Estimation of Partial Linear Error-in-Variables Models under Martingale Difference Sequence

DOI：10.3770/j.issn:2095-2651.2015.04.011

 作者 单位 于卓熙 吉林财经大学管理科学与信息工程学院 , 吉林 长春 130117 王德辉 吉林大学数学学院概率统计系, 吉林 长春 130021 黄娜 上海财经大学信息管理与工程学院, 上海 200433 吉林财经大学管理科学与信息工程学院 , 吉林 长春 130117

考虑部分线性模型$Y=x\beta+g(t)_e$,这里实验数据$x$具有测量误差, $Y$和$t$是精确测量的,误差项$e$构成鞅差序列.用$\tile{x}$ 表示原始观察数据中变量$x$的测量误差数据,假设原始数据有$N$组观测值,即样本$\{(Y_{j},\widetilde{x}_{j},t_{j})_{j=n+1}^{n+N}\}$,独立的核实数据共有$n$组观测值$\{(\widetilde{x}_{j},x_{j},t_{j})_{j=1}^{n}\}$.本文中,我们借助于核实数据,基于最小二乘准则,利用原始数据构造上述部分线性模型的参数$\beta$和非参数部分$g(\cdot)$的半参数估计量,证明估计量的相合性,并通过模拟计算验证我们所给出估计量的优良性.

Consider the partly linear model $Y=x\beta+g(t)+e$ where the explanatory $x$ is erroneously measured, and both $t$ and the response $Y$ are measured exactly, the random error $e$ is a martingale difference sequence. Let $\widetilde{x}$ be a surrogate variable observed instead of the true $x$ in the primary survey data. Assume that in addition to the primary data set containing $N$ observations of $\{(Y_{j},\widetilde{x}_{j},t_{j})_{j=n+1}^{n+N}\}$, the independent validation data containing $n$ observations of $\{(\widetilde{x}_{j},x_{j},t_{j})_{j=1}^{n}\}$ is available. In this paper, a semiparametric method with the primary data is employed to obtain the estimator of $\beta$ and $g(\cdot)$ based on the least squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. Finite sample behavior of the estimators is investigated via simulations too.