Rank Regression

Abstract

An estimation procedure for $(b, g)$ is developed for the transformation model $g(Y) = bz + \text{error, where} g$ is an unspecified strictly increasing function. The estimator for $b$ can be viewed as a hybrid between an $M$-estimator and an $R$-estimator. It differs from an $M$-estimator in that the dependent variable is replaced by a score based on ranks and from an $R$-estimator in that the ranks of dependent variable itself are used, not the ranks of the residuals. This provides robustness against the scale on which the variables are thought to be linearly related, as opposed to robustness against misspecification of the error distribution. Existence, uniqueness, consistency and asymptotic normality are studied.