Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (4) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (04) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (03) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (3) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

A Handbook of Second Order Expansions of Quantile Functions and Asymptotic Record Values Laws

2021 ◽  
Author(s):  
Gane Samb Lo ◽  
Moumouni Diallo ◽  
Modou Ngom
Keyword(s):  
Extreme Value Theory ◽  
Domain Of Attraction ◽  
Rates Of Convergence ◽  
Record Values ◽  
Value Theory ◽  
Extreme Value ◽  
Second Order ◽  
Record Value ◽  
Quantile Functions ◽  
The Right

In this monograph, our final objective is to provide second order expansions of quantile functions of as many probability laws as possible. Second order expansions of quantile functions are important tools for finding extreme value domain of attraction of probability laws and for discovering rates of convergence in extreme value theory. We hope that readers will make profit of the results in their works by using the right expansions of quantile functions from the monograph. In that spirit, we apply the quantiles expansions exposed here to deliver the corresponding asymptotic laws of records values. <br><br> In this first edition, fifty four distributions are concerned. For each of those probability laws, full computations for finding the expansion and the asymptotic record value theory are entirely justified. We will regularly update the handbook by adding probability laws in later editions.

Determination of the Probability Plotting Position for Type I Extreme Value Distribution

2012 ◽  
Vol 12 (14) ◽  
pp. 1501-1506 ◽  
Author(s):  
Ahmad Shukri Yah ◽  
Norlida Md. Nor ◽  
Nor Rohashikin ◽  
Nor Azam Ramli ◽  
Fauziah Ahmad ◽  
...  
Keyword(s):  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I

Tree-dependent extreme values: the exponential case

1990 ◽  
Vol 27 (01) ◽  
pp. 124-133 ◽  
Author(s):  
Vijay K. Gupta ◽  
Oscar J. Mesa ◽  
E. Waymire
Keyword(s):  
Branching Process ◽  
Extreme Values ◽  
Rooted Tree ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Extinction Time ◽  
Brownian Excursion ◽  
Weighted Trees

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

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