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.
For the pdf f(x) = (θ^2+θ)*(x^(θ-1))*(1-x), 0<x<1,
let Y_{1}<Y_{2}<...<Y_{n} denote the corresponding order statistics.
Find β so that W_{n} = (n^β)*Y_{1} converges in distribution.
Find the limiting distribution of W_{n}.
I know that F(x) = (x^θ)(θ-θx+1), so F_{Y1}(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 W_{n} converging in distribution.
Any help would be great!
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.