Abstract

We consider estimation and inference in a single-index regression model with an unknown convex link function. We introduce a convex and Lipschitz constrained least-square estimator (CLSE) for both the parametric and the nonparametric components given independent and identically distributed observations. We prove the consistency and find the rates of convergence of the CLSE when the errors are assumed to have only $q\ge 2$ moments and are allowed to depend on the covariates. When $q\ge 5$, we establish $n^{-1/2}$-rate of convergence and asymptotic normality of the estimator of the parametric component. Moreover, the CLSE is proved to be semiparametrically efficient if the errors happen to be homoscedastic. We develop and implement a numerically stable and computationally fast algorithm to compute our proposed estimator in the R package simest. We illustrate our methodology through extensive simulations and data analysis. Finally, our proof of efficiency is geometric and provides a general framework that can be used to prove efficiency of estimators in a wide variety of semiparametric models even when they do not satisfy the efficient score equation directly. Supplementary files for this article are available online.

摘要

我们考虑在具有未知凸链接函数的单指数回归模型中进行估计和推理。我们为给定独立且同分布的观测值的参数和非参数分量引入凸和 Lipschitz 约束最小二乘估计量 (CLSE)。当假设错误只有$q\ge \mathrm{2个}$时刻，并允许依赖于协变量。什么时候$q\ge \mathrm{5个}$, 我们建立$n^{-\mathrm{1个}/\mathrm{2个}}$- 参数分量估计器的收敛速度和渐近正态性。此外，如果误差恰好是同方差的，则 CLSE 被证明是半参数有效的。我们开发并实施了一种数值稳定且计算速度快的算法来计算我们在 R 包 simest 中提出的估计量。我们通过广泛的模拟和数据分析来说明我们的方法。最后，我们的效率证明是几何的，并提供了一个通用框架，可用于证明各种半参数模型中估计量的效率，即使它们不直接满足有效分数方程。本文的补充文件可在线获取。