Published

2015-01-01

Nonparametric Simultaneous Test Procedures

Procedimientos de pruebas simultáneas no paramétricas

DOI:

https://doi.org/10.15446/rce.v38n1.48805

Keywords:

Combining function, Multi-aspect Test, Permutation Principle, Two-sample Location-scale Problem (en)
Funciones de combinación, Parámetro de escala, Parámetro de localización, Principio de permutación, Problema de dos muestras, Pruebas simultáneas. (es)

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Authors

  • Hyo-Il Park Chongju University - Department of Statistics

In this research we propose several nonparametric simultaneous test procedures for location and scale parameters. We construct test statistics based on linear rank statistics choosing a suitable combining function. We obtain the overall p-values by applying the permutation principle. We compare the efficiency amongst combining functions by obtaining empirical powers through a simulation study. We discuss some interesting aspects of our procedure as concluding remarks.

En este artículo se propone un procedimiento de pruebas simultáneas no paramétricas para los paramétros de localización y escala. Se construyen los estadísticos de prueba basados en los estadísticos de rangos lineales para las subhipótesis nulas con la escogencia de una adecuada función de combinación; se obtienen los valores p al aplicar el principio de permutación; se compara la eficiencia entre las funciones de combinación mediante la obtención de las potencias empíricas a través de un estudio de simulación y por último se discuten algunos aspectos interesantes del procedimiento como conclusiones.

https://doi.org/10.15446/rce.v38n1.48805

Nonparametric Simultaneous Test Procedures

Procedimientos de pruebas simultáneas no paramétricas

HYO-IL PARK1

1Chongju University, Department of Statistics, Chongju, South Korea. Professor. Email: hipark@cju.ac.kr

Abstract

In this research we propose several nonparametric simultaneous test procedures for location and scale parameters. We construct test statistics based on linear rank statistics choosing a suitable combining function. We obtain the overall p-values by applying the permutation principle. We compare the efficiency amongst combining functions by obtaining empirical powers through a simulation study. We discuss some interesting aspects of our procedure as concluding remarks.

Key words: Combining function, Multi-aspect Test, Permutation Principle, Two-sample Location-scale Problem.

Resumen

En este artículo se propone un procedimiento de pruebas simultáneas no paramétricas para los paramétros de localización y escala. Se construyen los estadísticos de prueba basados en los estadísticos de rangos lineales para las subhipótesis nulas con la escogencia de una adecuada función de combinación; se obtienen los valores p al aplicar el principio de permutación; se compara la eficiencia entre las funciones de combinación mediante la obtención de las potencias empíricas a través de un estudio de simulación y por último se discuten algunos aspectos interesantes del procedimiento como conclusiones.

Palabras clave: funciones de combinación, parámetro de escala, parámetro de localización, principio de permutación, problema de dos muestras, pruebas simultáneas.

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References

1. Ansari, A. R. & Bradley, R. A. (1960), 'Rank-sum tests for dispersions', Annals of Mathematical Statistics 31, 1174-1189.

2. Brombin, C., Salmaso, L., Ferronato, G. & Galzignato, P.-F. (2011), 'Multi-aspect procedures for paired data with application to biometric morphing', Communications in Statistics-Simulation and Computation 40, 1-12.

3. Duran, B. S., Tsai, W. S. & Lewis, T. O. (1976), 'A class of location-scale nonparametric tests', Biometrika 63, 173-176.

4. Efron, B. (1979), 'Bootstrap methods: Another look at the jackknife', Annals of Statistics 7, 1-26.

5. Fisher, R. A. (1932), Statistical Methods for Research Workers, 4 edn, Oliver & Boyd, Edinburgh.

6. Fleming, T. R., Harrington, D. P. & O'Sullivan, M. (1987), 'Supremum version of the logrank and generalized ilcoxon statistics', Journal of the American Statistical Association 82, 312-320.

7. Good, P. (2000), Permutation Tests-A Practical Guide to Resampling Methods for Testing Hypotheses, 2 edn, Springer, New York.

8. Lepage, Y. (1971), 'A combination of Wilcoxon's and Ansari-Bradley's statistics', Biometrika 58, 213-217.

9. Lepage, Y. (1973), 'A table for a combined Wilcoxon Ansari-Bradley statistic', Biometrika 60, 113-116.

10. Liptak, I. (1958), 'On the combination of independent tests', Magyar Tudomanyos Akademia Matematikai Kutato Intezenek Kozlomenyei 3, 127-141.

