Unbiasedness or Biasedness and Approximations of Nonparametric Tests

Abstract

Finding the unbiasedness or biasedness of test statistics is important in testing a hypothesis. The unbiasedness or biasedness of the generalized Wilcoxon rank-sum test and the Jonckheere-Terpstra-type test are investigated. Deriving the exact critical value of the test statistic can be difficult when the sample sizes are increasing. In this situation, an approximation method to the distribution function of the test statistic can be useful with a higher order moment. We derive the expressions for the moment generating function of a linear rank test and a combination of two linear rank statistics. The accuracy of various approximations to the probability of various statistics are investigated. The normal approximation, the Edgeworth expansion, the saddlepoint approximation and the moment-based approximation with an adjusted specific distributional polynomial are used to evaluate the upper tail probability for various nonparametric tests.