[R] probability density function for maximum values in repeated finite samples from a normal distribution??

[David Winsemius]

On Sep 28, 2009, at 8:15 AM, Joerg van den Hoff wrote:

> this is probably not really a R specific question, if so apologies for
> off-topic posting:
>
> I'm interested in the probability density function of the maximum  
> values
> from repeated samples of size N from a normal distribution:
>
> smp <- rnorm(N, meanval, stdev)
>
> with some mean 'meanval' and standard deviation 'stdev'.
>
> I would like to know what is the frequency distribution of max(smp)  
> if I draw many such
> samples?
>
> if I investigate this simply via a simulation, I get of course  
> approximate
> results (and see that the resulting distribution is not quite normal
> anymore, that the mean increases with increasing N, etc.).
>
> my question: does somebody know whether there exists an analytical
> expression for the distribution of max(smp) (or where to look)?

Yes, there is an analytical description of the highest order statistic  
for Normal data. The question has been studied extensibly. You should  
be searching on "extreme value theory" or "extremal distributions".  
There is also an extRemes package for R, as well as an fExtremes  
package for financial applications and a SpatialExtremes package.
-- 

David Winsemius, MD
Heritage Laboratories
West Hartford, CT