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

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 70; no. 2; pp. 389 - 412
• Main Authors: Hallin, Marc, Vermandele, Catherine, Werker, Bas J. M.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Oxford, UK : Blackwell Publishing Ltd 01.01.2008; Blackwell Publishing Ltd; Blackwell Publishing; Blackwell; Royal Statistical Society; Oxford University Press
• Series: Journal of the Royal Statistical Society Series B
• Subjects: Consistent estimators; Density estimation; Distribution theory; Efficiency; Estimation; Estimators; Exact sciences and technology; General topics; Group invariance; Herbivores; Inference; Least absolute deviation estimation; Linear inference, regression; Linear regression; Local asymptotic normality; Mathematics; Median regression and auto-regression; Nonparametric inference; Parametric models; Preliminary estimates; Probability and statistics; Probability theory and stochastic processes; Quantile regression; Quantile regression and auto-regression; R-estimation; Regression analysis; Sciences and techniques of general use; Sign and rank tests; Signs; Statistical methods; Statistical models; Statistics; Studies; Time series; Time series models; Variance; Vector-autoregressive models; Density estimation; Statistical distribution; Error estimation; Least absolute deviation estimation; Non parametric estimation; Median; Stochastic process; Quantile regression; Consistent estimator; Pseudolikelihood; Distribution function; Sign and rank tests; Rank test; Quantile; Local asymptotic normality; Finance; Group invariance; Least absolute value method; Time series; Statistical estimation; Invariance; Semiparametric method; Sign test; Heteroscedasticity; Statistical method; Statistical regression; Median regression and auto-regression; Simulation; R- estimation; Asymptotic normality; Financial data; Quantile regression and auto-regression; Autocorrelation
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2007.00641.x

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.