Limiting distributions of linear programming estimators
A linear regression model \(Y_i=X_i^T\beta+W_i\) (\(i=1,\dots,n\)) is considered, where \(W_i\) are i.i.d. non-negative r.v.s. As an estimator for \(\beta\) the following is used: \[ \hat \beta_n=\arg\max_\phi \left\{\sum_{i=1}^n X_i^T\phi\quad\text{subject to} Y_i\geq X_i^T\phi\quad\forall i=1,\dots,n\right\}. \] The limit distribution of \(c_n(\hat\beta_n-\beta)\) is described as a solution of a linear programming problem with random restrictions. The proof is based on the idea of epi-convergence in distribution. A nonparametric regression model \(Y_i=f(x_i)+W_i\) with locally linear estimators for \(f\) is also discussed.
scientific article; zbMATH DE number 1833860