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

✍ Scribed by Hung Chen; Mong-Na Lo Huang; Wen-Jang Huang

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 567 KB
• Volume: 57
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0029

✦ Synopsis

In this paper, we consider the problem of approximating the location, x 0 # C, of a maximum of a regresion function, %(x), under certain weak assumptions on %.

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 %(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 %( } ) near x 0 . 1996 Academic Press, Inc.

1. Introduction

Let % be a real function defined on a bounded interval C # R, and suppose there is an x 0 # C with %(x 0 )>%(x) for any x{x 0 in C. It is further assumed that %( } ) is continuous. The objective is to determine x 0 based on n samples (X 1 , Y 1 ), ..., (X n , Y n ) with Y i =%(X i )+= i , where n is a predetermined number. Here [= i ] are independent and identically distributed (i.i.d.) random variables with zero expectation.

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