General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n pairs of independent draws from F(x)? Less technically, what is the distribution of the maximum associated with the kth greatest (of n) minima?
A specific example: Assume 8 independent draws from cdf F(x), which is defined over 0 to 1. Then, arbitrarily group the draws into 4 pairs. Compare the minimums of each pair. Label the maximum of these minimums as “a”. Label a’s pair (which is by definition > a) as “b”. Now, choose among the other three pairs arbitrarily, and label the two values in that pair as “c” and “d” (where c is the min of the pair and d is the max of the pair).
What are the distributions of b and d?
I know the distribution of a: F(a) = (1-(1-F(x))^2)^4 =Max of 4 draws of the Min of 2 draws of F(x).
I also know the distribution of c: F(c) = mixture of 1st , 2nd, and 3rd order statistics of 4 draws of Min of 2 draws of F(x). I get this by averaging the integrals (wrt x) for the pdfs that result from substituting (k=1, n=4), (k=2,n=4) and (k=3, n=4) into the following equation: (n!/((k - 1)!(n - k)!))(F(x)^(k - 1))*((1 - F(x))^(n - k))*F'(x)
I don’t know how to define F(b) or F(d)
And help would be greatly appreciated.