A Class of Signed Rank Estimators in Regression Models with Random Covariates

Abstract This note proves the asymptotic uniform linearity of a weighted empirical process of residual signed ranks and a class of linear residual signed rank statistics with bounded scores in nonlinear parametric regression models when covariates are random and independent of the errors. This result is used to derive limiting distributions of a class of signed rank estimators of the underlying regression parameters in these models. The latter result is applied to the errors in variables linear regression model to show that these estimators are robust against large measurement error in the sense that the asymptotic relative efficiency of a class of signed rank estimators against the bias corrected least square estimator tends to infinity as the measurement error variance tends to infinity (in some cases monotonically), when covariates and regression and measurement errors have Gaussian distributions.