Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful upper bounds would be helpful.



Sensitivity of inverse normal cdf
If $F$ is the standard normal CDF,
$$(Q^{-1})'(p) = \dfrac{1}{F'(t)} = \sqrt{2\pi} \exp(t^2/2)$$
where $p = F(t)$. The maximum for $\epsilon \le p \le 1-\epsilon$ is at the endpoints. So 
$$|Q^{-1}(p) - Q^{-1}(p')| \le \sqrt{2\pi} \exp(t^2/2) |p - p'|$$
where $t = Q^{-1}(1-\epsilon)$.
Asymptotically as $t \to +\infty$, $$F(t) \sim 1 - \dfrac{\exp(-t^2/2)}{\sqrt{2\pi} t}$$
so as $\epsilon \to 0+$
$$t \sim \sqrt{W(1/(2\pi \epsilon^2))}$$
where $W$ is the Lambert W function, and then
$$\sqrt{2\pi} \exp(t^2/2) \sim \dfrac{1}{\epsilon \sqrt{W(1/(2\pi \epsilon^2))}}$$


