Weighted Rank Estimators

Abstract

Rank-based estimators are important tools of robust estimation in popular semiparametric models under monotonicity constraints. Here we study weighted versions of such estimators. Optimally weighted monotone rank estimator (MR) of Cavanagh and Sherman (1998) attains the semiparametric efficiency bound in the nonlinear regression model and the binary choice model. Optimally weighted maximum rank correlation estimator (MRC) of Han (1987) has the asymptotic variance close to the semiparametric efficiency bound in single-index models under independence when the distribution of the errors is close to normal, and is consistent under deviations from the single index assumption. Under moderate nonlinearities and nonsmoothness in the data, the efficiency gains from weighting are likely to be small for MR and MRC in the binary choice model and for MRC in the transformation model, and can be large for MR and MRC in the monotone regression model.