Let \(X_ 1,X_ 2,..\). be i.i.d. nonnegative random variables and \(M_ n=\max (X_ 1,...,X_ n)\). Further, let Y be a random variable with distribution belonging to one of the three Gnedenko extreme value types (say \(P(Y\leq x)=\exp (-1/x)\), \(x>0)\). Given a continuous, increasing function \(\Psi\) one can define the probability metrics \[ \rho_{\Psi}(U,V)=\sup_{x\geq 0}\Psi (x)| P(U\leq x)-P(V\leq x)|,\quad and \] \[ \mu_{\Psi}(U,V)=\sup_{x\geq 0}\Psi (x)| \log (P(U\leq x)/P(V\leq x))|. \] The authors are interested in estimating the uniform distance \(\rho_ 1(M_ n/n,Y)\) in terms of \(\rho_{\Psi}(X_ 1,Y)\) and \(\mu_{\Psi}(X_ 1,Y)\) for a general class of functions \(\Psi\). The kind of results they obtained can be illustrated by the following assertion: Suppose \(\Psi\) is regularly varying with index \(r\geq 1\), such that \(\lim_{x\to \infty}x/\Psi (x)=0\). Then \[ (i)\quad \rho_{\Psi}(X_ 1,Y)1:\quad \lim_{x\to \infty}\Psi (x)(P(X_ 1\leq x)-P(Y\leq x))=0\quad iff\quad \lim_{x\to \infty}n^{-1}\Psi (n)\rho_ 1(M_ n/n,Y)=0. \] In addition, they also consider asymptotic expansions for \(P(M_ n/n\leq x)\).