Results 1 to 3 of 3

Math Help - Order Statistics/Change of Variable

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    218

    Order Statistics/Change of Variable

    Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf f(x) = e^{-x} 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  \Sigma a_i Y_i can be expressed as a linear function of independent random variables.

    a)

    so y_1 = z_1/n , y_2 = z_2/(n-1) +z_1/n , y_3 = z_3/(n-2) +  z_2/(n-1) +z_1/n, etc...

    So how would I find the jacobian?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    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).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    218
    so:

    h(y_1,y_2,...,y_n) = n! e^{-y_1 - y_2 - ... - y_n}

    The determinant of the Jacobian Matrix is 1/n!.

    g(z_1,z_2,...,z_n) = n! e^{-\frac{z_1}{n} - \frac{z_2}{n-1} - \frac{z_1}{n} - \frac{z_3}{n-2} - \frac{z_2}{n-1} - \frac{z_1}{n}...}\frac{1}{n!}

    but according to my textbook, the pdf of Z1, Z2...Zn is supposed to be (after the change of variable):

    g(z_1,z_2,...,z_n) =  e^{-z_1 - z_2 - ... - z_n}

    If I add up -\frac{z_1}{n} - \frac{z_2}{n-1} - \frac{z_1}{n} - \frac{z_3}{n-2} - \frac{z_2}{n-1} - \frac{z_1}{n}... would I just get z1, z2,...,zn?

    Okay, I think I got part A. But how would I do part B?
    Last edited by statmajor; November 5th 2009 at 06:16 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Change of Variable in a second-order PDE
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 14th 2011, 04:33 PM
  2. Second Order ODE - Change of Variable
    Posted in the Differential Equations Forum
    Replies: 11
    Last Post: August 21st 2010, 01:54 PM
  3. change of variable
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 17th 2009, 05:02 PM
  4. Change the variable
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 28th 2006, 11:26 AM
  5. Change of variable
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 22nd 2006, 08:50 AM

Search Tags


/mathhelpforum @mathhelpforum