A note on L-estimates for linear models (Q580842)

Let \(\rho_{\theta}(u)=\theta u-I_{(u<0)}\cdot u\). This note investigates the asymptotic distribution of the estimator of the vector \(\gamma\) of slope parameters in the model \(y_ i=\alpha +Z_ i\gamma +u\) determined by minimizing \(\sum_{j}\sum_{i}w_ j\rho_{\theta_ j}(y_ i-\alpha -Z_ i\gamma)\) for some choice of \(w_ j's\) and \(\theta_ j's\). It compares, in terms of asymptotic variance, this class of estimators and the class of averages of estimators \(\hat{\j}_{\theta_ j}\) obtained by minimizing \(\sum_{i}\rho_{\theta_ j}(y_ i-\alpha -Z_ i\gamma)\).