Estimation of the extreme value and the extreme points
Let f be a continuous function defined on some open measurable set \(A\subset R\) and let \(X_ 1,X_ 2,..\). be an i.i.d. sequence with an absolutely continuous distribution function G and \(P(A)=1\). Assume that \(y_ 0=\min \{f(x)\), \(x\in A\}<\infty\). The minimum points \(x_ 0\in A\) satisfying \(f(x_ 0)=y_ 0\) are not necessarily unique. The author gives sufficient conditions for the existence of norming constants \(a_ n>0\) for which \(a_ n^{-1}(\min \{f(X_ 1),...,f(X_ n)\}-y_ 0)\) converges weakly to a distribution of the type \(1-\exp (- cy^{\alpha})\), where c and \(\alpha\) are positive constants. Using these results, he constructs a confidence interval for \(y_ 0\). The discussion for \(A\subset R^ k\), i.e. the multidimensional case, is also provided. The case where G is the uniform distribution, \(A\subset R^ k\) and there exists a unique minimum point \(x_ 0\) was considered by \textit{L. de Haan}, J. Am. Stat. Assoc. 76, 467-469 (1981; Zbl 0462.62031)].
scientific article; zbMATH DE number 4007482