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Metrika

pp 1–33 | Cite as

A class of percentile modified Lepage-type tests

  • Amitava Mukherjee
  • Marco MarozziEmail author
Article
  • 24 Downloads

Abstract

The two-sample problem usually tests for a difference in location. However, there are many situations, for example in biomedicine, where jointly testing for difference in location and variability may be more appropriate. Moreover, heavy-tailed data, outliers and small-sample sizes are common in biomedicine and in other fields. These considerations make the use of nonparametric methods more appealing than parametric ones. The aim of the paper is to contribute to the literature about nonparametric simultaneous location and scale testing. More precisely, several existing tests are generalized and unified, and a new class of tests based on the Mahalanobis distance between the percentile modified test statistics for location and scale differences is introduced. The asymptotic distributions of the test statistics are obtained, and small-sample size behaviour of the tests is studied and compared to other tests via Monte Carlo simulations. It is shown that the proposed class of tests performs well when there are differences in both location and variability. A practical application is presented.

Keywords

Nonparametric tests Permutation tests Rank tests Location-scale tests Mahalanobis distance 

Mathematics Subject Classification

62G10 62G09 62P10 

Notes

Compliance with ethical standards

Conflict of interest

All authors declares that they have no conflict of interest.

Supplementary material

184_2018_700_MOESM1_ESM.docx (186 kb)
Supplementary material 1 (DOCX 185 kb)