11. Marozzi, M. (2004), 'A bi-aspect nonparametric test for the two-sample location problem', Computational Statistics and Data Analysis 44, 639-648.

12. Marozzi, M. (2007), 'Multivariate tri-aspect non-parametric testing', Journal of Nonparametric Statistics 19, 269-282.

13. Marozzi, M. (2009), 'Some notes on the location-scale Cucconi test', Journal of Nonparametric Statistics 21, 629-647.

14. Marozzi, M. (2012a), 'A modified Cucconi test for location and scale change alternatives', Revista Colombiana de Estadistica 35, 371-384.

15. Marozzi, M. (2012b), 'A distribution free test for the equality of scales', Communications in Statistics-Simulation and Computation 41, 878-889.

16. Marozzi, M. (2012b), 'A modified Hall-Padmanabhan test for the homogeneity of scales', Communications in Statistics-Theory and Methods 41, 3068-3078.

17. Marozzi, M. (2012b), 'A combined test for differences in scale based on the interquantile range', Statistical Papers 53, 61-72.

18. Marozzi, M. (2013), 'Nonparametric simultaneous tests for location and scale testing: A comparison of several methods', Communications in Statistics-Simulation and Computation 42, 1298-1317.

19. Marozzi, M. (2014), 'The multisample Cucconi test', Statistical Methods and Applications 23(2), 209-227.

20. Mood, A. M. (1954), 'On the asymptotic efficiency of certain nonparametric two-sample tests', Annals of Mathematical Statistics 25, 514-522.

21. Murakami, H. (2007), 'Lepage type statistic based on the modified Baumgartner statistic', Computational Statistics and Data Analysis 51, 5061-5067.

22. Neuhäuser, M., Leuchs, A.-K. & Ball, D. (2011), 'A new location-scale test based on a combination of the ideas of Levene and Lepage', Biometrical Journal 53(3), 525-534.

23. Park, H. I. (2011), 'A nonparametric test procedure based on a group of quantile tests', Communications in Statistics-Simulation and Computation 40, 759-783.

24. Pesarin, F. (2001), Multivariate Permutation Tests, Wiley, Chichester.

25. Pesarin, F. & Salmaso, L. (2010), Permutation Tests for Complex Data, Wiley, Chichester.

26. Pettitt, A. N. (1976), 'A two-sample Anderson-Darling rank statistic', Biometrika 63, 161-168.

27. Podgor, M. J. & Gastwirth, J. L. (1994), 'On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives', Statistics in Medicine 13, 747-758.

28. Randles, R. H. & Hogg, R. V. (1971), 'Certain uncorrelated and independent rank statistics', Journal of American Statistical Association 66, 569-574.

29. Randles, R. H. & Wolfe, D. A. (1979), Introduction to the Theory of Nonparametric Statistics, Wiley, New York.

30. Rublik, F. (2009), 'Critical values for testing location-scale hypothesis', Measurement Science Review 9, 9-15.

31. Salmaso, L. & Solari, A. (2005), 'Multiple aspect testing for case-control designs', Metrika 62, 331-340.

32. Shao, J. & Tu, D. (1995), The Jackknife and Bootstrap, Springer, New York.

33. Tippett, L. H. C. (1931), The Methods of Statistics, Williams and Norgate, London.

34. Zhang, J. (2006), 'Powerful two-sample tests based on the likelihood ratio', Technometrics 48, 95-103.

35. van Zwet, W. R. & Oostherhoff, J. (1967), 'On the combination of independent test statistics', Annals of Mathematical Statistics 38, 659-680.

[Recibido en diciembre de 2013. Aceptado en mayo de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n1a06,
    AUTHOR  = {Park, Hyo-Il},
    TITLE   = {{Nonparametric Simultaneous Test Procedures}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {1},
    pages   = {107-121}
}

References

Ansari, A. R. & Bradley, R. A. (1960), ‘Rank-sum tests for dispersions’, Annals of Mathematical Statistics 31, 1174–1189.

Brombin, C., Salmaso, L., Ferronato, G. & Galzignato, P.-F. (2011), ‘Multi-aspect procedures for paired data with application to biometric morphing’, Communications in Statistics-Simulation and Computation 40, 1–12.

Duran, B. S., Tsai, W. S. & Lewis, T. O. (1976), ‘A class of location-scale nonparametric tests’, Biometrika 63, 173–176.

Efron, B. (1979), ‘Bootstrap methods: Another look at the jackknife’, Annals of Statistics 7, 1–26.

