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Nonparametric regression expectiles *  

Author: Biao Zhang a
Affiliation:   a The University of Toledo,
DOI: 10.1080/10485259408832586
Publication Frequency: 8 issues per year
Published in: journal Journal of Nonparametric Statistics, Volume 3, Issue 3 & 4 1994 , pages 255 - 275
Formats available: PDF (English)
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Abstract

It is well known that a standard nonparametric regression analysis is to model the average behavior between the dependent variable Y and the explanatory variable x. But such an approach may not always be appropriate if one is interested in the extreme behavior of Y conditional on x. This paper considers the problem of estimating the expectile function of the conditional distribution of YY given x based on the observational data generated according to a nonparametric regression model. We proposed a kernel-type nonparametric regression estimator, called nonparametric regression expectile, using an asymmetric squared loss function. This estimator models not only the average behavior but also the extreme behavior of Y given x in the nonparametric regression setting. An iterative algorithm is presented to calculate the estimator. It is shown that the nonparametric regression expectile is consistent and asymptotically normally distributed. We also derive a lower bound for the asymptotic variance and the asymptotic expression for the mean square error and the optimal bandwidth. A simulation study is given to demonstrate the utility of the nonparametric regression expectile for understanding nonparametric regression data.
*This research was supported by NSF Grant DMS 89-02667. Computations were performed using computer facilities supported in part by the National Science Foundations Grants DMS 87-03942 and DMS 89-05292 awarded to the Department of Statistics at The University of Chicago, and by The University of Chicago Block Fund.
Keywords: Asymmetric squared loss; bandwidth; kernel-type; expectile function; nonparametric regression
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