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* = max_u 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 simulation-based samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x*.