $$ , where the order α tends...
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