Some c-sample rank tests of homogeneity against ordered alternatives based on U-statistics

Abstract

For the c-sample location problem with ordered alternatives we construct some test statistics, all of them are based on U-statistics. Several statistics from the literature are generalized and extended to our problem. In particular, the statistics of Xie and Priebe [Xie, J. and Priebe, C.E., 2002, A weighted generalization of the Mann–Whitney–Wilcoxon statistic. Journal of Statistical Planning and Inference, 102, 441–466.] are generalized from the two-sample problem. All the corresponding tests are based on different pairwise ranking methods, that of Puri [Puri, M.L., 1965, Some distribution-free k-sample rank tests of homogeneity against ordered alternatives. Communications on Pure and Applied Mathematics, 18, 51–63.], of Tryon and Hettmannsperger [Tryon, P.V. and Hettmansperger, T.P., 1973, A class of nonparametric tests for homogeneity against ordered alternatives. Annals of Statistics, 1, 1061–1070.], and of Büning and Kössler [Büning, H. and Kössler, W., 1999, The asymptotic power of Jonckheere-type tests for ordered alternatives. Australian and New Zealand Journal of Statistics, 41, 67–77.]. The asymptotic power and the asymptotic relative efficiency are derived. Some of these tests are used to construct adaptive tests. A simulation study shows that the asymptotic results can be used for sample sizes as small as n _i  = 10.

Keywords:

• Jonckheere-type test
• Puri-type test
• Tryon–Hettmansperger-type test
• Adaptive test
• Efficacy
• Asymptotic relative efficiency

Acknowledgements

The author is grateful to the referee for his valuable comments on the article.