Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics

In this paper, we consider the problem of approximating the location,x0∈C, of a maximum of a regresion function,θ(x), under certain weak assumptions onθ. HereCis a bounded interval inR. A specific algorithm considered in this paper is as follows. Taking a random sampleX1, …, Xnfrom a distribution overC, we have (Xi, Yi), whereYiis the outcome of noisy measurement ofθ(Xi). Arrange theYiʹs in nondecreasing order and take the average of ther Xiʹs which are associated with therlargest order statistics ofYi. This average,x0, will then be used as an estimate ofx0. The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates ofx0tox0are derived. Those rates will depend on the right tail of the noise distribution and the shape ofθ(·) nearx0.