• Mayer Alvo
  • Philip L. H. Yu
Part of the Springer Series in the Data Sciences book series (SSDS)


In this chapter, we consider the asymptotic efficiency of tests which requires knowledge of the distribution of the test statistics under both the null and the alternative hypotheses. In the usual cases such as for the sign and the Wilcoxon tests, the calculations are straightforward. However, the calculations become more complicated in the multi-sample situations and for these, we appeal to Le Cam’s lemmas. This is illustrated in the case of test statistics involving both the Spearman and Hamming distances. The smooth embedding approach is useful in that for a given problem, it leads to test statistics whose power function can be determined. The latter is then assessed against the power function of the optimal test statistic derived from Hoeffding’s formula for any given underlying distribution of the data.


  1. Alvo, M. and Pan, J. (1997). A general theory of hypothesis testing based on rankings. Journal of Statistical Planning and Inference, 61:219–248.MathSciNetCrossRefGoogle Scholar
  2. Hettmansperger, Thomas, P. (1994). Statistical Inference Based on Ranks. John Wiley.Google Scholar
  3. Hájek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, New York.zbMATHGoogle Scholar
  4. Serfling, Robert, J. (2009). Approximating Theorems of Mathematical Statistics. John Wiley and Sons.Google Scholar
  5. van der Vaart, A. (2007). Asymptotic Statistics. Cambridge University Press.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mayer Alvo
    • 1
  • Philip L. H. Yu
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina

Personalised recommendations