Limit Distributions of Multivariate Extreme Values in Nonstationary Sequences of Random Vectors

1989 ◽  
pp. 234-245 ◽  
Author(s):  
J. Hüsler
Keyword(s):  
Extreme Values ◽  
Random Vectors ◽  
Limit Distributions ◽  
Multivariate Extreme Values

scholarly journals Nonparametric Estimation of Multivariate Extreme Value Copulas with Known and Unknown Marginal Distributions

2021 ◽  
Vol 2068 (1) ◽  
pp. 012003
Author(s):  
Ayari Samia ◽  
Mohamed Boutahar
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Extreme Values ◽  
Empirical Distribution ◽  
Extreme Value ◽  
Dependence Function ◽  
Marginal Distributions ◽  
Nonparametric Estimators ◽  
Monte Carlo Experiments ◽  
Multivariate Extreme Values

Abstract The purpose of this paper is estimating the dependence function of multivariate extreme values copulas. Different nonparametric estimators are developed in the literature assuming that marginal distributions are known. However, this assumption is unrealistic in practice. To overcome the drawbacks of these estimators, we substituted the extreme value marginal distribution by the empirical distribution function. Monte Carlo experiments are carried out to compare the performance of the Pickands, Deheuvels, Hall-Tajvidi, Zhang and Gudendorf-Segers estimators. Empirical results showed that the empirical distribution function improved the estimators’ performance for different sample sizes.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (03) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (3) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

LetX(1)≦X(2)≦ ·· ·≦X(N(t))be the order statistics of the firstN(t) elements from a sequence of independent identically distributed random variables, where {N(t);t≧ 0} is a renewal counting process independent of the sequence ofX's. We give a complete description of the asymptotic distribution of sums made from the topktextreme values, for any sequencektsuch thatkt→ ∞,kt/t→ 0 ast→ ∞. We discuss applications to reinsurance policies based on large claims.

scholarly journals Limit theorems for generalized perimeters of random inscribed polygons. I

2020 ◽  
Vol 65 (4) ◽  
pp. 678-687
Author(s):  
Ekaterina N. Simarova ◽  
◽  
Keyword(s):  
Stochastic Geometry ◽  
Limit Theorems ◽  
Extreme Values ◽  
Limiting Behavior ◽  
Limit Distributions ◽  
Random Polygon ◽  
Generalized Perimeter ◽  
Definition Of ◽  
Inscribed Polygons

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

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