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

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  • Shaul Bar-Lev

Abstract

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→ 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*.

Suggested Citation

  • Shaul Bar-Lev, 2008. "Point and confidence interval estimates for a global maximum via extreme value theory," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(12), pages 1371-1381.
    • Handle: RePEc:taf:japsta:v:35:y:2008:i:12:p:1371-1381
    DOI: 10.1080/02664760802382442
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    References listed on IDEAS

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    1. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
    2. Einmahl, John H. J. & Magnus, Jan R., 2008. "Records in Athletics Through Extreme-Value Theory," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1382-1391.
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