Function of quantiles converges in law to a normally distributed random variable

**aznmaven** · March 29th 2011, 09:13 PM

The challenge here is to show that

$\left(\frac{n}{p(1-p)}\right)^{\frac{1}{2}}f\left(T_p\right)\left(\ch i _{(p)}-T_p\right)\to Z\sim N(0,1)$

Where convergence here is convergence in law, T_p is the pth quantile of f, where f is the PDF of the F distribution and r/n=p remains fixed. I don't even know where to start! Many thanks to whomever can solve this toughie...

**Sambit** · March 29th 2011, 11:11 PM

Use the fact that if $\chi_{(p)}$ is the sample quantile (sample size n) and $T_p$ is the population quantile (both are pth quantile) then asymptotically $\chi_{(p)}\sim N(T_p,\frac{p(1-p)}{n[f(\chi_{(p)})]^2})$ . You have to standardize it.

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