I have to show that this expression zero $\frac{\log(\bar{F}(p^{-1}(x_n))}{n(1-x_n)^2} \to 0$ for $n\to \infty$, where F is the cdf of a gamma distribution with scale $k\cdot \alpha$ and rate $\frac{\beta}{a}$. After $p^{-1}(x_n) = bs\sqrt{\log n}-d$, where $b$ and $s$ is postive constant between $0$ and $1$, and $d$ is a positive constants.
Let $x_n = \Phi(s\sqrt{\log n})$ , hence $x_n \to 1$ as $n\to \infty$.
Can anyone help me with that?
Here is my thoughts:
Using a result from my lecture notes we know that $n(1-x_n)^2 \sim \frac{c^2n}{s^2\log(n)}\exp(-s\sqrt{\log n})^2$ for $n\to \infty$.
($f(x_n)\sim g(x_n)$ means that $\frac{f(x_n)}{g(x_n)}\to 1$ for $n\to \infty$)
We have now an expression for the denominator (for large $n$). But I cant find a similar expression for the numerator, because we firstly have to find $\bar{F}=1-F$ and this is very challenging because of the integral of a gamma density is not easy to integrate.
Can anyone help me with the numerator or maybe just got a better idea? Lhospital could maybe be an easier way to show that the expression in first line goes to $0$?