Profile least squares estimators in the monotone single index model

Abstract

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

21 pages, 6 figures

References in corpus (3)

• The EFM approach for single-index models
• Score estimation in the monotone single index model
• Profile least squares estimators in the monotone single index model

Cited by in corpus (2)

• Profile least squares estimators in the monotone single index model
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