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 $\sum_i\{Y_i-\hat\psi_{n\alpha}(\alpha^TX_i)\}^2$ 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. 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.