Published

2014-01-01

Cramér-Von Mises Statistic for Repeated Measures

El estadístico de Cramér-Von Mises para medidas repetidas

DOI:

https://doi.org/10.15446/rce.v37n1.44357

Keywords:

Asymptotic Distribution, Bootstrap, Cramér-von Mises statistic, Hypothesis testing, Permutation test, Repeated Measures. (en)
Bootstrap, distribución asintótica, estadístico de Cramérvon Mises, medidas repetidas, test de hipótesis, test de permutaciones (es)

Authors

  • Pablo Martínez-Camblor Oficina de Investigación Biosanitaria (OIB), FICYT
  • Carlos Carleos Universidad de Oviedo
  • Norberto Corral Universidad de Oviedo

The Cramér-von Mises criterion is employed to compare whether the marginal distribution functions of a k-dimensional random variable are equal or not. The well-known Donsker invariance principle and the Karhunen-Loéve expansion is used in order to derive its asymptotic distribution. Two different resampling plans (one based on permutations and the other one based on the general bootstrap algorithm, gBA) are also considered to approximate its distribution. The practical behaviour of the proposed test is studied from a Monte Carlo simulation study. The statistical power of the test based on the Cramér-von Mises criterion is competitive when the underlying distributions are different in location and is clearly better than the Friedman one when the sole difference among the involved distributions is the spread or the shape. Both resampling plans lead to similar results although the gBA avoids the usual required interchangeability assumption. Finally, the method is applied on the study of the evolution inequality incomes distribution between some European countries along the years 2000 and 2011.

El criterio de Cramér-von Mises es empleado para comparar la igualdad entre las distribuciones marginales de una variable aleatoria k-dimensional. El conocido principio de invaranza de Donsker y la expansión de Karhunen- Loéve se usan para derivar su distribución asintótica. Dos planes de remuestreo diferentes (uno basado en permutaciones y el otro basado en el algoritmo bootstrap general, gBA) son usados para aproximar su distribución. El comportamiento práctico del test propuesto es estudiado mediante simulaciones de Monte Carlo. La potencia estadística del test basado en el criterio de Cramér-von Mises es competitiva cuando la distribuciones subyacentes difieren en el parámetro de localización. Este test es claramente superior al de Friedman cuando las únicas diferencias son en la dispersión o la forma. Ambos planes de remuestreo obtienen resultados similares aunque el gBA evita la hipótesis de intercambiabilidad. Finalmente, el método propuesto es aplicado al estudio de la evolución de las desigualdades en los  ingresos entre algunos países Europeos entre los años 2000 y 2011.

https://doi.org/10.15446/rce.v37n1.44357

Cramér-Von Mises Statistic for Repeated Measures

El estadístico de Cramér-Von Mises para medidas repetidas

PABLO MARTÍNEZ-CAMBLOR1, CARLOS CARLEOS2, NORBERTO CORRAL3

1FICYT, Oficina de Investigación Biosanitaria (OIB), Oviedo, Spain. Universidad de Oviedo, Departamento Estadística e IO y DM, Asturias, Spain. Biostatistician. Email: pmcamblor@hotmail.com
2Universidad de Oviedo, Departamento Estadística e IO y DM, Asturias, Spain. Professor. Email: carleos@uniovi.es
3Universidad de Oviedo, Departamento Estadística e IO y DM, Asturias, Spain. Lecturer. Email: norbert@uniovi.es


Abstract

The Cramér-von Mises criterion is employed to compare whether the marginal distribution functions of a k-dimensional random variable are equal or not. The well-known Donsker invariance principle and the Karhunen-Loéve expansion is used in order to derive its asymptotic distribution. Two different resampling plans (one based on permutations and the other one based on the general bootstrap algorithm, gBA) are also considered to approximate its distribution. The practical behaviour of the proposed test is studied from a Monte Carlo simulation study. The statistical power of the test based on the Cramér-von Mises criterion is competitive when the underlying distributions are different in location and is clearly better than the Friedman one when the sole difference among the involved distributions is the spread or the shape. Both resampling plans lead to similar results although the gBA avoids the usual required interchangeability assumption. Finally, the method is applied on the study of the evolution inequality incomes distribution between some European countries along the years 2000 and 2011.

