In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$.
Can someone help me how to prove this or give me a quick reference?
Thanks in advance.
Given that $X\sim N(0,1)$, we have: $$\int_{w}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx=\frac{1}{\sqrt{2\pi}}e^{-\frac{w^2}{2}}\int_{0}^{+\infty}\exp\left(-xw-\frac{x^2}{2}\right)dx\leq\frac{1}{w\sqrt{2\pi}}\exp\left(-\frac{w^2}{2}\right)$$ and, by assuming $w\gg 1$: $$\int_{w}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx\geq \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{w^2}{2}\right)\int_{0}^{\sqrt{2}}e^{-xw}\left(1-\frac{x^2}{2}\right)dx\geq\frac{w^2-1}{w^3\sqrt{2\pi}}\exp\left(-\frac{w^2}{2}\right),$$ hence: $$\Phi^{-1}(1-x)=\Theta\left(\sqrt{-\log x}\right)$$ as long as $x\to 0$. $$\Phi^{-1}(1-x)\approx\sqrt{W\left(\frac{1}{2\pi x^2}\right)}$$ is an even more precise approximation in terms of the Lambert W-function.