Published

2014-07-01

Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions

Herramientas gráficas para evaluar bondad de ajuste en distribuciones de no-localización-escala

DOI:

https://doi.org/10.15446/rce.v37n2spe.47941

Keywords:

Censored Data, Confidence Band, Data Analysis, Probability Plots (en)
Análisis de datos, Bandas de confianza, Datos censurados, Gráficos de probabilidad. (es)

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Authors

  • Claudia Castro-Kuriss Instituto Tecnológico de Buenos Aires, Argentina
  • Víctor Leiva Universidad Adolfo Ibáñez, Chile Universidad de Valparaíso, Chile
  • Emilia Athayde Universidade do Minho, Portugal

Goodness-of-fit (GOF) techniques are used for assessment whether a distribution is suitable to describe a data set or not. These techniques have been studied for distributions belonging to the location-scale family. However, one could be interested in making this assessment for distributions that do not belong to this family. We review the available GOF tests and propose graphical tools based on these tests for censored and uncensored data from non-location-scale distributions. Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov, Kuiper, Michael and Watson GOF statistics are considered. We apply the proposed results to real-world data sets to illustrate their potential, with emphasis on some Birnbaum-Saunders distributions.

Las técnicas de bondad de ajuste se usan para establecer si una distribución es apropiada o no para describir un conjunto de datos. Estas técnicas han sido estudiadas para distribuciones pertenecientes a la familia de locación y escala. Sin embargo, podríamos también estar interesados en establecer si una distribución que no pertenece a esta familia brinda un buen ajuste a los datos. Revisamos los tests de bondad de ajuste disponibles y proponemos herramientas gráficas basadas en estos tests para datos completos y censurados desde distribuciones de no locación y escala. Consideramos los estadísticos de Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov, Kuiper, Michael y Watson. Aplicamos los resultados propuestos a conjuntos de datos reales para ilustrar su potencial, particularmente en algunas distribuciones Birnbaum-Saunders

https://doi.org/10.15446/rce.v37n2spe.47941

Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions

Herramientas gráficas para evaluar bondad de ajuste en distribuciones de no-localización-escala

CLAUDIA CASTRO-KURISS1, VÍCTOR LEIVA2, EMILIA ATHAYDE3

1Instituto Tecnológico de Buenos Aires, Departamento de Matemática, Argentina. Associate Professor. Email: ccastro@itba.edu.ar
2Universidad Adolfo Ibáñez, Facultad de Ingeniería y Ciencias, Chile. Universidad de Valparaíso, Instituto de Estadística, Chile. Professor. Email: victorleivasanchez@gmail.com
3Universidade do Minho, Centro de Matemática, Portugal. Associate Professor. Email: mefqa@math.uminho.pt

Abstract

Goodness-of-fit (GOF) techniques are used for assessment whether a distribution is suitable to describe a data set or not. These techniques have been studied for distributions belonging to the location-scale family. However, one could be interested in making this assessment for distributions that do not belong to this family. We review the available GOF tests and propose graphical tools based on these tests for censored and uncensored data from non-location-scale distributions. Anderson-Darling, Cram\er-von Mises, Kolmogorov-Smirnov, Kuiper, Michael and Watson GOF statistics are considered. We apply the proposed results to real-world data sets to illustrate their potential, with emphasis on some Birnbaum-Saunders distributions.

Key words: Censored Data, Confidence Band, Data Analysis, Probability Plots.

Resumen

Las técnicas de bondad de ajuste se usan para establecer si una distribución es apropiada o no para describir un conjunto de datos. Estas técnicas han sido estudiadas para distribuciones pertenecientes a la familia de locación y escala. Sin embargo, podríamos también estar interesados en establecer si una distribución que no pertenece a esta familia brinda un buen ajuste a los datos. Revisamos los tests de bondad de ajuste disponibles y proponemos herramientas gráficas basadas en estos tests para datos completos y censurados desde distribuciones de no locación y escala. Consideramos los estadísticos de Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov, Kuiper, Michael y Watson. Aplicamos los resultados propuestos a conjuntos de datos reales para ilustrar su potencial, particularmente en algunas distribuciones Birnbaum-Saunders.

