Hodges-Lehmann Sign-based Estimators and Generalized Confidence Distributions in Linear Median Regressions with Moment-free Heterogenous Errors and Dependence of Unknown Form

This paper develops sign-based estimation methods for the parameters of a median regression in finite samples.We introduce p-value functions that give the confidence one may have in a certain value of the parameter giventhe realization of the sample and sign-based estimators that are the values associated with the highest confidence(p-value). The sign-based estimators are thus obtained using the Hodges-Lehmann principle of test inversion.They are expected to present the same robustness properties than the test statistics they come from and canstraightly be associated with the finite-sample-based inference procedure described in Coudin and Dufour (2007).We also show they are median unbiased (under symmetry and estimator unicity) and present equivariancefeatures similar to the LAD estimator. Consistency under point identification and asymptotic normality areprovided and hold under weaker assumptions than the LAD estimator. However, small sample behavior is ourfirst interest. By a Monte Carlo study of bias and RMSE, we show sign-based estimators perform better than theLAD in very heteroskedastic settings.