On the rate of convergence of normal extremes

1979 ◽  
Vol 16 (2) ◽  
pp. 433-439 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Rate Of Convergence ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Lévy Metric ◽  
Supremum Metric

Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn)/an to the extreme value distribution exp(–e–x), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.

On the rate of convergence of normal extremes

1979 ◽  
Vol 16 (02) ◽  
pp. 433-439 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Rate Of Convergence ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Lévy Metric ◽  
Supremum Metric

Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn )/an to the extreme value distribution exp(–e–x ), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (04) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ. If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u (x) converges to G γ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅ u (x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (4) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution Hγ. If Fu is the distribution function of the excesses over u and Gγ the distribution function of the generalized Pareto distribution, then it is well known that Fu(x) converges to Gγ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅u(x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

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