Palabras clave: análisis de datos, bandas de confianza, datos censurados, gráficos de probabilidad.

Texto completo disponible en PDF

References

1. Ahmed, S., Castro-Kuriss, C., Leiva, V., Flores, E. & Sanhueza, A. (2010), 'Truncated version of the Birnbaum-Saunders distribution with an application in financial risk', Pakistan Journal of Statistics 26, 293-311.

2. Athayde, E., Azevedo, C., Leiva, V. & Sanhueza, A. (2012), 'About Birnbaum-Saunders distributions based on the Johnson system', Communications in Statistics, Theory and Methods 41, 2061-2079.

3. Azevedo, C., Leiva, V., Athayde, E. & Balakrishnan, N. (2012), 'Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation.', Computational Statistics and Data Analysis 56, 3887-3897.

4. Balakrishnan, N., Gupta, R., Kundu, D., Leiva, V. & Sanhueza, A. (2011), 'On some mixture models based on the Birnbaum-Saunders distribution and associated inference', Journal of Statistical Planning and Inference 141, 2175-2190.

5. Balakrishnan, N., Leiva, V., Sanhueza, A. & Cabrera, E. (2009), 'Mixture inverse Gaussian distribution and its transformations, moments and applications', Statistics 43, 91-104.

6. Balakrishnan, N., Leiva, V., Sanhueza, A. & Vilca, F. (2009), 'Estimation in the Birnbaum-Saunders distribution based on scale-mixture of normals and the EM-algorithm', Statistics and Operations Research Transactions 33, 171-192.

7. Balakrishnan, N., Ng, H. & Kannan, N. (2004), 'Goodness-of-fit tests based on spacings for progressively type-II censored data from a general location-scale distribution.', IEEE Transactions on Reliability 53, 349-356.

8. Barr, D. & Davidson, T. (1973), 'A Kolmogorov-Smirnov test for censored samples', Technometrics 15, 739-757.

9. Barros, M., Leiva, V., Ospina, R. & Tsuyuguchi, A. (2014), 'Goodness-of-fit tests for the Birnbaum-Saunders distribution with censored reliability data', IEEE Transactions on Reliability 63, 543-554.

10. Barros, M., Paula, G. & Leiva, V. (2009), 'An R implementation for generalized Birnbaum-Saunders distributions', Computational Statistics and Data Analysis 53, 1511-1528.

11. Birnbaum, Z. & Saunders, S. (1969), 'Estimation for a family of life distributions with applications to fatigue', Journal of Applied Probability 6, 328-347.

12. Castillo, J. & Puig, P. (1997), 'Testing departures from Gamma, Rayleigh, and truncated normal distributions', Annals of the Institute of Statistical Mathematics 49, 255-269.

13. Castro-Kuriss, C. (2011), 'On a goodness-of-fit test for censored data from a location-scale distribution with application', Chilean Journal of Statistics 37, 15-136.

14. Castro-Kuriss, C., Kelmansky, D., Leiva, V. & Martinez, E. (2009), 'A new goodness-of-fit test for censored data with an application in monitoring processes', Communications in Statistics: Simulation and Computation 38, 1161-1177.

15. Castro-Kuriss, C., Kelmansky, D., Leiva, V. & Martinez, E. (2010), 'On a goodness-of-fit test for normality with unknown parameters and type-II censored data', Journal of Applied Statistics 37, 1193-1211.

16. Chen, G. & Balakrishnan, N. (1995), 'A general purpose approximate goodness-of-fit test', Journal of Quality Technology 27, 154-161.

17. Choulakian, V. & Stephens, M. (2001), 'Goodness-of-fit tests for the generalized Pareto distribution', Technometrics 43, 478-484.

18. Cohen, C. (1991), Truncated and Censored Samples, Marcel Dekker, NY.

19. D'Agostino, C. & Stephens, M. (1986), Goodness of Fit Techniques, Marcel Dekker, NY.

20. DePriest, D. (1983), 'Using the singly truncated normal distribution to analyse satellite data', Communications in Statistics, Theory and Methods 12, 263-272.

