Analysis of least absolute deviation

The least absolute deviation or L1 method is a widely known alternative to the classical least squares or L2 method for statistical analysis of linear regression models. Instead of minimizing the sum of squared errors, it minimizes the sum of absolute values of errors. Despite its long history and many ground-breaking works (cf. Portnoy and Koenker (1997) and references therein), the former has not been explored in theory as well as in application to the extent as the latter. This is largely due to the lack of adequate general inference procedures under the L1 approach. There is no counterpart to the simple and elegant analysis-of-variance approach, which is a standard tool in L2 method for testing linear hypotheses. The asymptotic variance of the L1 estimator involves the error density, or conditional densities in the case of heterogeneous errors, thereby making the usual standard error estimation difficult to obtain. This paper is an attempt to fill some of the gaps by developing a unified analysis-of-variance-type method for testing linear hypotheses. Like the classical L2-based analysis of variance, the method is coordinate free in the sense that it is invariant under any linear transformation of the covariates or regression parameters. Moreover, it does not 1 require the design matrix to be of full rank. It also allows the error terms to be heterogeneous