I showed and used this in a paper of mine.
The joint distribution of the order stats is n! the orginal density
WITH the restriction of y(1)<y(2)<....<y(n).
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf x ranging from 0 to infinity.
a) Show that Z1=nY1, Z2 = (n-1)(Y2 - y1) Z3= (n-2)(Y3-Y2)... Zn = Yn - Y_(n-1) are independent and that each Z has the exp distribution.
b) Demonstrate that all linear functions of Y1, Y2,...,Yn such as can be expressed as a linear function of independent random variables.
a)
so , , , etc...
So how would I find the jacobian?
so:
The determinant of the Jacobian Matrix is 1/n!.
but according to my textbook, the pdf of Z1, Z2...Zn is supposed to be (after the change of variable):
If I add up would I just get z1, z2,...,zn?
Okay, I think I got part A. But how would I do part B?