Skip to main content
SpringerLink
Log in

Adaptive Tests for the c-Sample Location Problem

  • Chapter
Statistical Inference, Econometric Analysis and Matrix Algebra

Abstract

This paper deals with the concept of adaptive tests and with an application to the c-sample location problem. Parametric tests like the ANOVA F-tests are based on the assumption of normality of the data which is often violated in practice. In general, the practising statistician has no clear idea of the underlying distribution of his data. Thus, an adaptive test should be applied which takes into account the given data set. We use the concept of Hogg [21], i.e. to classify, at first, the unknown distribution function with respect to two measures, one for skewness and one for tailweight, and then, at the second stage, to select an appropriate test for that classified type of distribution. It will be shown that under certain conditions such a two-staged adaptive test maintains the level. Meanwhile, there are a lot of proposals for adaptive tests in the literature in various statistical hypotheses settings. It turns out that all these adaptive tests are very efficient over a broad class of distributions, symmetric and asymmetric ones.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Behnen, K., Neuhaus, G.: Rank tests with estimated scores and their application. Teubner, Stuttgart (1989)

    MATH  Google Scholar 

  2. Beier, F., Büning, H.: An adaptive test against ordered alternatives. Comput. Stat. Data Anal. 25, 441–452 (1997)

    Article  MATH  Google Scholar 

  3. Büning, H.: Robuste und adaptive tests. De Gruyter, Berlin (1991)

    MATH  Google Scholar 

  4. Büning, H.: Robust and adaptive tests for the two-sample location problem. OR Spektrum 16, 33–39 (1994)

    Article  MATH  Google Scholar 

  5. Büning, H.: Adaptive tests for the c-sample location problem — the case of two-sided alternatives. Commun. Stat. Theor. Meth. 25, 1569–1582 (1996)

    Article  MATH  Google Scholar 

  6. Büning, H.: Adaptive Jonckheere-type tests for ordered alternatives. J. Appl. Stat. 26, 541–551 (1999)

    Article  MATH  Google Scholar 

  7. Büning, H.: An adaptive distribution-free test for the general two-sample problem. Comput. Stat. 17, 297–313 (2002)

    Article  MATH  Google Scholar 

  8. Büning, H.: An adaptive test for the two-sample scale problem. Diskus-sionsbeiträge des Fachbereichs Wirtschaftswissenschaft der Freien Univer-sität Berlin, Nr. 2003/10 (2003)

    Google Scholar 

  9. Büning, H., Kössler, W.: Adaptive tests for umbrella alternatives. Biom. J. 40, 573–587 (1998)

    Article  MATH  Google Scholar 

  10. Büning, H., Rietz, M.: Adaptive bootstrap tests and its competitors in the c-sample location problem. J. Stat. Comput. Sim. 73, 361–375 (2003)

    Article  MATH  Google Scholar 

  11. Büning, H., Thadewald, T.: An adaptive two-sample location-scale test of Lepage-type for symmetric distributions. J. Stat. Comput. Sim. 65, 287–310 (2000)

    Article  MATH  Google Scholar 

  12. Büning, H., Trenkler, G.: Nichtparametrische statistische Methoden. De Gruyter, Berlin (1994)

    MATH  Google Scholar 

  13. Chatfield, C.: Problem-solving — A statistician's guide. Chapman & Hall, London (1988)

    Google Scholar 

  14. Gastwirth, J.L.: Percentile modifications of two-sample rank tests. J. Am. Stat. Assoc. 60, 1127–1140 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gibbons, J.D., Chakraborti, S.: Nonparametric statistical inference, 3rd ed. Dekker, New York (1992)

    MATH  Google Scholar 

  16. Hájek, J., Sidák, Z.S., Sen, P.K.: Theory of rank tests. Academic, New York (1999)

    MATH  Google Scholar 

  17. Hall, P., Padmanabhan, A.R.: Adaptive inference for the two-sample scale problem. Technometrics 39, 412–422 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hand, D.J., Daly, F., Lunn, A.D., McConway, K.J., Ostrowski, E.: A handbook of small data sets. Chapman & Hall, London (1994)

    Google Scholar 

  19. Handl, A.: Maßzahlen zur Klassifizierung von Verteilungen bei der Konstruk-tion adaptiver verteilungsfreier Tests im unverbundenen Zweistichproben-Problem. Unpublished Dissertation, Freie Universität Berlin (1986)

