Robust Wald-Type Tests of One-Sided Hypotheses in the Linear Model

Abstract

Abstract Let us consider the linear model Y = Xθ + E in the usual matrix notation, where the errors are iid. Our main objective is to develop robust Wald-type tests for a large class of hypotheses on θ; by robust we mean robustness in terms of size and power against long-tailed error distributions. First assume that the error distribution is symmetric about the origin. Let L and be the least squares and a robust estimator of θ. Assume that they are asymptotically normal about θ with covariance matrices σ 2(XtX)–1 and τ 2(XtX)–1, respectively. So could be an M estimator or a high breakdown point estimator. Robust Wald-type tests based on (denoted by RW) are studied here for testing a large class of one-sided hypotheses on θ. It is shown that the asymptotic null distribution of RW and that of the usual Wald-type statistic based on L (denoted by W) are the same. This is a useful result since the critical values and procedures for computing the p values for W are directly applicable to RW as well. A more important result is that the Pitman asymptotic efficiency of RW relative to W is (σ 2 /τ 2), which is precisely the asymptotic efficiency of relative to L . In other words, the efficiency robustness properties of relative to L translate to power robustness of RW relative to W. These results hold for asymmetric error distributions as well, with a minor modification. The general theory presented incorporates Wald-type statistics based on a large class of estimators that includes M estimators, bounded influence estimators, and high breakdown point estimators. The main requirement is that the estimator under consideration be asymptotically normal about 0; hence it does not explicitly require the errors to be iid or symmetric. If the asymptotic covariance of is not proportional to (XtX)–1, the asymptotic null distribution of RW and of W are unlikely to be the same. The results of a simulation study show that in realistic situations RW based on an M estimator is likely to have at least as much power as W, and more if the errors have long tails. A simple example illustrates the application of RW and its advantages over W.