Tests of Linear Hypotheses and l"1 Estimation

statistics of a linear hypothesis in the standard linear model. These test statistics, which correspond to Wald, likelihood ratio, and Lagrange multiplier tests, are shown to have the same limiting chi-square behavior under mild regularity conditions on design and the distribution of errors. The asymptotic theory of the tests is derived for a large class of error distributions; thus in Huber's 10 terminology we investigate the behavior of the likelihood ratio test under non-standard conditions. The asymptotic efficiency of the 11 tests involves a modest sacrifice of power compared to classical tests in cases of strictly Gaussian errors but may yield large efficiency gains in non-Gaussian situations. The Lagrange multiplier test seems particularly attractive from a computational standpoint. We derive the asymptotic distribution of the three alternative 11 test statistics for a simple linear exclusion hypothesis. Extension of these results to hypotheses of the form R,8 = r is a straightforward exercise. When the density of the error distribution is strictly positive at the median, all three test statistics have the same limiting central x2 behavior at the null and noncentral x2 behavior for local alternatives to the null. When the variance of the error distribution is bounded, analogous results are well known for classical forms of the Wald, likelihood ratio, and Lagrange multipler tests based on least-squares methods. See, for example, Silvey 18 and the discussion in Section 4 below.