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Let $Y_2$ denote the second smallest item of a random sample of size $n$ from a distribution of the continuous type that has $\text{cdf }F(x)$ and $\text{pdf }f(x) = F'(x)$. Find the limiting distribution of $Wn = nF(Y_2)$.

I am not sure where to start.

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    $Pr[W_n\le w]=Pr[F(Y_2)\le\frac{w}{n}]=Pr[Y_2\le F^{-1}(\frac{w}{n})]$. Now $Y_2$ is sometimes called an ordinal statistic and IIRC you can use a binomial argument to calculate this probability: how many ways of choosing one of $n$ of the $X_i$ (if $Y_2$ is the second smallest of $\{X_1,\dots,X_n\}$) to be less than $Y_2$... [Ross](http://books.google.ch/books/about/A_first_course_in_probability.html?id=Bc1FAQAAIAAJ&redir_esc=y) has a good treatment of it.2012-04-18
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    @bgins Isn't $W_n=nF(Y_2)$? In which case, your first equality is a bit off?2012-04-18
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    @KannappanSampath: thank you, just in time!2012-04-18

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