21. Dufour, R. & Maag, U. (1978), 'Distribution results for modified Kolmogorov-Smirnov statistics for truncated or censored samples', Technometrics 20, 29-32.

22. Ferreira, M., Gomes, M. & Leiva, V. (2012), 'On an extreme value version of the Birnbaum-Saunders distribution', REVSTAT - Statistical Journal 10, 181-210.

23. Fierro, R., Leiva, V., Ruggeri, F. & Sanhueza, A. (2013), 'On a Birnbaum-Saunders distribution arising from a non-homogeneous Poisson process', Statistics & Probability Letters 83, 1233-1239.

24. Fischer, T. & Kamps, U. (2011), 'On the existence of transformations preserving the structure of order statistics in lower dimensions', Journal of Statistical Planning and Inference 141, 536-548.

25. Hubert, M. & Vanderveeken, S. (2008), 'Outlier detection for skewed data', Journal of Chemometrics 22, 235-246.

26. Hubert, M. & Vandervieren, E. (2008), 'An adjusted boxplot for skewed distributions', Computational Statistics and Data Analysis 52, 5186-5201.

27. Hui, W., Gel, Y. & Gastwirth, J. (2008), 'Lawstat: An r package for law, public policy and biostatistics', Journal Statistical Software 28, 1-26.

28. Johnson, N., Kotz, S. & Balakrishnan, N. (1994), Continuous Univariate Distributions, Wiley, NY.

29. Kotz, S., Leiva, V. & Sanhueza, A. (2010), 'Two new mixture models related to the inverse Gaussian distribution', Methodology and Computing in Applied Probability 12, 199-212.

30. Lawless, J. (2003), Statistical Models and Methods for Lifetime Data, Wiley, NY.

31. Leiva, V., Athayde, E., Azevedo, C. & Marchant, C. (2011), 'Modeling wind energy flux by a Birnbaum-Saunders distribution with unknown shift parameter', Journal of Applied Statistics 38, 2819-2838.

32. Leiva, V., Marchant, C., Saulo, H., Aslam, M. & Rojas, F. (2014), 'Capability indices for Birnbaum-Saunders processes applied to electronic and food industries', Journal of Applied Statistics 41, 1881-1902.

33. Leiva, V., Ponce, M., Marchant, C. & Bustos, O. (2012), 'Fatigue statistical distributions useful for modeling diameter and mortality of trees', Revista Colombiana de Estadística 35, 349-367.

34. Leiva, V., Rojas, E., Galea, M. & Sanhueza, A. (2014), 'Diagnostics in Birnbaum-Saunders accelerated life models with an application to fatigue data', Applied Stochastic Models in Business and Industry 30, 115-131.

35. Leiva, V., Sanhueza, A., Kotz, S. & Araneda, N. (2010), 'A unified mixture model based on the inverse Gaussian distribution', Pakistan Journal of Statistic 26, 445-460.

36. Leiva, V., Santos-Neto, M., Cysneiros, F. & Barros, M. (2014), 'Birnbaum-Saunders statistical modelling: A new approach', Statistical Modelling 14, 21-48.

37. Leiva, V., Saulo, H., Leao, J. & Marchant, C. (2014), 'A family of autoregressive conditional duration models applied to financial data', Computational Statistics and Data Analysis 79, 175-191.

38. Leiva, V., Soto, G., Cabrera, E. & Cabrera, G. (2011), 'New control charts based on the Birnbaum-Saunders distribution and their implementation', Revista Colombiana de Estadística 34, 147-176.

39. Lin, C.-T., Huang, Y.-L. & Balakrishnan, N. (2008), 'A new method for goodness-of-fit testing based on type-II right censored samples', IEEE Transactions on Reliability 57, 633-642.

40. Malmquist, S. (1950), 'On a property of order statistics from a rectangular distribution', Scandinavian Actuarial Journal 33, 214-222.

41. Marden, J. (2004), 'Positions and QQ-plots', Statistical Sciences 19, 606-614.

42. Marshall, A. & Olkin, I. (2007), Life Distributions, Springer, NY..

43. Michael, J. (1983), 'The stabilized probability plot', Biometrika 70, 11-17.

44. Michael, J. & Schucany, W. (1979), 'A new approach to testing goodness of fit for censored samples', Technometrics 21, 435-441.

