Permutation Test For Location Parameters When Sample Size Is Small

To test the equality of two means,the parametric t-test has been widely used.However the sample size is often small and the population is nonnormal.Welch(1942) proposed another test statistics T_w,and approximated its distribution by scaled t distribution with suitable degrees.Xu(2008) obtained the asymptotic expansions of distribution function of T_w up to n^-1.Wilcoxon(1963) provided a nonparametric rank sum test method.Bootstrap is often recommended in terms of test(Efron,1979).Therefor Permutation test methods are often recommended and used too.In this paper we study the size and the power of permutation test and compare them with those of other test methods under small sample size.In this test,it is indicated that the permutation test and other test methods are identical in size while the permutation test gives larger power than other test methods. A technique is developed to decrease the actual size when the distribution is Discrete. Further more,we extent our results to one sample and multi-sample tests.Numerical studies are carried out to demonstrate these results.