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!
Re: Finding the Limiting Distribution
You have computed the distribution function of and it's given that therefore
If converges in distribution then there exists a random variable : (and continuous in )
Try to compute for which values of the limit will converge.