Fisher, R. A. (1932), Statistical Methods for Research Workers, 4 edn, Oliver & Boyd, Edinburgh.

Fleming, T. R., Harrington, D. P. & O’Sullivan, M. (1987), ‘Supremum versión of the logrank and generalized ilcoxon statistics’, Journal of the American Statistical Association 82, 312–320.

Good, P. (2000), Permutation Tests-A Practical Guide to Resampling Methods for Testing Hypotheses, 2 edn, Springer, New York.

Lepage, Y. (1971), ‘A combination of Wilcoxon’s and Ansari-Bradley’s statistics’, Biometrika 58, 213–217.

Lepage, Y. (1973), ‘A table for a combined Wilcoxon Ansari-Bradley statistic’, Biometrika 60, 113–116.

Liptak, I. (1958), ‘On the combination of independent tests’, Magyar Tudomanyos Akademia Matematikai Kutato Intezenek Kozlomenyei 3, 127–141.

Marozzi, M. (2004), ‘A bi-aspect nonparametric test for the two-sample location problem’, Computational Statistics and Data Analysis 44, 639–648.

Marozzi, M. (2007), ‘Multivariate tri-aspect non-parametric testing’, Journal of Nonparametric Statistics 19, 269–282.

Marozzi, M. (2009), ‘Some notes on the location-scale Cucconi test’, Journal of Nonparametric Statistics 21, 629–647.

Marozzi, M. (2012a), ‘A modified Hall-Padmanabhan test for the homogeneity of scales’, Communications in Statistics-Theory and Methods 41, 3068–3078.

Marozzi, M. (2012b), ‘A combined test for differences in scale based on the interquantile range’, Statistical Papers 53, 61–72.

Marozzi, M. (2012c), ‘A distribution free test for the equality of scales’, Communications in Statistics-Simulation and Computation 41, 878–889.

Marozzi, M. (2012d), ‘A modified Cucconi test for location and scale change alternatives’, Revista Colombiana de Estadistica 35, 371–384.

Marozzi, M. (2013), ‘Nonparametric simultaneous tests for location and scale testing: A comparison of several methods’, Communications in Statistics- Simulation and Computation 42, 1298–1317.

Marozzi, M. (2014), ‘The multisample Cucconi test’, Statistical Methods and Applications 23(2), 209–227.

Mood, A. M. (1954), ‘On the asymptotic efficiency of certain nonparametric two-sample tests’, Annals of Mathematical Statistics 25, 514–522.

Murakami, H. (2007), ‘Lepage type statistic based on the modified Baumgartner statistic’, Computational Statistics and Data Analysis 51, 5061–5067.

Neuhäuser, M., Leuchs, A.-K. & Ball, D. (2011), ‘A new location-scale test based on a combination of the ideas of Levene and Lepage’, Biometrical Journal 53(3), 525–534.

Park, H. I. (2011), ‘A nonparametric test procedure based on a group of quantile tests’, Communications in Statistics-Simulation and Computation 40, 759–783.

Pesarin, F. (2001), Multivariate Permutation Tests, Wiley, Chichester. Pesarin, F. & Salmaso, L. (2010), Permutation Tests for Complex Data, Wiley, Chichester.

Pettitt, A. N. (1976), ‘A two-sample Anderson-Darling rank statistic’, Biometrika 63, 161–168.

Podgor, M. J. & Gastwirth, J. L. (1994), ‘On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives’, Statistics in Medicine 13, 747–758.

Randles, R. H. & Hogg, R. V. (1971), ‘Certain uncorrelated and independent Rank statistics’, Journal of American Statistical Association 66, 569–574.

Randles, R. H. &Wolfe, D. A. (1979), Introduction to the Theory of Nonparametric Statistics, Wiley, New York.

Rublik, F. (2009), ‘Critical values for testing location-scale hypothesis’, Measurement Science Review 9, 9–15.

Salmaso, L. & Solari, A. (2005), ‘Multiple aspect testing for case-control designs’, Metrika 62, 331–340.

Shao, J. & Tu, D. (1995), The Jackknife and Bootstrap, Springer, New York.

Tippett, L. H. C. (1931), The Methods of Statistics, Williams and Norgate, London.

van Zwet, W. R. & Oostherhoff, J. (1967), ‘On the combination of independent test statistics’, Annals of Mathematical Statistics 38, 659–680.