Key words: Asymptotic Distribution, Bootstrap, Cramér-von Mises statistic, Hypothesis testing, Permutation test, Repeated Measures.


Resumen

El criterio de Cramér-von Mises es empleado para comparar la igualdad entre las distribuciones marginales de una variable aleatoria k-dimensional. El conocido principio de invaranza de Donsker y la expansión de Karhunen-Loéve se usan para derivar su distribución asintótica. Dos planes de remuestreo diferentes (uno basado en permutaciones y el otro basado en el algoritmo bootstrap general, gBA) son usados para aproximar su distribución. El comportamiento práctico del test propuesto es estudiado mediante simulaciones de Monte Carlo. La potencia estadística del test basado en el criterio de Cramér-von Mises es competitiva cuando la distribuciones subyacentes difieren en el parámetro de localización. Este test es claramente superior al de Friedman cuando las únicas diferencias son en la dispersión o la forma. Ambos planes de remuestreo obtienen resultados similares aunque el gBA evita la hipótesis de intercambiabilidad. Finalmente, el método propuesto es aplicado al estudio de la evolución de las desigualdades en los ingresos entre algunos países Europeos entre los años 2000 y 2011.

Palabras clave: Bootstrap, distribución asintótica, estadístico de Cramér-von Mises, medidas repetidas, test de hipótesis, test de permutaciones.


Texto completo disponible en PDF


References

1. Adler, R. J. (1990), An introduction to continuity, extrema and related topics for general gaussian processes, 'IMS Lecture Notes-Monograph Series', Vol. 12, Institute of Mathematical Statistics, , , Hayward, California.

2. Alkarni, S. H. & Siddiqui, M. M. (2001), 'An upper bound for the distribution function of a positive definite quadratic form', Journal of Statistical Computation and Simulation 69(1), 51-56.

3. Anderson, T. W. (1962), 'On the distribution of the two-sample cramér-von mises criterion', Annals of Mathematical Statistics 33(3), 1148-1159.

4. Arcones, M. A. & Gine, E. (1992), 'On the bootstrap of U and V statistics', Annals of Statistics 20(2), 655-674.

5. Beran, R. (1982), 'Estimated sampling distributions: the bootstrap and competitors', Annals of Statistics 10, 212-225.

6. Ciba-Geigy, L. S. & Olsson, B. (1982), 'A nearly distribution-free test for comparing dispersion in paired samples', Biometrika 69(2), 484-485.

7. Cowel, F. A. (2009), Measuring inequality. Accesed 16/04/2013. *http://darp.lse.ac.uk/papersdb/cowell_measuringinequality3.pdf

8. Cramér, H. (1928), 'On the composition of elementary errors: II statistical applications', Skandinavisk Aktuarietidskrift 11, 141-180.

9. Csörgo, S. & Faraway, J. J. (1996), 'The exact and asymptotic distributions of Cramér-von Mises statistics', Journals of the Royal Statistical Society B 58(1), 221-234.

10. Deheuvels, P. (2005), 'Weighted multivariate Cramér-von Mises-type statistics', Afrika Statistika 1(1), 1-14.

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

12. Efron, E. (1982), The Jackknife, the Bootstrap and Other Resampling Plans, Society for Industrial and Applied Mathematics. *http://epubs.siam.org/doi/abs/10.1137/1.9781611970319

13. Freitag, G., Czado, C. & Munk, A. (2007), 'A nonparametric test for similarity of marginals-with applications to the assessment of population bioequivalence', Journal of Statistical Planning & Inference 137(3), 697-711.

14. Good, P. (2000), Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses, Springer Verlag, New York.

15. Govindarajulu, Z. (1995), 'A class of asymptotically distribution free test procedures for equality of marginals under multivariate dependence', American Journal of Mathematical and Management Sciences 15, 375-394.

16. Govindarajulu, Z. (1997), 'A class of asymptotically distribution free tests for equality of marginals in multivariate populations', Mathematical Methods of Statistics 6, 92-111.

