We consider least squares estimators of the finite regression parameter α in the single index regression model Y = ψ(α^T X) + ε, where X is a d-dimensional random vector, E(Y|X) = ψ(α^T X), and ψ is a monotone. It has been suggested to estimate α by a profile least squares estimator, minimizing ±∑ⁿ_i=1(Y_i - ψ(α^T X_i))² over monotone ψ and α on the boundary S_d-1 of the unit ball. Although this suggestion has been around for a long time, it is still unknown whether the estimate is √n-convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed α, but using a different global sum of squares, is √n-convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.

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 that 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 and the penalized least-squares estimator methods, both available on CRAN as R packages, and compare with link-free methods, where the covariates are elliptically symmetric.