    Google Scholar 

  20. Hill, N.J., Padmanabham, A.R., Puri, M.L.: Adaptive nonparametric procedures and applications. Appl. Stat. 37, 205–218 (1988)

    Article  MATH  Google Scholar 

  21. Hogg, R.V.: Adaptive robust procedures. A partial review and some suggestions for future applications and theory. J. Am. Stat. Assoc. 69, 909–927 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hogg, R.V.: A new dimension to nonparametric tests. Commun. Stat. Theor. Meth. 5, 1313–1325 (1976)

    Article  MathSciNet  Google Scholar 

  23. Hogg, R.V., Fisher, D.M., Randles, R.H.: A two-sample adaptive distribution-free test. J. Am. Stat. Assoc. 70, 656–661 (1975)

    Article  Google Scholar 

  24. Hothorn, L., Liese, F.: Adaptive Umbrellatests — Simulationsuntersuchungen. Rostocker Mathematisches Kolloquium 45, 57–74 (1991)

    MATH  Google Scholar 

  25. Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  26. Husková, M.: Partial review of adaptive procedures. In: Sequential Methods in Statistics 16, Banach Center Publications, Warschau (1985)

    Google Scholar 

  27. Jonckheere, A.R.: A distribution-free k-sample test against ordered alternatives. Biometrika 41, 133–145 (1954)

    MATH  MathSciNet  Google Scholar 

  28. Mack, G.A., Wolfe, D.A.: K-sample rank tests for umbrella alternatives. J. Am. Stat. Assoc. 76, 175–181 (1981)

    Article  MathSciNet  Google Scholar 

  29. Neuhäuser, M., Büning, H., Hothorn, L.: Maximum test versus adaptive tests for the two-sample location problem. J. Appl. Stat. 31, 215–227 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. O'Gorman, T.W.: A comparison of an adaptive two-sample test to the t-test, rank-sum, and log-rank tests. Commun. Stat. Simul. Comp. 26, 1393–1411 (1997)

    Article  MATH  Google Scholar 

  31. O'Gorman, T.W.: Applied adaptive statistical methods — tests of significance and confidence intervals. ASA-SIAM Ser. Stat. Appl. Prob., Philadelphia (2004)

    Google Scholar 

  32. Randles, R.H., Wolfe, D.A.: Introduction to the theory of nonparametric statistics. Wiley, New York (1979)

    MATH  Google Scholar 

  33. Roussas, G.G.: A first course in mathematical statistics. Addison-Wesley, Reading, MA (1973)

    MATH  Google Scholar 

  34. Ruberg, S.J.: A continuously adaptive nonparametric two-sample test. Com-mun. Stat. Theor. Meth. 15, 2899–2920 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Simpson, D.G., Margolin, B.H.: Recursive nonparametric testing for dose-response relationships subject to downturns at high doses. Biometrika 73, 589–596 (1986)

    MATH  MathSciNet  Google Scholar 

  36. Sun, S.: A class of adaptive distribution-free procedures. J. Stat. Plan. Infer. 59, 191–211 (1997)

    Article  MATH  Google Scholar 

  37. Tiku, M.L., Tan, W.Y., Balakrishnan, N.: Robust inference. Dekker, New York (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Freie Universität Berlin, Berlin, 14195, Germany

    Herbert Büning

Authors
  1. Herbert Büning
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Herbert Büning .

Editor information

Editors and Affiliations

  1. TU Dresden Fakultät Wirtschaftswissenschaften, Professur für Quantitative Verfahren, insb. Ökonometrie, Mommsenstr. 12, Dresden, 01062, Germany

    Bernhard Schipp

  2. TU Dortmund Fakultät Statistik, Lehrstuhl für Wirtschaftsund Sozialstatistik, Vogelpothsweg 78, Dortmund, 44221, Germany

    Walter Kräer

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Physica-Verlag Heidelberg

About this chapter

Cite this chapter

Büning, H. (2009). Adaptive Tests for the c-Sample Location Problem. In: Schipp, B., Kräer, W. (eds) Statistical Inference, Econometric Analysis and Matrix Algebra. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2121-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation