Journal of the Italian Statistical Society

, Volume 1, Issue 1, pp 87–101 | Cite as

A resampling procedure for nonparametric combination of several dependent tests

  • Fortunato Pesarin
Article

Summary

This paper deals with nonparametric methods for combining dependent permutation or randomization tests. Particularly, they are nonparametric with respect to the underlying dependence structure. The methods are based on a without replacement resampling procedure (WRRP) conditional on the observed data, also called conditional simulation, which provide suitable estimates, as good as computing time permits, of the permutational distribution of any statistic. A class C of combining functions is characterized in such a way that all its members, under suitable and reasonable conditions, are found to be consistent and unbiased. Moreover, for some of its members, their almost sure asymptotic equivalence with respect to best tests, in particular cases, is shown. An applicational example to a multivariate permutationalt-paired test is also discussed.

Keywords and phrases

without replacement resampling procedures conditional inference conditional simulation bootstrap nonparametric combination of tests permutation or randomization methods multidimensionalt-paired 

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References

  1. Bailin M. andPesarin F., (1990) ‘Una Procedura di Ricampionamento e di Combinazione Nonparametrica per il Problema di Behrens-Fisher Multivariato’. InAtti della Società Italiana di Statistica, XXXV Riunione Scientifica. Vol. 2, CEDAM, Padova, 351–358.Google Scholar
  2. Barndorff-Nielsen O. E. andCox D. R., (1979), Edgeworth and Saddle-point Approximation with Statistical Applications.J.R.S.S., 41, 279–312.MATHMathSciNetGoogle Scholar
  3. Bell C. B. andSen P. K., (1984), Randomization Procedures. InHandbook of Statistics, N. 4, Krishnaiah P. R. and Sen P. K. eds. 1–29, Elsevier Sciences Publishers, North-Holland, Amsterdam.Google Scholar
  4. Berk R. H. andJones D. H., (1978), Relatively Optimal Combination of Tests.Scand. J. of Statis., 5, 158–162.MathSciNetGoogle Scholar
  5. Chatterjee S. K., (1984), Restricted Alternatives, InHandbook of Statistics, N. 4, Krishnaiah P. R. and Sen P. K. eds. 327–345, Elsevier Sciences Publishers, North-Holland, Amsterdam.Google Scholar
  6. Dwass M., (1957), Modified Randomization Tests for Nonparametric Hypotheses.Ann. Math. Statistics, 28, 181–187.MathSciNetGoogle Scholar
  7. Eddington E. S. (1987),Randomization Tests, 2nd Ed., M. Dekker, New York.Google Scholar
  8. Folks J. L., (1984), Combination of Independent Tests. InHandbook of Statistics, N. 4, Krishnaiah P. R. and Sen P. K. eds. 113–121, Elsever Sciences Publishers, North-Holland, Amsterdam.Google Scholar
  9. Hinkley D. V., (1989), Bootstrap Significance Tests.Proceedings of the ISI 47th Session, Paris, 53, 3, 65–74.MathSciNetGoogle Scholar
  10. Lehmann E. L., (1986),Testing Statistical Hypotheses. J. Wiley, New York.MATHGoogle Scholar
  11. Liptak I. (1958), On the Combination of Independent Tests,A Magyar Tudomanyos Akademia Matematikai Kutato Intezetenek Kozlomenyei, III, 127–141.Google Scholar
  12. Littell R. C. andFolks J. L., (1971), Asymptotic Optimality of Fisher's Method of Combining Independent Test.J. A. S. A., 66, 802–806.MATHMathSciNetGoogle Scholar
  13. Littell R. C. andFolks J. L., (1973), Asymptotic Optimality of Fisher's Method of Combining Independent Test, II.J. A. S. A, 68, 193–194.MATHMathSciNetGoogle Scholar
