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

• Published in: Journal of the American Statistical Association 1992-03, Vol.87 (417), p.156-161
• Main Author: Silvapulle, Mervyn J.
• Format: Article
• Language: English
• Subjects: Asymmetric errors; Asymptotic efficiency; Bounded influence; Breakdown point; Composite Hypothesis; Error rates; Estimators; Exact sciences and technology; Inequality constraints; Inequality Constraints: M estimator; Least squares; Linear inference, regression; Linear models; Linear regression; Mathematics; Outliers; P values; Pitman efficiency; Point estimators; Power efficiency; Probability and statistics; R estimator; Sciences and techniques of general use; Statistics; Theory and Methods
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1992.10475187

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 (X t X) -1 and τ 2 (X t X) -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 (X t X) -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.