Connections between two classes of estimators for single-index models

Single-index model is a very popular and powerful semiparametric model. As an improvement of the maximum rank correlation estimator, Shen et al. proposed the linearized maximum rank correlation estimator. We show that this estimator has some interesting connections with the distribution-transformed least-squares estimator for single-index models. We also propose a rescaled distribution-transformed least-squares estimator, which is mathematically equivalent to the linearized maximum rank correlation estimator when the distribution of the response is absolutely continuous. Despite some nontrivial connections, the two estimation procedures are different in terms of motivations, interpretations, and applications. We discuss some of the differences between the two estimation procedures.