Zhang, J. (2006), ‘Powerful two-sample tests based on the likelihood ratio’, Technometrics 48, 95–103

How to Cite

APA

Park, H.-I. (2015). Nonparametric Simultaneous Test Procedures. Revista Colombiana de Estadística, 38(1), 107–121. https://doi.org/10.15446/rce.v38n1.48805

ACM

[1]
Park, H.-I. 2015. Nonparametric Simultaneous Test Procedures. Revista Colombiana de Estadística. 38, 1 (Jan. 2015), 107–121. DOI:https://doi.org/10.15446/rce.v38n1.48805.

ACS

(1)
Park, H.-I. Nonparametric Simultaneous Test Procedures. Rev. colomb. estad. 2015, 38, 107-121.

ABNT

PARK, H.-I. Nonparametric Simultaneous Test Procedures. Revista Colombiana de Estadística, [S. l.], v. 38, n. 1, p. 107–121, 2015. DOI: 10.15446/rce.v38n1.48805. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/48805. Acesso em: 29 jan. 2025.

Chicago

Park, Hyo-Il. 2015. “Nonparametric Simultaneous Test Procedures”. Revista Colombiana De Estadística 38 (1):107-21. https://doi.org/10.15446/rce.v38n1.48805.

Harvard

Park, H.-I. (2015) “Nonparametric Simultaneous Test Procedures”, Revista Colombiana de Estadística, 38(1), pp. 107–121. doi: 10.15446/rce.v38n1.48805.

IEEE

[1]
H.-I. Park, “Nonparametric Simultaneous Test Procedures”, Rev. colomb. estad., vol. 38, no. 1, pp. 107–121, Jan. 2015.

MLA

Park, H.-I. “Nonparametric Simultaneous Test Procedures”. Revista Colombiana de Estadística, vol. 38, no. 1, Jan. 2015, pp. 107-21, doi:10.15446/rce.v38n1.48805.

Turabian

Park, Hyo-Il. “Nonparametric Simultaneous Test Procedures”. Revista Colombiana de Estadística 38, no. 1 (January 1, 2015): 107–121. Accessed January 29, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/48805.

Vancouver

1.
Park H-I. Nonparametric Simultaneous Test Procedures. Rev. colomb. estad. [Internet]. 2015 Jan. 1 [cited 2025 Jan. 29];38(1):107-21. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/48805

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CrossRef citations9

1. Hyo-Il Park. (2016). Test procedures for the mean and variance simultaneously under normality. Communications for Statistical Applications and Methods, 23(6), p.563. https://doi.org/10.5351/CSAM.2016.23.6.563.

2. Hyo-Il Park. (2020). Simultaneous multivariate tests under the normality assumption. Communications in Statistics - Simulation and Computation, 49(7), p.1886. https://doi.org/10.1080/03610918.2018.1508702.

3. Dominika Polko-Zając. (2019). ON PERMUTATION LOCATION–SCALE TESTS. Statistics in Transition New Series, 20(4), p.153. https://doi.org/10.21307/stattrans-2019-039.

4. Z. L. Chong, A. Mukherjee, Michael B. C. Khoo. (2020). Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity. Quality Engineering, 32(1), p.91. https://doi.org/10.1080/08982112.2019.1578974.

5. Rosa Arboretti Giancristofaro, Stefano Bonnini, Livio Corain, Luigi Salmaso. (2016). Dependency and truncated forms of combinations in multivariate combination-based permutation tests and ordered categorical variables. Journal of Statistical Computation and Simulation, 86(18), p.3608. https://doi.org/10.1080/00949655.2016.1177826.

6. Hyo-Il Park. (2017). On tests detecting difference in means and variances simultaneously under normality. Communications in Statistics - Theory and Methods, 46(20), p.10025. https://doi.org/10.1080/03610926.2016.1228964.

7. Hidetoshi Murakami, Markus Neuhäuser. (2024). A two-sample nonparametric test for one-sided location-scale alternative. Journal of Applied Statistics, , p.1. https://doi.org/10.1080/02664763.2024.2392119.

8. Amitava Mukherjee, Rudra Sen. (2018). Optimal design of Shewhart–Lepage type schemes and its application in monitoring service quality. European Journal of Operational Research, 266(1), p.147. https://doi.org/10.1016/j.ejor.2017.09.013.

9. Atul Rajaram Chavan, Digambar Tukaram Shirke. (2019). Simultaneously Testing for Location and Scale Parameters of Two Multivariate Distributions. Revista Colombiana de Estadística, 42(2), p.185. https://doi.org/10.15446/rce.v42n2.70815.

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