From: Charles Metz 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" 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 , "Amir H. Salek" 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