Robust Tests of Inequality Constraints and One-sided Hypotheses in the Linear Model

For the linear regression model Yi = x~P + fi' 1 :::; i :::; n, where f1, ..., f n are independent and identically distributed, robust tests of various hypotheses on Pare developed; by 'robust', we mean robustness of size and power against long tailed error distributions which may not be symmetric. The proposed test statistic, denoted MR, resembles the likelihood ratio statistic since, ignoring scaling and normalizing factors, MR is essentially the same as {L(po) L(,81)}, where L(b) = LP(Yi x~b) is the usual loss function used in M-estimation, P is convex, and ,80 and ,81 are the M-estimates under the null and alternative hypotheses respectively. Thus, L(b) takes the place of log likelihood in these tests. The two main results are (i) under the null hypothesis, the asymptotic distribution of MR and that of the normal theory likelihood ratio (ML) statistic are the same; and (ii) in terms of asymptotic efficiency, the behaviour of MR relative to ML is similar to that of the corresponding M-estimator relative to the least squares estimator. The first result is useful since it enables us to make use of an extensive literature on ML tests; in particular, the critical values and procedures for computing p-values for ML are applicable for MR as well. The second is useful since it tells us that MR is in fact power-robust against long tailed errors. The general results incorporate the important situations such as Ho: RP = 0 against HI: RP ~ 0, RP =j:. 0 and H2: RP ~ 0 against H3: Rp 1:. 0, where R is a matrix. An example and simulation results illustrate that MR has desirable robustness properties compared to ML, and that the implementation of the former is not any more difficult than that of the latter.