17. Hall, P. & Wilson, S. R. (1991), 'Two guidelines for bootstrap hypothesis testing', Biometrics 47, 757-762.

18. Horváth, L. & Steinebach, J. (1999), 'On the best approximation for bootstrapped empirical processes', Statistical & Probability Letters 41, 117-122.

19. Kiefer, J. (1959), 'K-Sample analogues of the Kolmogorov-Smirnov, Cramér-von Mises tests', Annals of Mathematical Statistis 30, 420-447.

20. Lam, F. C. & Longnecker, M. T. (1983), 'Modified Wilcoxon rank sum test for paired data', Biometrika 70(2), 510-513.

21. Martin, M. A. (2007), 'Bootstrap hypothesis testing for some common statistical problems: a critical evaluation of size and power properties', Computational Statistics & Data Analysis 51, 6321-6342.

22. Martínez-Camblor, P. (2007), 'Central limit theorems for S-Gini and Theil inequality coefficients', Revista Colombiana de Estadística 30(2), 287-300.

23. Martínez-Camblor, P. (2010), 'Nonparametric k-sample test based on kernel density estimator for paired design', Computational Statistics & Data Analysis 54, 2035-2045.

24. Martínez-Camblor, P. (2011), 'Testing the equality among distribution functions from independent and right censored samples via Cramér-von Mises criterion', Journal of Applied Statistics 38(6), 1117-1131.

25. Martínez-Camblor, P., Carelos, C. & Corral, N. (2012), 'Sobre el estadístico de Cramér-von Mises', Revista de Matemáticas: Teoría y Aplicaciones 19, 89-101.

26. Martínez-Camblor, P., Corral, N. & Vicente, D. (2011), 'Statistical comparison of the genetic sequence type diversity of invasive Neisseria meningitidis isolates in northern Spain (1997-2008)', Ecological Informatics 6(6), 391-398.

27. Martínez-Camblor, P. & Uña-Álvarez, J. (2009), 'Non-parametric k-sample tests: density functions vs distribution functions', Computational Statistics & Data Analysis 53(9), 3344-3357.

28. Munzel, U. (1999a), 'Nonparametric methods for paired samples', Statistica Neerlandica 53(3), 277-286.

29. Munzel, U. (1999b), 'Linear rank score statistics when ties are present', Statistics & Probability Letters 41, 389-395.

30. Nelsen, R. B. (2007), 'Extremes of non-exchangeability', Statistical Papers 48, 329-336.

31. Podgor, M. J. & Gastwirth, J. L. (1996), Efficiency robust rank tests for stratified data, 'Research Developments in Probability and Statistics', Festschrift in honor of Madan L. Puri. VSP International Science Publishers, Leiden, Netherlands.

32. Quessy, J. F. & Éthier, F. (2012), 'Cramér-von Mises and characteristic function tests for the two and k-sample problems with dependent data', Computational Statistics and Data Analysis 56, 2097-2111.

33. Van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press, Cambridge.

34. Venkatraman, E. S. & Begg, C. B. (1996), 'A distribution-free procedure for comparing receiver operating characteristic curves from a paired experiment', Biometrika 83(4), 835-848.

35. Von Mises, R. E. (1991), Wahrscheinlichkeitsrechnung, Deuticke, Vienna.

36. Westfall, P. H. & Young, S. S. (1993), Resampling-Based Multiple Testing: Examples and Methods for p-value Adjustment, Wiley, New York.


[Recibido en septiembre de 2013. Aceptado en diciembre de 2013]

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

@ARTICLE{RCEv37n1a04,
    AUTHOR  = {Martínez-Camblor, Pablo and Carleos, Carlos and Corral, Norberto},
    TITLE   = {{Cramér-Von Mises Statistic for Repeated Measures}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {1},
    pages   = {45-67}
}

References

Adler, R. J. (1990), An introduction to continuity, extrema and related topics for general gaussian processes, in ‘IMS Lecture Notes-Monograph Series’, Vol. 12, Institute of Mathematical Statistics, Hayward, California.

Alkarni, S. H. & Siddiqui, M. M. (2001), ‘An upper bound for the distribution function of a positive definite quadratic form’, Journal of Statistical Computation and Simulation 69(1), 51–56.

