Papers › Profile least squares estimators in the monotone single index model
Profile least squares estimators in the monotone single index model
Fadoua Balabdaoui, Piet Groeneboom
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We consider least squares estimators of the finite regression parameter α in the single index regression model Y=ψ(αᵀ X)+ϵ, where X is a d-dimensional random vector, (Y|X)=ψ(αᵀ X), and where ψ is monotone. It has been suggested to estimate α by a profile least squares estimator, minimizing ∑ᵢ₌₁ⁿ(Yᵢ-ψ(αᵀ Xᵢ))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 $\sqrt{n}$ convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed $\alpha$, 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.
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