NA样本部分线性模型估计的强相合性

考虑回归模型:Y~((j))(x_(in),t_(in))=t_(in)β+g(x_(in))+σ_(in)e~((j))(x_(in)),1≤j≤m,1≤i≤n,其中σ_(in)~2=f(u_(in)),(x_(in),t_(in),u_(in))为固定非随机设计点列,β是未知待估参数,g(·)和f(·)是未知函数,误差{e~((j))(x_(in))}是均值为零的NA变量.给出基于g(·)和f(·)一类非参数估计的β的最小二乘估计和加权最小二乘估计,并在适当条件下得到了它们的强相合性.

STRONG CONSISTENCY OF ESTIMATORS IN PARTIAL LINEAR MODEL UNDER NA SAMPLES

Consider the heteroscedastic regression model:$Y^{(j)}(x_{\rm in},t_{\rm in})=t_{\rm in}\beta+g(x_{\rm in})+\sigma_{\rm in}e^{(j)}(x_{\rm in}), 1\leq j\leq m, 1\leq i\leq n$, where $\sigma_{\rm in}^{2}=f(u_{\rm in})$, $(x_{\rm in},t_{\rm in},u_{\rm in})$ are fixed design points, $\beta$ is an unknown parameter, $g(\cdot)$ and $f(\cdot)$ are unknown functions, and the errors $\{e^{(j)}(x_{\rm in})\}$ are mean zero NA random variables. The strong consistency for least-squares estimator and weighted least-squares estimator of $\beta$ is studied based on the family of nonparametric estimates of $g(\cdot)$ and $f(\cdot)$.