Let $z(q)$ be the quantile of a standard normal random variable $Z$, i.e., $z(q) = k$ when $\Pr(Z\geq k) = q$. Then I would like to know why the following two results hold.
(a) If we hold $\alpha$ fixed, then asymptotically $$ |z(\alpha / n)|=\sqrt{2 \log (n)}\left(1-\frac{\log \log n}{4 n}\right) . $$ (b) For finite samples, one can use the excellent approximation $$ |z(\alpha / n)| \approx \sqrt{B\left(1-\frac{\log B}{B}\right)} \text { where } B:=2 \log (n/\alpha )-\log (2 \pi) $$
Could someone explain why (a) and (b) hold, or perhaps guide me some references to them? Thanks.