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Smoothed empirical likelihood methods for quantile regression models. (English) Zbl 1138.62017
Summary: This paper considers an empirical likelihood method to estimate the parameters of quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve higher order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order \(n^{ - 1}\) and may be Bartlett corrected to produce regions with errors of order \(n^{ - 2}\), where n denotes the sample size. Our result is an extension of the previous result of S. X. Chen and P. Hall [Ann. Stat. 21, No. 3, 1166–1181 (1993; Zbl 0786.62053)] to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method suggested by J. L. Horowitz [Econometrica 66, No. 6, 1327–1351 (1998; Zbl 1056.62517)].

MSC:
62G05 Nonparametric estimation
62G15 Nonparametric tolerance and confidence regions
65C05 Monte Carlo methods
62G20 Asymptotic properties of nonparametric inference
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