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

Author: Chen H.   Huang M.N.L.   Huang W.J.  

Publisher: Academic Press

ISSN: 0047-259X

Source: Journal of Multivariate Analysis, Vol.57, Iss.2, 1996-05, pp. : 191-214

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Abstract

In this paper, we consider the problem of approximating the location, x 0∈ C , of a maximum of a regresion function, theta ( x ), under certain weak assumptions on theta . Here C is a bounded interval in R . A specific algorithm considered in this paper is as follows. Taking a random sample X 1 , , X n from a distribution over C , we have ( X i , Y i ), where Y i is the outcome of noisy measurement of theta ( X i ). Arrange the Y i 's in nondecreasing order and take the average of the r X i 's which are associated with the r largest order statistics of Y i . This average, ^x 0 , will then be used as an estimate of x 0 . The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates of ^x 0 to x 0 are derived. Those rates will depend on the right tail of the noise distribution and the shape of theta (·) near x 0 .