Distribution of order statistics.

(i) If Z1,..., ZN is a sample from a cumulative distribution function F with density f , the joint density of Yi = Z(si), i = 1,..., n, is N! f (y1)... f (yn)

(s1 − 1)!(s2 − s1 − 1)! ...(N − sn)! (6.67)

×[F(y1)]

s1−1

[F(y2) − F(y1)]

s2−s1−1 ... [1 − F(yn)]

N−sn for y1 < ··· < yn.
(ii) For the particular case that the Z’s are a sample from the uniform distribution on (0,1), this reduces to N!
(s1 − 1)!(s2 − s1 − 1)! ...(N − sn)! (6.68)
ys1−1 1 (y2 − y1)
s2−s1−1 ...(1 − yn)N−sn .
For n = 1, (6.68) is the density of the beta distribution Bs,N−s+1, which therefore is the distribution of the single order statistic Z(s) from U(0, 1).
(iii) Let the distribution of Y1,..., Yn be given by (6.68), and let Vi be defined by Yi = ViVi+1 ... Vn for i = 1,..., n. Then the joint distribution of the Vi is N!
(s1 − 1)! ...(N − sn)!
&n i=1 vsi−1 i (1 − vi)
si+1−si−1 (sn+1 = N + 1), so that the Vi are independently distributed according to the beta distribution Bsi−i+1,si+1−si .
[(i): If Y1 = Z(s1),..., Yn = Z(sn ) and Yn+1,..., YN are the remaining Z’s in the original order of their subscripts, the joint density of Y1,..., Yn is N(N − 1)...(N −
n + 1)
 ...  f (yn+1)... f (yN ) dyn+1 ... dyN integrated over the region in which s1 − 1 of the y’s are < y1,s2 − s1 − 1 between y1 and y2,..., and N − sn > yn.
Consider any set where a particular s1 − 1 of the y’s is < y1, a particular s2 − s1 −
1 of them is between y1 and y2, and so on, There are N!/(s1 − 1)! ...(N − sn)!
of these regions, and the integral has the same value over each of them, namely [F(y1)]
s1−1[F(y2) − F(y1)]
s2−s1−1 ... [1 − F(yn)]N−sn .]