Profile Least Squares Estimators in the Monotone Single Index Model

We consider least squares estimators of the finite regression parameter $$\boldsymbol{\alpha }$$ α in the single index regression model $$Y=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})+\varepsilon $$ Y = ψ ( α T X ) + ε , where $$\boldsymbol{X}$$ X is a d-dimensional random vector, $${\mathbb E}(Y|\boldsymbol{X})=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})$$ E ( Y | X ) = ψ ( α T X ) , and $$\psi $$ ψ is a monotone. It has been suggested to estimate $$\boldsymbol{\alpha }$$ α by a profile least squares estimator, minimizing $$\sum _{i=1}^n(Y_i-\psi (\boldsymbol{\alpha }^T\boldsymbol{X}_i))^2$$ ∑ i = 1 n ( Y i - ψ ( α T X i ) ) 2 over monotone $$\psi $$ ψ and $$\boldsymbol{\alpha }$$ α on the boundary $$\mathcal {S}_{d-1}$$ 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 $$\sqrt{n}$$ n -convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed $$\boldsymbol{\alpha }$$ α , but using a different global sum of squares, is $$\sqrt{n}$$ n -convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.