Results 1 to 2 of 2

Math Help - Finding the Limiting Distribution

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    United States
    Posts
    3

    Finding the Limiting Distribution

    For the pdf f(x) = (θ^2+θ)*(x^(θ-1))*(1-x), 0<x<1,
    let Y1<Y2<...<Yn denote the corresponding order statistics.

    Find β so that Wn = (n^β)*Y1 converges in distribution.
    Find the limiting distribution of Wn.

    I know that F(x) = (x^θ)(θ-θx+1), so FY1(y) = 1-(1-(y^θ)(θ-θy+1))^n, but plugging in w/(n^β) for y does not get me anything that I can see would lead to Wn converging in distribution.

    Any help would be great!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Finding the Limiting Distribution

    You have computed the distribution function F_{Y_1} of Y_1 and it's given that W_n = n^\beta Y_1 therefore
    F_{W_n}(y) = P(W_n\leq y)=P(n^\beta Y_1\leq y)=P\left(Y_1\leq \frac{y}{n^\beta}\right)=F_{Y_1}\left(\frac{y}{n^{  \beta}}\right).
    If W_1,W_2,\ldots converges in distribution then there exists a random variable W: \lim_{n \to +\infty} F_{W_n}(x) = F_{W}(x) (and F_{W} continuous in x)
    Try to compute for which values of \beta the limit will converge.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Limiting distribution = Initial Distribution MC
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 16th 2011, 11:17 PM
  2. Limiting distribution involving chi square distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 20th 2009, 05:22 PM
  3. limiting distribution
    Posted in the Advanced Statistics Forum
    Replies: 10
    Last Post: May 4th 2009, 11:47 PM
  4. What are the Limiting Distribution and Limiting Probabilities
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 3rd 2009, 01:49 PM
  5. limiting distribution
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: December 4th 2007, 02:18 PM

Search Tags


/mathhelpforum @mathhelpforum