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

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  • Chen, Hung
  • Huang, Mong-Na Lo
  • Huang, Wen-Jang

Abstract

In this paper, we consider the problem of approximating the location,x0[set membership, variant]C, of a maximum of a regresion function,[theta](x), under certain weak assumptions on[theta]. 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 of[theta](Xi). Arrange theYi's in nondecreasing order and take the average of ther Xi's 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 of[theta](·) nearx0.

Suggested Citation

  • Chen, Hung & Huang, Mong-Na Lo & Huang, Wen-Jang, 1996. "Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics," Journal of Multivariate Analysis, Elsevier, vol. 57(2), pages 191-214, May.
    • Handle: RePEc:eee:jmvana:v:57:y:1996:i:2:p:191-214
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    Citations

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    Cited by:

    1. Bai, Zhidong & Chen, Zehua & Wu, Yaohua, 2003. "Convergence rate of the best-r-point-average estimator for the maximizer of a nonparametric regression function," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 319-334, February.
    2. V. Bartkutė & L. Sakalauskas, 2009. "Statistical Inferences for Termination of Markov Type Random Search Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 141(3), pages 475-493, June.
    3. Facer, Matthew R. & Müller, Hans-Georg, 2003. "Nonparametric estimation of the location of a maximum in a response surface," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 191-217, October.
    4. Carlos A. Flores, 2007. "Estimation of Dose-Response Functions and Optimal Doses with a Continuous Treatment," Working Papers 0707, University of Miami, Department of Economics.

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