45. O'Reilly, F. & Stephens, M. (1988), 'Transforming censored samples for testing fit', Technometrics 30, 79-86.

46. Pakyari, R. & Balakrishnan, N. (2012), 'A general purpose approximate goodness-of-fit test for progressively type-II censored data', IEEE Transactions on Reliability 61, 238-244.

47. Paula, G., Leiva, V., Barros, M. & Liu, S. (2012), 'Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance', Applied Stochastic Models in Business and Industry 28, 16-34.

48. Rad, A., Yousefzadeh, F. & Balakrishnan, N. (2011), 'Goodness-of-fit test based on Kullback-Leibler information for progressively type-II censored data', IEEE Transactions on Reliability 60, 570-579.

49. Rojas, F., Leiva, V., Wanke, P. & Marchant, C. (2015), 'Optimization of contribution margins in food services by modeling independent component demand', Revista Colombiana de Estadística 38, 1-30.

50. Sanhueza, A., Leiva, V. & Lopez-Kleine, L. (2011), 'On the student-t mixture inverse Gaussian model with an application to protein production', Revista Colombiana de Estadística 34, 177-195.

51. Stehlík, M., Strelec, L. & Thulin, M. (2014), 'On robust testing for normality in chemometrics', Chemometrics and Intelligent Laboratory Systems 130, 98-108.

52. Vilca, F., Sanhueza, A., Leiva, V. & Christakos, G. (2010), 'An extended Birnbaum-Saunders model and its application in the study of environmental quality in Santiago, Chile', Stochastic Environmental Research and Risk Assessment 24, 771-782.

[Recibido en mayo de 2014. Aceptado en noviembre de 2014]

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

@ARTICLE{RCEv37n2a06,
    AUTHOR  = {Castro-Kuriss, Claudia and Leiva, Víctor and Athayde, Emilia},
    TITLE   = {{Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {2},
    pages   = {341-365}
}

References

Ahmed, S., Castro-Kuriss, C., Leiva, V., Flores, E. & Sanhueza, A. (2010), ‘Truncated version of the Birnbaum-Saunders distribution with an application in financial risk’, Pakistan Journal of Statistics 26, 293–311.

Athayde, E., Azevedo, C., Leiva, V. & Sanhueza, A. (2012), ‘About Birnbaum- Saunders distributions based on the Johnson system’, Communications in Statistics, Theory and Methods 41, 2061–2079.

Azevedo, C., Leiva, V., Athayde, E. & Balakrishnan, N. (2012), ‘Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation.’, Computational Statistics and Data Analysis 56, 3887–3897.

Balakrishnan, N., Gupta, R., Kundu, D., Leiva, V. & Sanhueza, A. (2011), ‘On some mixture models based on the Birnbaum-Saunders distribution and associated inference’, Journal of Statistical Planning and Inference 141, 2175– 2190.

Balakrishnan, N., Leiva, V., Sanhueza, A. & Cabrera, E. (2009), ‘Mixture inverse Gaussian distribution and its transformations, moments and applications’, Statistics 43, 91–104.

Balakrishnan, N., Leiva, V., Sanhueza, A. & Vilca, F. (2009), ‘Estimation in the Birnbaum-Saunders distribution based on scale-mixture of normals and the EM-algorithm’, Statistics and Operations Research Transactions 33, 171–192.

Balakrishnan, N., Ng, H. & Kannan, N. (2004), ‘Goodness-of-fit tests based on spacings for progressively type-II censored data from a general location-scale distribution.’, IEEE Transactions on Reliability 53, 349–356.

Barr, D. & Davidson, T. (1973), ‘A Kolmogorov-Smirnov test for censored samples’, Technometrics 15, 739–757.

Barros, M., Leiva, V., Ospina, R. & Tsuyuguchi, A. (2014), ‘Goodness-of-fit tests for the Birnbaum-Saunders distribution with censored reliability data’, IEEE Transactions on Reliability 63, 543–554.

Barros, M., Paula, G. & Leiva, V. (2009), ‘An R implementation for generalized Birnbaum-Saunders distributions’, Computational Statistics and Data Analysis 53, 1511–1528.