Anderson, T. W. (1962), ‘On the distribution of the two-sample cramér-von mises criterion’, Annals of Mathematical Statistics 33(3), 1148–1159.

Arcones, M. A. & Gine, E. (1992), ‘On the bootstrap of u and v statistics’, Annals of Statistics 20(2), 655–674.

Beran, R. (1982), ‘Estimated sampling distributions: The bootstrap and competitors’, Annals of Statistics 10, 212–225.

Ciba-Geigy, L. S. & Olsson, B. (1982), ‘A nearly distribution-free test for comparing dispersion in paired samples’, Biometrika 69(2), 484–485.

Cowel, F. A. (2009), Measuring inequality. Accesed 16/04/2013. *http://darp.lse.ac.uk/papersdb/cowell_measuringinequality3.pdf

Cramér, H. (1928), ‘On the composition of elementary errors: II statistical applications’, Skandinavisk Aktuarietidskrift 11, 141–180.

Csörgo, S. & Faraway, J. J. (1996), ‘The exact and asymptotic distributions of Cramér-von Mises statistics’, Journals of the Royal Statistical Society B 58(1), 221–234.

Deheuvels, P. (2005), ‘Weighted multivariate Cramér-von Mises-type statistics’, Afrika Statistika 1(1), 1–14.

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

Efron, E. (1982), The Jackknife, the Bootstrap and Other Resampling Plans, Society for Industrial and Applied Mathematics.

*http://epubs.siam.org/doi/abs/10.1137/1.9781611970319

Freitag, G., Czado, C. & Munk, A. (2007), ‘A nonparametric test for similarity of marginals–with applications to the assessment of population bioequivalence’, Journal of Statistical Planning & Inference 137(3), 697–711.

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

Govindarajulu, Z. (1995), ‘A class of asymptotically distribution free test procedures for equality of marginals under multivariate dependence’, American Journal of Mathematical and Management Sciences 15, 375–394.

Govindarajulu, Z. (1997), ‘A class of asymptotically distribution free tests for equality of marginals in multivariate populations’, Mathematical Methods of Statistics 6, 92–111.

Hall, P. & Wilson, S. R. (1991), ‘Two guidelines for bootstrap hypothesis testing’, Biometrics 47, 757–762.

Horváth, L. & Steinebach, J. (1999), ‘On the best approximation for bootstrapped empirical processes’, Statistical & Probability Letters 41, 117–122.

Kiefer, J. (1959), ‘k-Sample analogues of the Kolmogorov-Smirnov, Cramér-von Mises tests’, Annals of Mathematical Statistis 30, 420–447.

Lam, F. C. & Longnecker, M. T. (1983), ‘Modified Wilcoxon rank sum test for paired data’, Biometrika 70(2), 510–513.

Martin, M. A. (2007), ‘Bootstrap hypothesis testing for some common statistical problems: A critical evaluation of size and power properties’, Computational Statistics & Data Analysis 51, 6321–6342.

Martínez-Camblor, P. (2007), ‘Central limit theorems for S-Gini and Theil inequality coefficients’, Revista Colombiana de Estadística 30(2), 287–300.

Martínez-Camblor, P. (2010), ‘Nonparametric k-sample test based on kernel density estimator for paired design’, Computational Statistics & Data Analysis 54, 2035–2045.

Martínez-Camblor, P. (2011), ‘Testing the equality among distribution functions from independent and right censored samples via Cramér-von Mises criterion’, Journal of Applied Statistics 38(6), 1117–1131.

Martínez-Camblor, P., Carelos, C. & Corral, N. (2012), ‘Sobre el estadístico de Cramér-von Mises’, Revista de Matemáticas: Teoría y Aplicaciones 19, 89– 101.

Martínez-Camblor, P., Corral, N. & Vicente, D. (2011), ‘Statistical comparison of the genetic sequence type diversity of invasive Neisseria meningitidis isolates in northern Spain (1997-2008)’, Ecological Informatics 6(6), 391–398.

Martínez-Camblor, P. & Uña-Álvarez, J. (2009), ‘Non-parametric k-sample tests: density functions vs distribution functions’, Computational Statistics & Data Analysis 53(9), 3344–3357.

