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Semiparametrically efficient inference based on signs and ranks for median-restricted models

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Since the pioneering work of Koenker and Bassett, median-restricted models have attracted considerable interest. Attention in these models, so far, has focused on least absolute deviation (auto-)regression quantile estimation and the corresponding sign tests. These methods use a pseudolikelihood that is based on a double-exponential reference density and enjoy quite attractive properties of root n consistency (for estimators) and distribution freeness (for tests). The paper extends these results to general, i.e. not necessarily double-exponential, reference densities. Using residual signs and ranks (not signed ranks) and a general reference density f, we construct estimators that remain root n consistent, irrespective of the true underlying density g (i.e. also for g  /=f). However, instead of reaching semiparametric efficiency bounds under double-exponential g, they reach these bounds when g coincides with the chosen reference density f. Moreover, we show that choosing reference densities other than the double-exponential in applications can lead to sizable gains in efficiency. The particular case of median regression is treated in detail; extensions to general quantile regression, heteroscedastic errors and time series models are briefly described. The performance of the method is also assessed by simulation and illustrated on financial data.

Keywords: Group invariance; Least absolute deviation estimation; Local asymptotic normality; Median regression and auto-regression; Quantile regression and auto-regression; R-estimation; Sign and rank tests

Document Type: Research Article


Affiliations: 1: Université Libre de Bruxelles, Belgium 2: Tilburg University, The Netherlands

Publication date: April 1, 2008

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