  We consider estimation in the single index model where the link function is
monotone. For this model a profile least squares estimator has been proposed to
estimate the unknown link function and index. Although it is natural to propose
this procedure, it is still unknown whether it produces index estimates which
converge at the parametric rate. We show that this holds if we solve a score
equation corresponding to this least squares problem. Using a Lagrangian
formulation, we show how one can solve this score equation without any
reparametrization. This makes it easy to solve the score equations in high
dimensions. We also compare our method with the Effective Dimension Reduction
(EDR) and the Penalized Least Squares Estimator (PLSE) methods, both available
on CRAN as R packages, and compare with link-free methods, where the covariates
are ellipticallly symmetric.
