🚀 go-pugleaf

Thanks to everyone who responded to my question about the
density of the max order statistic for binomially distributed
random variables, including:

B. Narasimhan (naras@cda.mrs.umn.edu)
Herman Rubin  (hrubin@pop.stat.purdue.edu)
Tom Permutt   (tpermutt@medinfo.ab.umd.edu)
Terry Moore   (T.Moore@massey.ac.nz)
Fouad Younan  (sasfgy@unx.sas.com)
Ernie Scheuer (VCMGT001@vax.csun.edu)

By the third message, I had enough info to work it out, and the
final message from Ernie Scheuer confirmed my answer.
The last treatment was the most complete, so I will include it
here for anyone who also wanted to know the answer.

Thanks again!
/Keith

------------------------------------------------
From: VCMGT001@vax.csun.edu (Ernie Scheuer)
Subject: Dist. of the max of several binomials

Here's a treatment of the query you posted in sci.math.stat.

Y1,...,YK are iid binom(n,p).

Let:
S(y;n,p) = sum(j=0 to j=y)[(n-binomial-j)*(p^j)*((1-p)^(n-j))]

That is, S(y;n,p) = P[Yi <= y], i = 1,..., K .

Let Y = max(Y1,...,YK) .  Then
                              
P[Y <= y] = P[max(Y1,...,YK) <= y] = P[Y1 <= y,...,YK <= y]
          
          = {P[Yi <= y]}^K = {S(y;n,p)}^K


You wanted the density for Y:

P[Y = y] = P[Y <= y] - P[Y <= y-1] for 0 < y <= n;
P[Y = 0] = (1-p)^(nK).

For y > 0 the expression for P[Y = y] is the difference of two
polynomials of degree K*n;  if you want this closed form, I 
suppose you could get it, but it wouldn't be "pretty."


There are tables of, and computer programs for, S(y;n,p).  


Caution:  S(y;n,p) could properly be called the cumulative
          distribution function for the binom(n,p) random
          variable.  The term cumulative binomial distribution
          is sometimes used for 1 - S(y-1;n,p), the probability
          that a binom(n,p) random variable is greater than or
          equal to y.  Moral: look carefully at binomial tables 
          or programs to see exactly what they provide.

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