In this paper, we consider the single-index measurement error model with mismeasured covariates in the nonparametric part. To solve the problem, we develop a simulation-extrapolation (SIMEX) algorithm based on the local linear smoother and the estimating equation. For the proposed SIMEX estimation, it is not needed to assume the distribution of the unobserved covariate. We transform the boundary of a unit ball in \({\mathbb {R}}^p\) to the interior of a unit ball in \({\mathbb {R}}^{p-1}\) by using the constraint \(\Vert \beta \Vert =1\). The proposed SIMEX estimator of the index parameter is shown to be asymptotically normal under some regularity conditions. We also derive the asymptotic bias and variance of the estimator of the unknown link function. Finally, the performance of the proposed method is examined by simulation studies and is illustrated by a real data example.

中文翻译：

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具有协变量测量误差的单指标模型的SIMEX估计

在本文中，我们考虑了在非参数部分中带有协变量估计错误的单指标测量误差模型。为了解决该问题，我们基于局部线性平滑器和估计方程，开发了一种模拟外推（SIMEX）算法。对于建议的SIMEX估计，不需要假设未观察到的协变量的分布。我们在变换的边界的单元球的\（{\ mathbb {R}} ^ P \）到单元球在内部\（{\ mathbb {R}} ^ {P-1} \）通过使用约束\（\ Vert \ beta \ Vert = 1 \）。所提出的索引参数的SIMEX估计量在某些规则性条件下被证明是渐近正态的。我们还导出了未知链接函数的估计量的渐近偏差和方差。最后，通过仿真研究检查了所提出方法的性能，并通过一个实际数据示例进行了说明。