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Exact asymptotics of component-wise extrema of two-dimensional Brownian motion
Springer Science and Business Media LLC - Tập 23 - Trang 569-602 - 2020
We derive the exact asymptotics of
$ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty , $
where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 lead......
On the regular variation of ratios of jointly Fréchet random variables
Springer Science and Business Media LLC - - 2012
Extreme value theory for anomaly detection – the GPD classifier
Springer Science and Business Media LLC - Tập 23 - Trang 501-520 - 2020
Classification tasks usually assume that all possible classes are present during the training phase. This is restrictive if the algorithm is used over a long time and possibly encounters samples from unknown new classes. It is therefore fundamental to develop algorithms able to distinguish between normal and abnormal test data. In the last few years, extreme value theory has become an important to......
Consistency of Hill estimators in a linear preferential attachment model
Springer Science and Business Media LLC - Tập 22 - Trang 1-28 - 2018
Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Statistical estimates of the tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics. The consistency of the Hill estimator for network data has not been explored and the major goal in this paper is to prov......
A Location Invariant Hill-Type Estimator
Springer Science and Business Media LLC - Tập 4 - Trang 199-217 - 2001
The Hill's estimator (Hill, 1975) has been largely used in extreme value theory in order to estimate the tail index associated to a distribution function with a positive index. One of the criticisms to its use is the possible associated bias, dependent on the top portion of the original sample used, and also the fact that it is not location invariant. Here, a new Hill-type estimator is studied, wh......
A formula for hidden regular variation behavior for symmetric stable distributionsAbstract We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation . We give... ...
Springer Science and Business Media LLC - - 2020
A general estimator for the extreme value index: applications to conditional and heteroscedastic extremes
Springer Science and Business Media LLC - Tập 18 - Trang 479-510 - 2015
The tail behavior of a survival function is controlled by the extreme value index. The aim of this paper is to propose a general procedure for the estimation of this parameter in the case where the observations are not necessarily distributed from the same distribution. The idea is to estimate in a consistent way the survival function and to apply a general functional to obtain a consistent estima......
On convergence of extremes under power normalization
Springer Science and Business Media LLC - Tập 16 - Trang 285-301 - 2012
In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown t......
Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology*
Springer Science and Business Media LLC - Tập 3 - Trang 207-229 - 2000
In this paper, and in a context of regularly varying tails, we propose different alternatives to a well-known estimator of the tail index—the Hill estimator (Hill, 1975). These alternatives have essentially in mind a reduction in bias, preferably without increasing Mean Square Error, by the use of suitable Generalized Jackknife methodologies (Gray and Schucany, 1972). The first estimate obtained t......
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