Finding the Limiting Distribution

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!

Re: Finding the Limiting Distribution

You have computed the distribution function $\displaystyle F_{Y_1}$ of $\displaystyle Y_1$ and it's given that $\displaystyle W_n = n^\beta Y_1$ therefore

$\displaystyle F_{W_n}(y) = P(W_n\leq y)=P(n^\beta Y_1\leq y)=P\left(Y_1\leq \frac{y}{n^\beta}\right)=F_{Y_1}\left(\frac{y}{n^{ \beta}}\right)$.

If $\displaystyle W_1,W_2,\ldots$ converges in distribution then there exists a random variable $\displaystyle W$: $\displaystyle \lim_{n \to +\infty} F_{W_n}(x) = F_{W}(x)$ (and $\displaystyle F_{W}$ continuous in $\displaystyle x$)

Try to compute for which values of $\displaystyle \beta$ the limit will converge.