# Finding the Limiting Distribution

• Nov 27th 2012, 04:23 PM
marty6491
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!
• Dec 2nd 2012, 02:09 PM
Siron
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.