Asymmetric least squares regression estimation: A nonparametric approach^∗

^∗ Research partially supported by the Science and Engineering Research Council (U.K.)

Abstract

This paper considers the nonparametric estimation of regression expectiles and percentiles by using an asymmetric least squares (ALS) approach, in which the squared error loss function is given different weights depending on whether thc residual is positive or negative. The kernel method based on locally linear fit is adopted, which also provides an estimator of the derivative of the regression function. Under the assumption that the observations are strictly stationary and ρ-mixing the asymptotic normality for the estimators of conditional expectiles is established by using the convexity lemma. For a large class of regression models, the ALS approach can be adapted to estimate the conditional percentiles directly. Further, we show that these ALS estimators for conditional percentiles are consistent.

Keywords:

• asymmetric least squares estimator
• expectile
• kernel estimation
• local linear regression
• percentile
• ρ-mixing

^∗ Research partially supported by the Science and Engineering Research Council (U.K.)

^∗ Research partially supported by the Science and Engineering Research Council (U.K.)

Notes

^∗ Research partially supported by the Science and Engineering Research Council (U.K.)