Asymptotic Efficiency of a Class of $c$-Sample Tests

Abstract

For testing the equality of $c$ continuous probability distributions on the basis of $c$ independent random samples, the test statistics of the form $\mathscr{L} = \sum^c_{j = 1} m_j\lbrack(T_{N,j} - \mu_{N,j})/A_N\rbrack^2$ are considered. Here $m_j$ is the size of the $j$th sample, $\mu_{N,j}$ and $A_N$ are normalizing constants, and $T_{N,j} = (1/m_j) \sum^N_{i = 1} E_{N,i}Z^{(j)}_{N,i}$ where $Z^{(j)}_{N,i} = 1$, if the $i$th smallest of $N = \sum^N_{j = 1} m_j$ observations is from the $j$th sample and $Z^{(j)}_{N,i} = 0$ otherwise. Sufficient conditions are given for the joint asymptotic normality of $T_{N,j}; j = 1, \cdots, c$. Under suitable regularity conditions and the assumption that the $i$th distribution function is $F(x + \theta_i/N^{\frac{1}{2}})$, the limiting distribution of $\mathscr{L}$ is derived. Finally, the asymptotic relative efficiencies in Pitman's sense of the $\mathscr{L}$ test relative to some of its competitors viz. the Kruskal-Wallis $H$ test (which is a particular case of the $\mathscr{L}$ test) and the classical $F$ test are obtained and shown to be independent of the number $c$ of samples.