Score estimation in the monotone single index model

We consider estimation of the regression parameter in the single index model where the link function $\psi$ is monotone. For this model it has been proposed to estimate the link function nonparametrically by the monotone least square estimate $\hat\psi_{n\alpha}$ for a fixed regression parameter $\alpha$ and to estimate the regression parameter by minimizing the sum of squared deviations $\sum_i\{Y_i-\hat\psi_{n\alpha}(\alpha^T X_i)\}^2$ over $\alpha$, where $Y_i$ are the observations and $ X_i$ the corresponding covariates. Although it is natural to propose this least squares procedure, it is still unknown whether it will produce $\sqrt{n}$-consistent estimates of $\alpha$. We show that the latter property will hold if we solve a score equation corresponding to this minimization problem. By using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization, which makes it easy to solve the score equations in high dimensions. We also compare our method with other methods such as Han's maximum rank correlation estimate, which has been proved to be $\sqrt{n}$-consistent.