Selection of the K Largest Order Statistics for the Domain of Attraction of the Gumbel Distribution

1995 ◽  
Vol 90 (431) ◽  
pp. 1055 ◽  
Author(s):  
Julian Z. Wang
Keyword(s):  
Order Statistics ◽  
Domain Of Attraction ◽  
Gumbel Distribution ◽  
Selection Of

scholarly journals A Double-indexed Functional Hill Process and Applications

2016 ◽  
Vol 8 (4) ◽  
pp. 144
Author(s):  
Modou Ngom ◽  
Gane Samb Lo
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Order Statistics ◽  
Finite Dimension ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Extreme Value Index

<div>Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} ,  \label{fme}<br />\end{equation}</div><div> </div><div>indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.</div>

scholarly journals Extremes of autoregressive threshold processes

2009 ◽  
Vol 41 (2) ◽  
pp. 428-451 ◽  
Author(s):  
Claudia Brachner ◽  
Vicky Fasen ◽  
Alexander Lindner
Keyword(s):  
Stationary Solution ◽  
Domain Of Attraction ◽  
Gumbel Distribution ◽  
Stationary Solutions ◽  
Threshold Autoregressive ◽  
Extremal Behavior ◽  
Process Convergence ◽  
Dimensional Distribution ◽  
Regularly Varying ◽  
Noise Sequence

In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ > 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

scholarly journals Aggregation of rapidly varying risks and asymptotic independence

2009 ◽  
Vol 41 (3) ◽  
pp. 797-828 ◽  
Author(s):  
Abhimanyu Mitra ◽  
Sidney I. Resnick
Keyword(s):  
Optimization Problem ◽  
Domain Of Attraction ◽  
Gumbel Distribution ◽  
Asymptotic Independence ◽  
Tail Behavior ◽  
Maximal Domain ◽  
Linear Combinations ◽  
Marginal Distributions ◽  
Positive Coefficients ◽  
Portfolio Design

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X,Y) such that P(X+Y>x) ∼ (constant) P(X>x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions ofXandYare subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

scholarly journals Automated selection of r for the r largest order statistics approach with adjustment for sequential testing

2016 ◽  
Vol 27 (6) ◽  
pp. 1435-1451 ◽  
Author(s):  
Brian Bader ◽  
Jun Yan ◽  
Xuebin Zhang
Keyword(s):  
Order Statistics ◽  
Sequential Testing ◽  
Selection Of
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