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.

  • $\begingroup$ @Jennifer Did you look into extreme value theory books? $\endgroup$
    – user16007
    Jul 18 '11 at 23:39
  • 1
    $\begingroup$ You could also look into asking this question at stats.stackexchange.com $\endgroup$ Jul 19 '11 at 4:17

This question has been answered by Bogdan Lataianu at this link:



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.