, Volume 81, Issue 8, pp 891–930 | Cite as

Efficiency comparison of the Wilcoxon tests in paired and independent survey samples

  • Ludwig Baringhaus
  • Daniel GaigallEmail author


The efficiency concepts of Bahadur and Pitman are used to compare the Wilcoxon tests in paired and independent survey samples. A comparison through the length of corresponding confidence intervals is also done. Simple conditions characterizing the dominance of a procedure are derived. Statistical tests for checking these conditions are suggested and discussed.


Wilcoxon tests Pitman efficiency Bahadur efficiency Length of confidence intervals U-statistics Kernel density estimator 

Mathematics Subject Classification

62G10 62K05 



The authors thank the editor and the referees for constructive comments and suggestions. The second author was supported by a doctoral scholarship from the Hans-Böckler-Stiftung.


  1. Bahadur RR (1971) Some limit theorems in statistics. SIAM, PhiladelphiaCrossRefGoogle Scholar
  2. Baringhaus L, Gaigall D (2017) Hotelling’s \(T^2\) tests in paired and independent survey samples—an efficiency comparison. J Multivar Anal 154:177–198CrossRefGoogle Scholar
  3. Callaert H, Veraverbeke N (1981) The order of the normal approximation for a studentized \(U\)-statistic. Ann Stat 9:194–200MathSciNetCrossRefGoogle Scholar
  4. Gibbons JD, Chakraborti S (2011) Nonparametric statistical inference. CRC Press, Boca RatonCrossRefGoogle Scholar
  5. Groeneboom P, Oosterhoff J (1977) Bahadur efficiency and probabilities of large deviations. Stat Neerl 31:1–24MathSciNetCrossRefGoogle Scholar
  6. Hájek J, Šidak Z, Sen PK (1999) Theory of rank tests. Academic Press, San DiegoCrossRefGoogle Scholar
  7. Hettmansperger TP (1984) Statistical inference based on ranks. Wiley, New YorkzbMATHGoogle Scholar
  8. Hodges JL, Lehmann EL (1956) The efficiency of some nonparametric competitors of the \(t\)-test. Ann Math Stat 27:324–335MathSciNetCrossRefGoogle Scholar
  9. Hoeffding W (1940) Maßstabinvariante Korrelationstheorie. Schriften des Mathematischen Institiuts und des Instituts für Angewandte Mathematik der Universität Berlin 5:179–233Google Scholar
  10. Hollander M (1967) Asymptotic efficiency of two nonparametric competitors of Wilcoxon’s two-sample test. J Am Stat Assoc 62:939–949MathSciNetCrossRefGoogle Scholar
  11. Jammalamadaka SR, Janson S (1986) Limit theorems for a triangular scheme of \(U\)-statistics with applications to inter-point distances. Ann Probab 14:1347–1358MathSciNetCrossRefGoogle Scholar
  12. Nikitin Y (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  13. Nikitin Y, Ponikarov E (2001) Rough asymptotics of probabilities of Chernoff type large deviations for von Mises functionals and \(U\)-statistics. Am Math Soc Transl 203:107–146MathSciNetzbMATHGoogle Scholar
  14. Pratt J (1961) Length of confidence intervals. J Am Stat Assoc 56:549–567MathSciNetCrossRefGoogle Scholar
  15. Serfling RS (1980) Approximation theorems of mathematical statistics. Wiley, New YorkCrossRefGoogle Scholar
  16. Shirahata S, Sakamoto Y (1992) Estimate of variance of U-statistics. Commun Stat Theory Methods 10:2969–2981CrossRefGoogle Scholar
  17. Weber NC (1980) Rates of convergence for U-statistics with varying kernels. Bull Austral Math Soc 21:1–5MathSciNetCrossRefGoogle Scholar
  18. Wieand HS (1976) A condition under which the Pitman and Bahadur approaches to efficiency coincide. Ann Stat 4:1003–1011MathSciNetCrossRefGoogle Scholar
  19. Witting H, Müller-Funk U (1995) Mathematische Statistik II. Teubner, StuttgartCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Mathematische StochastikLeibniz Universität HannoverHannoverGermany

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