Strong representations for LAD estimators in linear models

Abstract

Consider the standard linear modely i =z i β+e i ,i=1, 2,...,n, where zi denotes theith row of ann x p design matrix,β∈ℝ p is an unknown parameter to be estimated ande i are independent random variables with a common distribution functionF. The least absolute deviation (LAD) estimate $$\tilde \beta $$ of β is defined as any solution of the minimization problem $$\sum\limits_{i = 1}^n { \left| {y_i - z_i \tilde \beta } \right| = \inf \left\{ {\sum\limits_{i = 1}^n { \left| {y_i - z_i \beta } \right|:\beta \in \mathbb{R}^p } } \right\}} .$$ In this paper Bahadur type representations are obtained for $$\tilde \beta $$ under very mild conditions onF near zero and onz i,i=1,...,n. These results are extended to the case, when {e i} is a mixing sequence. In particular the results are applicable when the residualse i form a simple autoregressive process.