Nonparametric regression expectiles *
Author:
Biao Zhang a
| Affiliation: | a The University of Toledo, |
DOI:
10.1080/10485259408832586
Publication Frequency:
8 issues per year
Subjects:
Mathematical Economics;
Mathematical Finance;
Medical Statistics;
Statistical Theory & Methods;
Statistics;
Statistics for the Biological Sciences;
Stochastic Models & Processes;
Formats available:
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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.
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*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.
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| Keywords: Asymmetric squared loss; bandwidth; kernel-type; expectile function; nonparametric regression |
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