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

Abstract

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.Then we establish exact three term asymptotic expansions for the mean and the median of X(n). This is achieved by reducing the extreme value problem to a problem about sample means. This technique is general and should apply to suitable other distributions. One of our main conclusions is that the popular approximation E[X(n)]≈2logn should be discontinued, unless n is fantastically large. Better approximations are suggested in this article. An application of some of our results to the Donoho–Jin sparse signal recovery model is made.The standard Cauchy case is touched on at the very end.