Birnbaum, Z. & Saunders, S. (1969), ‘Estimation for a family of life distributions with applications to fatigue’, Journal of Applied Probability 6, 328–347.

Castillo, J. & Puig, P. (1997), ‘Testing departures from Gamma, Rayleigh, and truncated normal distributions’, Annals of the Institute of Statistical Mathematics 49, 255–269.

Castro-Kuriss, C. (2011), ‘On a goodness-of-fit test for censored data from a location-scale distribution with application’, Chilean Journal of Statistics 37, 15–136.

Castro-Kuriss, C., Kelmansky, D., Leiva, V. & Martinez, E. (2009), ‘A new goodness-of-fit test for censored data with an application in monitoring processes’, Communications in Statistics: Simulation and Computation 38, 1161–1177.

Castro-Kuriss, C., Kelmansky, D., Leiva, V. & Martinez, E. (2010), ‘On a goodness-of-fit test for normality with unknown parameters and type-II censored data’, Journal of Applied Statistics 37, 1193–1211.

Chen, G. & Balakrishnan, N. (1995), ‘A general purpose approximate goodness- of- fit test’, Journal of Quality Technology 27, 154–161.

Choulakian, V. & Stephens, M. (2001), ‘Goodness-of-fit tests for the generalized Pareto distribution’, Technometrics 43, 478–484.

Cohen, C. (1991), Truncated and Censored Samples, Marcel Dekker, NY.

D’Agostino, C. & Stephens, M. (1986), Goodness of Fit Techniques, Marcel Dekker, NY.

DePriest, D. (1983), ‘Using the singly truncated normal distribution to analyse satellite data’, Communications in Statistics, Theory and Methods 12, 263–272.

Dufour, R. & Maag, U. (1978), ‘Distribution results for modified Kolmogorov- Smirnov statistics for truncated or censored samples’, Technometrics 20, 29–32.

Ferreira, M., Gomes, M. & Leiva, V. (2012), ‘On an extreme value version of the Birnbaum-Saunders distribution’, REVSTAT - Statistical Journal 10, 181–210.

Fierro, R., Leiva, V., Ruggeri, F. & Sanhueza, A. (2013), ‘On a Birnbaum-Saunders distribution arising from a non-homogeneous Poisson process’, Statistics & Probability Letters 83, 1233–1239.

Fischer, T. & Kamps, U. (2011), ‘On the existence of transformations preserving the structure of order statistics in lower dimensions’, Journal of Statistical Planning and Inference 141, 536–548.

Hubert, M. & Vanderveeken, S. (2008), ‘Outlier detection for skewed data’, Journal of Chemometrics 22, 235–246.

Hubert, M. & Vandervieren, E. (2008), ‘An adjusted boxplot for skewed distributions’, Computational Statistics and Data Analysis 52, 5186–5201.

Hui, W., Gel, Y. & Gastwirth, J. (2008), ‘lawstat: An r package for law, public policy and biostatistics’, Journal Statistical Software 28, 1–26.

Johnson, N., Kotz, S. & Balakrishnan, N. (1994), Continuous Univariate Distributions, Wiley, NY.

Kotz, S., Leiva, V. & Sanhueza, A. (2010), ‘Two new mixture models related to the inverse Gaussian distribution’, Methodology and Computing in Applied Probability 12, 199–212.

Lawless, J. (2003), Statistical Models and Methods for Lifetime Data, Wiley, NY. Leiva, V., Athayde, E., Azevedo, C. & Marchant, C. (2011), ‘Modeling wind energy flux by a Birnbaum-Saunders distribution with unknown shift parameter’, Journal of Applied Statistics 38, 2819–2838.

Leiva, V., Marchant, C., Saulo, H., Aslam, M. & Rojas, F. (2014), ‘Capability indices for Birnbaum-Saunders processes applied to electronic and food industries’, Journal of Applied Statistics 41, 1881–1902.

Leiva, V., Ponce, M., Marchant, C. & Bustos, O. (2012), ‘Fatigue statistical distributions useful for modeling diameter and mortality of trees’, Revista Colombiana de Estadística 35, 349–367.

