On consistency of the weighted least squares estimators in a semiparametric regression model

This paper is concerned with the semiparametric regression model \(y_i=x_i\beta +g(t_i)+\sigma _ie_i,~~i=1,2,\ldots ,n,\) where \(\sigma _i^2=f(u_i)\), \((x_i,t_i,u_i)\) are known fixed design points, \(\beta \) is an unknown parameter to be estimated, \(g(\cdot )\) and \(f(\cdot )\) are unknown functions, random errors \(e_i\) are widely orthant dependent random variables. The p-th (\(p>0\)) mean consistency and strong consistency for least squares estimators and weighted least squares estimators of \(\beta \) and g under some more mild conditions are investigated. A simulation study is also undertaken to assess the finite sample performance of the results that we established. The results obtained in the paper generalize and improve some corresponding ones of negatively associated random variables.