Sharp fixed n bounds and asymptotic expansions for the mean and the median of a Gaussian sample maximum, and applications to the Donoho–Jin model

We are interested in the sample maximum X(n) of an i.i.d standard normal sample of size n. First, we derive two-sided bounds on the mean and the median of X(n) that are valid for any fixed n≥n0, where n0 is ‘small’, e.g. n0=7. These fixed n bounds are established by using new very sharp bounds on the standard normal quantile function Φ−1(1−p). The bounds found in this paper are currently the best available explicit nonasymptotic bounds, and are of the correct asymptotic order up to the number of terms involved.