[For the entire collection see Zbl 0608.00013.]
Suppose the sample maximum of a random sample from the distribution function (df)F has a nondegenerate limit distribution with df G; that is, there exist \(a_ n>0\) and \(b_ n\) such that \(F^ n(a_ nx+b_ n)\to G(x)\) as \(n\to \infty\). Then G(x) is of the form \(G_{\theta}(x)=\exp (- (1-\theta x)^{1/\theta})\), \(x>\theta^{-1}\), where \(G_{\theta}\) is a Fréchet type distribution, Gumbel distribution or a Weibull type distribution according to \(\theta <0\), \(\theta =0\) or \(\theta >0\), respectively. When \(G(x)=G_{\theta}(x)\), F is said to exhibit a penultimate phenomenon, if there exists a sequence \(\theta_ n\) such that \(G_{\theta_ n}(x)\) is closer (in some sense) to \(F^ n(a_ nx+b_ n)\) than \(G_{\theta}(x).\)
For \(\theta\leq 0\) the authors present a large class of dfs for which such a situation arises. They also present some numerical evidence of the penultimate behavior in the form of the Kolmogorov-Smirnov distance function. This phenomenon for the Gumbel df case \((\theta =0)\) has been known for quite sometime. Similar results hold for \(\theta >0\) as well.