From: Charles Metz <c-metz@uchicago.edu>
Subject: Re: Distribution function
Date: Fri, 19 Feb 1999 04:44:07 GMT
Newsgroups: sci.stat.math,sci.math,alt.sci.math.probability,alt.sci.math.statistics.prediction
To: "Amir H. Salek" <shayegan@aludra.usc.edu>
Keywords: extreme-value distributions

Amir H. Salek wrote:

 > Can anyone kindly help me with the following question?
 > 
 > ------
 > Is there any distribution function that remains invariant 
 > with respect to the MAX operation? That means, if x and y 
 > have the same type distribution functions (e.g. normal 
 > distribution, etc) MAX{x,y} has that type of distribution
 > function as well.

With b > 0, three such distribution functions are:
 
 F(x) = exp(-exp(-(x-a)/b)) for -infinity <= x <= +infinity;

 F(x) = exp(-(b/x)**k) for 0 < x <= +infinity
      = 0 for x <= 0; 

and

 F(x) = exp((x-a)/b) for -inf <= x <= a
      = 1 for x > a.
  
It's easy to show that all three of the distribution functions above
have the property that you described.

These forms are sometimes known as "extreme-value distributions,"
because each can arise as the asymptotic form of the distribution of the
maximum (or, with a change of sign, the minimum) of N iid random
variables as N becomes large.  Parent distributions that give rise to
the first asymptotic form are known as "exponential type," whereas those
that give rise to the second are known as "Cauchy type."  The third form
above, which I've chosen for simplicity, is actually a special case of a
more general family now often known as the Weibull distribution.

For further details see, Volume 1 of "The Advanced Theory of
Statistics," by Kendall and Stewart (pages 352-356 in my 1979 edition),
which cites a 1928 paper by Fisher and Tippett as the origin of all
three asymptotic forms.

   Charles Metz
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Distribution function
Date: 19 Feb 1999 00:57:26 GMT
Newsgroups: sci.stat.math,sci.math,alt.sci.math.probability,alt.sci.math.statistics.prediction

In article <Pine.GSO.4.02.9902181546450.15820-100000@aludra.usc.edu>, "Amir H. Salek" <shayegan@aludra.usc.edu> writes:

|> ------
|> Is there any distribution function that remains invariant with respect to
|> the MAX operation? That means, if x and y have the same type distribution
|> functions (e.g. normal distribution, etc) MAX{x,y} has that type of
|> distribution function as well.
|> ------

A very important assumption is being left out here, namely that x and y are
independent.  With that assumption, the answer is yes: consider random variables
x such that -x has an exponential distribution.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2