# Profile least squares estimators in the monotone single index model

We consider least squares estimators of the finite dimensional regression parameter α in the single index regression model Y=ψ(α^TX)+ϵ, where X is a d-dimensional random vector, E(Y|X)=ψ(α^TX), and where ψ is monotone. It has been suggested to estimate α by a profile least squares estimator, minimizing ∑_i=1^n(Y_i-ψ(α^TX_i))^2 over monotone ψ and α on the boundary S_d-1of 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. An augmented Lagrange method, embedded in the Hooke-Jeeves pattern search algorithm, is implemented in R to compute the profile least squares estimators.

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