References

  1. Ansari AR, Bradley RA (1960) Rank-sum tests for dispersions. Ann Math Stat 31:1174–1189MathSciNetCrossRefGoogle Scholar
  2. Box GEP (1953) Non-normality and tests on variances. Biometrika 40:318–335MathSciNetCrossRefGoogle Scholar
  3. Brownie C, Boos DD, Hughes-Oliver J (1990) Modifying the t and ANOVA F tests when treatment is expected to increase variability relative to controls. Biometrics 46:259–266MathSciNetCrossRefGoogle Scholar
  4. Büning H (1991) Robuste und adaptive tests. DeGruyter, BerlinCrossRefGoogle Scholar
  5. Büning H, Kössler W (1999) The asymptotic power of Jonckheere-Type tests for ordered alternatives. Aust N Z J Stat 41:67–78MathSciNetCrossRefGoogle Scholar
  6. Büning H, Thadewald T (2000) An adaptive two-sample location-scale test of Lepage-type for symmetric distributions. J Stat Comput Simul 65:287–310MathSciNetCrossRefGoogle Scholar
  7. Cheng SW, Thaga K (2006) Single variables control charts: an overview. Qual Reliab Eng Int 22:811–820CrossRefGoogle Scholar
  8. Chernoff H, Savage IR (1958) Asymptotic normality and efficiency of certain nonparametric test statistics. Ann Math Stat 29:972–994MathSciNetCrossRefGoogle Scholar
  9. Choi KS, Moon JY, Kim DW, Byun HR, Kripalani RH (2010) The significant increase of summer rainfall occurring in Korea from 1998. Theoret Appl Climatol 102:275–286CrossRefGoogle Scholar
  10. Chowdhury S, Mukherjee A, Chakraborti S (2014) A new distribution-free control chart for joint monitoring of location and scale parameters of continuous distributions. Qual Reliab Eng Int 30:191–204CrossRefGoogle Scholar
  11. Chowdhury S, Mukherjee A, Chakraborti S (2015) Qual Reliab Eng Int 31:135–151CrossRefGoogle Scholar
  12. Cucconi O (1968) Un nuovo test non parametrico per il confronto tra due gruppi campionari. Giornale degli Economisti 27:225–248Google Scholar
  13. Duran BS, Tsai WS, Lewis TO (1976) A class of location-scale nonparametric tests. Biometrika 63:173–176MathSciNetzbMATHGoogle Scholar
  14. Friedrich S, Brunner E, Pauly M (2017) Permuting longitudinal data in spite of the dependencies. J Multivar Anal 153:255–265MathSciNetCrossRefGoogle Scholar
  15. Gastwirth JL (1965) Percentile modifications of two sample rank tests. J Am Stat Assoc 60:1127–1141MathSciNetCrossRefGoogle Scholar
  16. Gibbons JD, Chakraborti S (2011) Nonparametric statistical inference, 5th edn. CRC Press, Boka Raton FloridazbMATHGoogle Scholar
  17. Groeneveld RA, Meeden G (1984) Measuring skewness and kurtosis. The Statistician 33:391–399CrossRefGoogle Scholar
  18. Hájek J, Šidák Z, Sen PK (1998) Theory of rank tests, 2nd edn. Academic Press, New YorkzbMATHGoogle Scholar
  19. Janssen A (1997) Studentized permutation permutation tests for non-i.i.d hypotheses and the generalized Behrens-Fisher problem. Stat Probab Lett 36:9–21MathSciNetCrossRefGoogle Scholar
  20. Janssen A (1999) Testing nonparametric statistical functionals with applications to rank tests. J Stat Plan Inference 81:71–93MathSciNetCrossRefGoogle Scholar
  21. Kontopantelis E, Reeves D (2012) Performance of statistical methods for meta-analysis when true study effects are non-normally distributed: a simulation study. Stat Methods Med Res 21:409–426MathSciNetCrossRefGoogle Scholar
  22. Kössler, W (2006) Asymptotic Power and Efficiency of Lepage-Type Tests for the Treatment of Combined Location-Scale Alternatives. Informatik-Bericht Nr. 200, Humboldt-Universität zu Berlin, 1-26. https://edoc.hu-berlin.de/handle/18452/3114. Accessed 22 Oct 2016
  23. Kössler W, Kumar N (2010) An adaptive test for the two-sample scale problem based on U-statistics. Commun Stat Simul Comput 39:1785–1802MathSciNetCrossRefGoogle Scholar
  24. Lehmann EL (2009) Parametric versus nonparametrics: two alternative methodologies. J Nonparametric Stat 21:397–405MathSciNetCrossRefGoogle Scholar
  25. Lepage Y (1971) A combination of Wilcoxon’s and Ansari-Bradley’s statistics. Biometrika 58:213–217MathSciNetCrossRefGoogle Scholar
  26. Ludbrook J, Dudley H (1998) Why permutation tests are superior to t and F tests in biomedical research. The Am Stat 52:127–132Google Scholar
  27. Marozzi M (2004) A Bi-aspect nonparametric test for the two-sample location problem. Comput Stat Data Anal 44:639–648MathSciNetCrossRefGoogle Scholar
  28. Marozzi M (2013) Nonparametric simultaneous tests for location and scale testing: a comparison of several methods. Commun Stat Simul Comput 42:1298–1317MathSciNetCrossRefGoogle Scholar
  29. Marozzi M (2014) The multisample cucconi test. Stat Methods Appl 23:209–227MathSciNetCrossRefGoogle Scholar
  30. Marozzi M (2015) Multivariate multidistance tests for high-dimensional low sample size case-control studies. Stat Med 34:1511–1526MathSciNetCrossRefGoogle Scholar
  31. McCracken AK, Chakraborti S (2013) Control charts for joint monitoring of mean and variance: an overview. Qual Technol Quant Manag 10:17–35CrossRefGoogle Scholar
  32. Miller RG (1968) Jackknifing variances. Ann Math Stat 39:567–582MathSciNetCrossRefGoogle Scholar
  33. Muccioli C, Belford R, Podgor M, Sampaio P, de Smet M, Nussenblatt R (1996) The diagnosis of intraocular inflammation and cytomegalovirus retinitis in HIV-infected patients by laser flare photometry. Ocular Immunol Inflamm 4:75–81CrossRefGoogle Scholar
  34. Mukherjee A (2017) Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples. Int J Adv Manuf Technol 92:101–116CrossRefGoogle Scholar
  35. Murakami H (2007) Lepage type statistic based on the modified Baumgartner statistic. Comput Stat Data Anal 51:5061–5067MathSciNetCrossRefGoogle Scholar
  36. Neuhäuser M (2000) An exact two-sample test based on the Baumgartner-Weiss-Schindler statistic and a modification of Lepage’s test. Commun Stat Theory Methods 29:161–168MathSciNetCrossRefGoogle Scholar
  37. Neuhäuser M, Senske R (2004) The Baumgartner-Weiss-Schindler test for the detection of differentially expressed genes in replicated microarray experiments. Bionformatics 20:3553–3564CrossRefGoogle Scholar
  38. Niu C, Guo X, Xu W, Zhu L (2014) Testing equality of shape parameters in several inverse Gaussian populations. Metrika 77:795–809MathSciNetCrossRefGoogle Scholar
  39. Pauly M (2011) Discussion about the quality of F-ratio resampling tests for comparing variances. Test 20:163–179MathSciNetCrossRefGoogle Scholar
  40. Pauly M, Asendorf T, Konietschke F (2016) Permutation-based inference for the AUC: a unified approach for continuous and discontinuous data. Biometrical J 58:1319MathSciNetCrossRefGoogle Scholar
  41. Pesarin F, Salmaso L (2010) Permutation tests for complex data. Wiley, ChichesterCrossRefGoogle Scholar
  42. Podgor MJ, Gastwirth JL (1994) On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives. Stat Med 13:747–758CrossRefGoogle Scholar
  43. Puri ML, Sen PK (1971) Nonparametric methods in multivariate analysis. Wiley, New YorkzbMATHGoogle Scholar
  44. Randles RH, Hogg RV (1971) Certain uncorrelated and independent rank statistics. J Am Stat Assoc 66:569–574CrossRefGoogle Scholar
  45. Rice KL, Rubins JB, Lebahn F, Parenti CM, Duane PG, Kuskowski M, Joseph AM, Niewoehner DE (2000) Withdrawal of chronic systemic corticosteroids in patients with COPD. Am J Respir Crit Care Med 162:174–178CrossRefGoogle Scholar
  46. Romano JP, Wolf M (1999) Subsampling inference for the mean in the heavy-tailed case. Metrika 50:55–69MathSciNetCrossRefGoogle Scholar
  47. Sheil J, O’Muircheartaigh I (1977) Algorithm AS106: the distribution of non-negative quadratic forms in normal variables. Appl Stat 26:92–98CrossRefGoogle Scholar
  48. Tiku ML, Tan WY, Balakrishnan N (1986) Robust inference. Marcel Dekker, New YorkzbMATHGoogle Scholar
  49. Umlauft M, Konietschke F, Pauly M (2017) Rank-based permutation approaches for non-parametric factorial designs. Br J Math Stat Psychol 70:368–390CrossRefGoogle Scholar
  50. Zhang J, Mu X, Xia Y, Martin FL, Hang W, Liu L, Tian M, Huang Q, Shen H (2014) Metabolomic analysis reveals a unique urinary pattern in normozoospermic infertile men. J Proteome Res 13:3088–3099CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.XLRI-Xavier School of ManagementXLRI JamshedpurIndia
  2. 2.Department of Environmental Sciences, Informatics and StatisticsUniversity of VeniceVeniceItaly

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