Suppose the number of observations associated with rank j is
nj , 1 ≤ j ≤ n, and the corresponding proportion is
qj = nj/N
, where N = Summation( nj )
For each rank j, the observations xi(j), i= 1, ..., nj ,
are independent, which can be taken as realizations of the j-th order statistics.
The PDF and CDF of the j-th order statistics are given by fj (·) and Fj (·), respectively, where
fj (x) = g(F(x); j, n + 1 − j),
Fj (x) = G(F(x); j, n + 1 − j),
where F(·) is the CDF of the population from which samples are being taken, and g(·) and
G(·) are the PDF and CDF of Beta distribution, respectively.