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

  1. #1
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    Function of quantiles converges in law to a normally distributed random variable

    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...
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  2. #2
    Senior Member Sambit's Avatar
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    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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