Least squares estimation in the monotone single index model
Least squares estimation in the monotone single index model
复制标题
单调单指标模型中的最小二乘估计
作者：
F. Balabdaoui;C. Durot;H. Jankowski
我们研究了单调单指数模型，其中一个真实的响应变量$Y$通过关系$E[Y|X]=Psi_0(α^T_0X)$几乎必然地与一个$d$维协变量$X$相联系。岭函数$Psi_0$和指数参数$α_0$都是未知的，并且假设岭函数在其支撑区间上是单调的。在某些正则性条件下，在不对回归误差施加特定分布的情况下，我们分别证明了捆绑函数$psi_0({α}^T_0CDOT)的最小二乘估计，以及岭函数和指数的最小二乘估计在$ell_2范数下的收敛速度。此外，我们还证明了最小二乘估计量对分段常数岭函数是参数速率自适应的。
We study the monotone single index model where a real response variable $Y $ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y | X] = Psi_0(alpha^T_0 X)$ almost surely. Both the ridge function, $Psi_0$, and the index parameter, $alpha_0$, are unknown and the ridge function is assumed to be monotone on its interval of support. Under some regularity conditions, without imposing a particular distribution on the regression error, we show the $n^{-1/3}$ rate of convergence in the $ell_2$-norm for the least squares estimator of the bundled function $psi_0({alpha}^T_0 cdot),$ and also that of the ridge function and the index separately. Furthermore, we show that the least squares estimator is nearly parametrically rate-adaptive to piecewise constant ridge functions.