Summary: Gumbel Distribution

["Lambert.Winnie" <lambert.winnie@ensco.com>]

Thanks much to Sam Buttrey, Ravi Varadhan, Rolf Turner, and Simon Rosenfeld.  All inputs were helpful, but the suggestion that held the clue needed by my co-worker was from Rolf:

 

Your co-worker may have omitted the normalizing constants. The relevant fact is that if

X_(n) = max{X_1, ..., X_n}

where the X_i are iid random variables from a distribution

``in the domain of attraction'' of the Gumbel distribution then

P((X_(n) - a_n)/b_n <= x) ---> G(x) = exp(-exp(-x))

as n ---> infinity, where the a_n and b_n are ``normalizing constants'' which depend on the particular distribution that the X_i come from.

The normalizing constants for the standard normal distribution are:

ln(4 pi) + ln ln n

a_n = sqrt(2 ln n) - -------------------

2(sqrt(2 ln n))

1

b_n = ------------------

sqrt(2 ln n)

That is: If you repeatedly generate samples of size n (where n is

``large'') from a N(mu,sigma^2) distribution, standardize these samples (form Z_i = (X_i - mu)/sigma), take the max of the Z_i, and then form G = (Z_max - a_n)/b_n and thereby get a whole bunch of G's, then the histogram of the G's will look very much like the graph of G(x) = exp(-exp(-x)).

 

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Winifred C. Lambert   Senior Scientist/Meteorologist
ENSCO, Inc.
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