Munzel, U. (1999a), ‘Linear rank score statistics when ties are present’, Statistics & Probability Letters 41, 389–395.

Munzel, U. (1999b), ‘Nonparametric methods for paired samples’, Statistica Neerlandica 53(3), 277–286.

Nelsen, R. B. (2007), ‘Extremes of non-exchangeability’, Statistical Papers 48, 329– 336.

Podgor, M. J. & Gastwirth, J. L. (1996), Efficiency robust rank tests for stratified data, in E. Brunner & M. Denker, eds, ‘Research Developments in Probability and Statistics’, Festschrift in honor of Madan L. Puri. VSP International Science Publishers, Leiden, Netherlands.

Quessy, J. F. & Éthier, F. (2012), ‘Cramér-von Mises and characteristic function tests for the two and k-sample problems with dependent data’, Computational Statistics and Data Analysis 56, 2097–2111.

Van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press, Cambridge.

Venkatraman, E. S. & Begg, C. B. (1996), ‘A distribution-free procedure for comparing receiver operating characteristic curves from a paired experiment’, Biometrika 83(4), 835–848.

Von Mises, R. E. (1991), Wahrscheinlichkeitsrechnung, Deuticke, Vienna.

Westfall, P. H. & Young, S. S. (1993), Resampling-Based Multiple Testing: Examples and Methods for p-value Adjustment, Wiley, New York.

How to Cite

APA

Martínez-Camblor, P., Carleos, C. and Corral, N. (2014). Cramér-Von Mises Statistic for Repeated Measures. Revista Colombiana de Estadística, 37(1), 45–67. https://doi.org/10.15446/rce.v37n1.44357

ACM

[1]
Martínez-Camblor, P., Carleos, C. and Corral, N. 2014. Cramér-Von Mises Statistic for Repeated Measures. Revista Colombiana de Estadística. 37, 1 (Jan. 2014), 45–67. DOI:https://doi.org/10.15446/rce.v37n1.44357.

ACS

(1)
Martínez-Camblor, P.; Carleos, C.; Corral, N. Cramér-Von Mises Statistic for Repeated Measures. Rev. colomb. estad. 2014, 37, 45-67.

ABNT

MARTÍNEZ-CAMBLOR, P.; CARLEOS, C.; CORRAL, N. Cramér-Von Mises Statistic for Repeated Measures. Revista Colombiana de Estadística, [S. l.], v. 37, n. 1, p. 45–67, 2014. DOI: 10.15446/rce.v37n1.44357. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44357. Acesso em: 29 mar. 2024.

Chicago

Martínez-Camblor, Pablo, Carlos Carleos, and Norberto Corral. 2014. “Cramér-Von Mises Statistic for Repeated Measures”. Revista Colombiana De Estadística 37 (1):45-67. https://doi.org/10.15446/rce.v37n1.44357.

Harvard

Martínez-Camblor, P., Carleos, C. and Corral, N. (2014) “Cramér-Von Mises Statistic for Repeated Measures”, Revista Colombiana de Estadística, 37(1), pp. 45–67. doi: 10.15446/rce.v37n1.44357.

IEEE

[1]
P. Martínez-Camblor, C. Carleos, and N. Corral, “Cramér-Von Mises Statistic for Repeated Measures”, Rev. colomb. estad., vol. 37, no. 1, pp. 45–67, Jan. 2014.

MLA

Martínez-Camblor, P., C. Carleos, and N. Corral. “Cramér-Von Mises Statistic for Repeated Measures”. Revista Colombiana de Estadística, vol. 37, no. 1, Jan. 2014, pp. 45-67, doi:10.15446/rce.v37n1.44357.

Turabian

Martínez-Camblor, Pablo, Carlos Carleos, and Norberto Corral. “Cramér-Von Mises Statistic for Repeated Measures”. Revista Colombiana de Estadística 37, no. 1 (January 1, 2014): 45–67. Accessed March 29, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/44357.

Vancouver

1.
Martínez-Camblor P, Carleos C, Corral N. Cramér-Von Mises Statistic for Repeated Measures. Rev. colomb. estad. [Internet]. 2014 Jan. 1 [cited 2024 Mar. 29];37(1):45-67. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44357

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