Random Weighting Approximation For M-method In Linear Models

The least squares method is a common method for the parameter estimator in analysis of the linear regression. But the LS method is not desirable in some cases. Statisticians have provided many other options. Among them, the M-method is the most important and fruitful. The LS method and the LAD (Least absolute deviation) are two special cases of the M method. But the asymptotic variance of the M-estimator and the critical value for the test in the linear hypotheses of linear models are usually related to the estimate of the nuisance parameters, which are not easy to be estimated accurately. We propose the random weighting methods to solve the difficulty.We use two random weighting methods for approximating the distribution of the M-test statistic when the covariates are fixed in linear models. We also suggest random weighting methods for approximating the null distribution of the Wald-test statistic and Rao-test statistic. It is shown that, under both the null and the local alternatives, the random weighting statistics have the same asymptotic distribution as the original test statistic under the null hypotheses. So the critical values of the M-test can be obtained by the random weighting method without estimating the nuisance parameters and the power evaluation is possible under the local alternatives.We study the randomly weighting approximation of the distribution of the M-estimator, when the covariates are random in linear models. We show that the random weighting estimation of the distribution of the M-estimator is uniformly consistent. We can estimate the asymptotic variance of the M-estimator without estimating the nuisance parameters by the randomly weighting approximation of the asymptotic distribution of M-estimator.We investigate the results by Monte Carlo simulations. The proposed methods are shown to work well by the simulation studies. An interesting feature for Poisson random weights and binary random weights can attain similar efficiency as other positive weighting variables.