Asymptotic Efficiency of a Class of c-Sample Tests

For testing the equality of c continuous probability distributions on the basis of c independent random samples, the test statistics of the form L = ∑ᶜj = 1 mⱼ(TN,j - μN,j)/AN² are considered. Here mⱼ is the size of the jth sample, μN,j and AN are normalizing constants, and TN,j = (1/mⱼ) ∑Ni = 1 EN,iZ⁽ʲ⁾N,i where Z⁽ʲ⁾N,i = 1, if the ith smallest of N = ∑Nj = 1 mⱼ observations is from the jth sample and Z⁽ʲ⁾N,i = 0 otherwise. Sufficient conditions are given for the joint asymptotic normality of TN,j; j = 1, ⋯, c. Under suitable regularity conditions and the assumption that the ith distribution function is F(x + θᵢ/N1/2), the limiting distribution of L is derived. Finally, the asymptotic relative efficiencies in Pitman's sense of the L test relative to some of its competitors viz. the Kruskal-Wallis H test (which is a particular case of the L test) and the classical F test are obtained and shown to be independent of the number c of samples.