Approximating Null Distribution of the M-Test Statistics in Linear Models

Linear hypotheses in linear models can be tested by the M-method. The M-test, the Wald-type test (W-test) and the Rao's score-type test (R-test) are the three most commonly used testing methods. However, the critical values for these tests are usually related to the unknown error distribution. In this paper, we propose random weighting resampling methods for approximating the null distribution of these tests. It is shown that under both the null and the local alternatives these random weighting test statistics all have the same asymptotic null distributions as that for the original test statistic. The critical values of these tests can therefore be obtained by the Monte Carlo random weighting method. An important feature of the proposed methods is that the approximation are valid even the null hypothesis is not true and the power evaluation is possible under the local alternatives. We conduct extensive simulations under different error distribution specifications and different choices of random weighting variables to assess performance of proposed method. The results show that the random weighting M-testing method can provide pretty accurate approximation of the null distribution.