[R] Extreme value distributions (Long.)

[Jorge Luis Ojeda Cabrera]

Rolf Turner escribió:
> 
> This may not actually be an R/Splus problem, but it started
> off that way .....
> 
> ===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===
> Executive summary:
> ==================
> 
> Simulations involving extreme value distributions seem to ``work''
> when the underlying distribution is exponential(1) or exponential(2)
> == chi-squared_2, but NOT when the underlying distribution is
> chi-squared_1.
> 
> Can anyone make an educated conjecture as to why?
> ===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===
> 
> More (much more!) detail:
> =========================
> 
> I have recently been doing some simulations which relate to extreme
> value distributions.  I have observed a phenomenon which puzzles me,
> and I would appreciate it if anyone could shed some light on the
> puzzle.  The phenomenon occurs in both R and Splus.  (Also, I have
> now discovered, in stand-alone Fortran.)
> 

> ...
> ...

>                                         cheers,
> 
>                                                 Rolf Turner
>                                                 rolf at math.unb.ca
> 

It is a quite disappointing phenomena involving extremal value
distribution.
We have experienced the same behaviour with normal variates and gaussian
processes.
In "Extremes and Related Properties of Random Sequences and Processes",
Leadbetter et Al. (1983), pointed out that the convergence rate was
$O(\log^{-1} n)$, at least in the case of normal variates. I guess it is
almost the same in case of the other random variates.
On the other hand, "On convergence rates of supreme" Hall, P. (1991),
suggest that bootstrap approximation should work better, and as far I
know this has been also pointed
in a recent article from Cs\''orgo\'' M. I guess. These articles are
devoted to processes, and they are developed in a rather theoretical
framework. Although I am not sure if bootstrap work for a size n sample,
I guess should do. 

In order to do bootstrap, I found useful 'boot' package.

Let me know if you find further references for the non gaussian case,
and if bootstrap worked.

Jorge Ojeda.
Dep. Métodos Estadísticos.
U. Zaragoza.
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