Statistics & Probability Letters 6 (1988) 251-256 March 1988 North-Holland A CLASS OF DISTRIBUTION-FREE TESTS FOR TESTING HOMOGENEITY AGAINST ORDERED ALTERNATIVES S. CHAKRABORTI * Department of Management Science and Statistics, University of Alabama, Tuscaloosa, AL 35487, USA M.M. DESU Department of Statistics, SUNY at Buffalo, Buffalo, NY 14261, USA Received February 1987 Revised June 1987 Abstract: A class of distribution-free tests for homogeneity of several populations against ordered alternatives is proposed. These are particularly useful when testing time is expensive so that an early termination of an experiment is desirable. The Pitman efficacy is derived and the optimal member of the class, the one with maximum efficacy, is identified. This optimal test is then compared with some other popular tests in terms of the asymptotic relative efficiency. A modification of the proposed test is considered when no additional information about the alternatives is available so that the optimal test can not be used. The modified test is compared to the optimal test through asymptotic relative efficiency. Keywords: Distribution-free tests for homogeneity, ordered alternatives, early termination, Pitman efficacy. 1. Introduction Let Xil, Si2 ..... Sin i, i = 1, 2 ..... K, be K in- dependent random samples, where Xij, j = 1, 2 ..... n i is a sample from the i th population with an absolutely continuous distribution func- tion F,(x)= F(x- Oi). In other words the distri- butions of the K populations are the same except for a possible difference in their location parame- ters. Without any loss of generality let F(0)='/3, so that 0~, the location parameter of the ith popu- lation is its /3th quantile. In this paper we are concerned with testing the null hypothesis H0: 01=02 ..... O K against the ordered alternatives /41: 01<02<-" <0,~, * Research supported in part by grant No. 1357 from the Research Grants Committee, University of Alabama. with at least one strict inequality. This testing problem is considered without making any para- metric model assumption about F. The above problem has received considerable attention in the literature. Terpstra (1952) and Jonckheere (1954) proposed a rank test for this problem based on the Mann-Whitney U-statis- tics. Their results were generalized by Puri (1965) to a class of linear rank tests using an arbitrary score function and any Chernoff-Savage statistic. Following the approach suggested by Hogg (1965) Tryon and Hettmansperger (1973) generalized Puri's class of test statistics by including weighting coefficients, to form arbitrary linear combinations of Chernoff-Savage statistics. The weighting coef- ficients were obtained, when the relative spacings among the 0's were specified, by maximizing the Pitman efficacy of the test. Recently Rao and Gore (1984) considered weighted linear combina- tions of rank statistics and determined the weights such that the Pitman efficacy is maximized in case the 0's are equi-spaced. 0167-7152/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland) 251