  14. Mielke P. W. andIyer H. K., (1982), Permutation Techniques for Analyzing Multi-Response Data from Randomized Block Experiments.Commun. Statist. Theory Meth., 11, 1427–1437.MATHGoogle Scholar
  15. Miller R. (1986), Multiple Comparisons. InEncyclopedia of Statistical Sciences, N. 5, Johnson N. L. and Kotz S., eds. 679–689; Wiley, New York.Google Scholar
  16. Oosterhoff J., (1969),Combination of One Sided Statistical Tests. Amsterdam Mathematical Center, Amsterdam, Tract. N. 28.MATHGoogle Scholar
  17. Pallini A. (1990a), Asymptotically Optimal Combinations of Dependent Test Statistics. InProceedings of the Workshop on Small Sample Asymptotics and Related Problems, Padova, Department of Statistics, 149–161.Google Scholar
  18. Pallini A. (1990b), Bahadur Exact Slopes for a Class of Combinations of Dependent Tests. (submitted for publication).Google Scholar
  19. Pallini A. andPesarin F., (1990), Combinazione di Test Dipendenti: una Soluzione basata su Tecniche di Ricampionamento. InAtti della Società Italiana di Statistica, XXXV Riunione Scientifica. Vol. 2, CEDAM, Padova, 367–364.Google Scholar
  20. Pallini A. andPesarin F. (1992), A class of Combinations of Dependent Tests by a Resampling Procedure. In Bootstrapping and Related Techniques, Proceedings Tier, FRG, 1990; Jöckel, Rothe, Sendler Eds., 93–98. Springer-Verlag Lecture Notes in Economics and Mathematical Systems 376, Berlin.Google Scholar
  21. Pesarin F., (1988), Combinazione Non Parametrica di Test Dipendenti. InGiornate di Metodologia Statistica di Bressanone, Dipart. di Sc. Statistische, Univ. di Padova, Padova, 241–248.Google Scholar
  22. Pesarin F. (1989),A Nonparametric Combination Method for Dependent Permutation Tests. Dipartimento di Scienze Statistische, Univ. di Padova, Padova, Technical Report, N. 3-1989.Google Scholar
  23. Pesarin F., (1990a). On a Nonparametric Combination Method for Dependent Permutation Tests with Applications.Psychotherapy and Psychosomatics, Vol. 54, 172–179.CrossRefGoogle Scholar
  24. Pesarin F., (1990b), Tecniche di Ricampionamento e Verifica Multidimensionale delle Ipotesi.Statistica, Vol. L, No. 3, 483–501.MathSciNetGoogle Scholar
  25. Pesarin F. (1990 c), Multidimensional Testing for Mixed Variables by Resampling Procedures. InLa Statistica nella Ricerca Epidemiologica e Farmacologica, (under publication byRivista di Statistica Applicata).Google Scholar
  26. Puri M. L. andSen P. K., (1971),Nonparametric Methods in Multivariate Analysis. Wiley, New York.MATHGoogle Scholar
  27. Romano J. P., (1989), Bootstrap and Randomization Tests of Some Nonparametric Hypotheses.The Annals of Statistics, 17, 141–159.MATHMathSciNetGoogle Scholar
  28. Vadiveloo J., (1983), On the Theory of Modified Randomization Tests for Nonparametric Hypotheses.Comm. Statist. Theor. Meth. 11, 1581–1596.MathSciNetGoogle Scholar
  29. Westberg M., (1986),A Tippet-adaptive Method of Combining Independent Statistical Test. Department. of Statistics, University of Göteborg, Göteborg, Research Report, N. 3-1986.Google Scholar
  30. Zanella A., (1973), Sulle procedure di Classificazione Simultanea.Statistica, XXXIII, 63–120.MathSciNetGoogle Scholar

Copyright information

© Societa Italiana di Statistica 1992

Authors and Affiliations

  • Fortunato Pesarin
    • 1
  1. 1.Department of StatisticsUniversity of PaduaPadovaItaly

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