Point and confidence interval estimates for a global maximum via extreme value theory

The aim of this paper is to provide some practical aspects of point and interval estimates of the global maximum of a function using extreme value theory. Consider a real-valued function f : D → R defined on a bounded interval D such that f is either not known analytically or is known analytically but has rather a complicated analytic form.We assume that f possesses a global maximum attained, say, at u∗ ∈ D with maximal value x∗ = maxu f (u) ·= f (u∗). The problem of seeking the optimum of a function which is more or less unknown to the observer has resulted in the development of a large variety of search techniques. In this paper we use the extreme-value approach as appears in Dekkers et al. [A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), pp. 1833–1855] and de Haan [Estimation of the minimum of a function using order statistics, J. Amer. Statist. Assoc. 76 (1981), pp. 467–469]. We impose some Lipschitz conditions on the functions being investigated and through repeated simulationbased samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x∗.