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, $\E(Y|X)=\psi(\alpha^T X)$, 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.

翻译：我们考虑单指标回归模型 $Y=\psi(\alpha^T X)+\epsilon$ 中有限回归参数 $\alpha$ 的最小二乘估计量，其中 $X$ 是 $d$ 维随机向量，$\E(Y|X)=\psi(\alpha^T X)$，且 $\psi$ 是单调函数。已有文献建议通过剖面最小二乘估计量来估计 $\alpha$，即在单位球面边界 $S_{d-1}$ 上关于单调函数 $\psi$ 和参数 $\alpha$ 最小化 $\sum_{i=1}^n(Y_i-\psi(\alpha^T X_i))^2$。尽管这一建议已提出多年，但该估计量是否具有 $\sqrt{n}$ 收敛性尚不明确。我们证明：采用固定 $\alpha$ 下的同一点态最小二乘估计量，但使用不同的全局平方和构造的剖面最小二乘估计量具有 $\sqrt{n}$ 收敛性且渐近正态。本文还研究了相应损失函数之间的差异，并与其他方法进行了比较。