On a $c$-Sample Test Based on Trimmed Samples

Abstract

Let $X_{i1} < \cdots < X_{in_i}(i = 1,\cdots, c)$ be the order statistics from an absolutely continuous $\operatorname{cdf} F_i(x) = F(x - \theta_i)$ where $F(x)$ has symmetric density. The problem of testing the hypothesis $H_0:\theta_1 = \cdots = \theta_c$, which has been discussed by many authors, will be considered in this paper. We are concerned with the tests based on only the middle $n_i - 2k_i$ random variables $X_{ik_i+1} < \cdots < X_{in_i-k_i}$ where $k_i = \lbrack n_i\alpha \rbrack$ is the largest integer not exceeding $n_i\alpha$ for any $\alpha, 0 < \alpha < \frac{1}{2}$. A test of Bhapkar's type [Bhapkar, V. P. (1961). A nonparametric test for the problem of several samples. Ann. Math. Statist. 32 1108-1117] is proposed for this problem and it is shown that, for some distributions with heavy tails, the asymptotic relative efficiency of the proposed test relative to Bhapkar's test, which is based on the complete samples, is larger than one. The work presented in this paper is an attempt toward generalizing Hettmansperger's results [Hettmansperger, T. P. (1968). On the trimmed Mann-Whitney statistics. Ann. Math. Statist. 39 1610-1614] to the $c$-sample problem.