Profile least squares estimators in the monotone single index model

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.