I'm having trouble with the subscripts of
Are these two order stats that differ by 2?
I'm starting with
and I'm trying to find the density of
I look at the joint pdf, then do a Jacobian transformation and get very close to the expression of a beta function, but it's not quite it (but should be):
Any suggestions on how I can get it into the form of a beta function or where I may have gone awry?
Thanks. My 2-2 transformation ends up looking very similar. I set one variable (I'll rename them from my original post so we don't run into confusion with the variables that you use) to be
I then get that
The Jacobian turns out to be equal to 1 in that transformation and I plug in the new variables and get
I'm not sure how to manipulate the variables to get it into the form of a beta eventually (before or after integration).
Thanks! I do owe you. I didn't see the forest for the trees (or maybe I banged my head against the wall one too many times).
I found another reference as well (Order Statistics, 3rd ed., David & Nagajara) - they use
t = v * (1-s) and integrate over v.
I'll still have to mull it over for a while on how I can get it to be more intuitive. (Still don't quite get the beta with the easier substitution since the s outside the integral comes back into the integral for me when I do this.)
Order statistics - Google Books
you can read this as well
SpringerLink - Journal Article
I wanted you to try that other substitution
and see if you obtained the same result I did.
After dinner I may post my work.............
BACK to my variables... we have
Let s=y-z but now t=z, the smaller of the two order stats. The jacobian should be one again
That doesn't need checking in these cases.
The density of these two is...
That region becomes t>0, s>0 and s+t<1
So the density of s becomes
NOW use calc one, to make the bounds go from 0 to 1, let
which gives you and
and all I see are beta constants now...........
It looks like your reference did what I did.
I have David's book somewhere in my office, but I usually derive these things from scratch.
I can do most things faster by myself than looking it up.
Usually I can't find it or it may never have been done before.
So I just do it.
You're welcome, this is what I do for a living
(if you saw that paper).
I was trying to help another person this morning and he became so rude.
I don't have to do this for free.
I gave him plenty of help and I won't ever again.
All I expect is a thank you.