ON SCALE-INVARIANT M-STATISTICS IN MULTIVARIATE K SAMPLES

抄録

Scale invariant tests based on M-statistics are proposed in order to test homogeneity in a multivariate k sample model. Asymptotic noncentral x²-distributions are drawn under a contiguous sequence of location-alternatives without assuming Fisher consistency, and asymptotic robustness is derived. Permutation tests based on the proposed M-test statistics are considered. Using a Monte Carlo simulation, the power of these tests is compared with permutation tests based on parametric test statistics. Next, robust estimators for location parameters are proposed, based on scale-invariant M-statistics, and the asymptotic normality of these estimators is drawn. After a simple algorithm is studied, the risks of the M-estimators and the least squares estimators are compared in a simulation. For the univariate case, it is found that (i) the asymptotic relative efficiency (ARE) of the proposed M-procedures relative to parametric procedures agrees with the ARE of one-sample M-estimator proposed by Huber (1964) relative to the sample mean, and that (ii) for small sample sizes, the M-procedures are more efficient than parametric procedures except for the case where the underlying distribution is normal.