Leiva, V., Rojas, E., Galea, M. & Sanhueza, A. (2014), ‘Diagnostics in Birnbaum- Saunders accelerated life models with an application to fatigue data’, Applied Stochastic Models in Business and Industry 30, 115–131.

Leiva, V., Sanhueza, A., Kotz, S. & Araneda, N. (2010), ‘A unified mixture model based on the inverse Gaussian distribution’, Pakistan Journal of Statistic 26, 445–460.

Leiva, V., Santos-Neto, M., Cysneiros, F. & Barros, M. (2014), ‘Birnbaum- Saunders statistical modelling: A new approach’, Statistical Modelling 14, 21– 48.

Leiva, V., Saulo, H., Leao, J. & Marchant, C. (2014), ‘A family of autoregressive conditional duration models applied to financial data’, Computational Statistics and Data Analysis 79, 175–191.

Leiva, V., Soto, G., Cabrera, E. & Cabrera, G. (2011), ‘New control charts based on the Birnbaum-Saunders distribution and their implementation’, Revista Colombiana de Estadística 34, 147–176.

Lin, C.-T., Huang, Y.-L. & Balakrishnan, N. (2008), ‘A new method for goodness- of- fit testing based on type-II right censored samples’, IEEE Transactions on Reliability 57, 633–642.

Malmquist, S. (1950), ‘On a property of order statistics from a rectangular distribution’, Scandinavian Actuarial Journal 33, 214–222.

Marden, J. (2004), ‘Positions and QQ-plots’, Statistical Sciences 19, 606–614.

Marshall, A. & Olkin, I. (2007), Life Distributions, Springer, NY

Michael, J. (1983), ‘The stabilized probability plot’, Biometrika 70, 11–17.

Michael, J. & Schucany, W. (1979), ‘A new approach to testing goodness of fit for censored samples’, Technometrics 21, 435–441.

O’Reilly, F. & Stephens, M. (1988), ‘Transforming censored samples for testing fit’, Technometrics 30, 79–86.

Pakyari, R. & Balakrishnan, N. (2012), ‘A general purpose approximate goodness- of- fit test for progressively type-II censored data’, IEEE Transactions on Reliability 61, 238–244.

Paula, G., Leiva, V., Barros, M. & Liu, S. (2012), ‘Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance’, Applied Stochastic Models in Business and Industry 28, 16–34.

Rad, A., Yousefzadeh, F. & Balakrishnan, N. (2011), ‘Goodness-of-fit test based on Kullback-Leibler information for progressively type-II censored data’, IEEE Transactions on Reliability 60, 570–579.

Rojas, F., Leiva, V., Wanke, P. & Marchant, C. (2015), ‘Optimization of contribution margins in food services by modeling independent component demand’, Revista Colombiana de Estadística 38, 1–30.

Sanhueza, A., Leiva, V. & Lopez-Kleine, L. (2011), ‘On the student-t mixture inverse Gaussian model with an application to protein production’, Revista Colombiana de Estadística 34, 177–195.

Stehlík, M., Strelec, L. & Thulin, M. (2014), ‘On robust testing for normality in chemometrics’, Chemometrics and Intelligent Laboratory Systems 130, 98– 108.

Vilca, F., Sanhueza, A., Leiva, V. & Christakos, G. (2010), ‘An extended Birnbaum-Saunders model and its application in the study of environmental quality in Santiago, Chile’, Stochastic Environmental Research and Risk Assessment 24, 771–782.

How to Cite

APA

Castro-Kuriss, C., Leiva, V. and Athayde, E. (2014). Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions. Revista Colombiana de Estadística, 37(2Spe), 341–365. https://doi.org/10.15446/rce.v37n2spe.47941

ACM

[1]
Castro-Kuriss, C., Leiva, V. and Athayde, E. 2014. Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions. Revista Colombiana de Estadística. 37, 2Spe (Jul. 2014), 341–365. DOI:https://doi.org/10.15446/rce.v37n2spe.47941.

ACS

(1)
Castro-Kuriss, C.; Leiva, V.; Athayde, E. Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions. Rev. colomb. estad. 2014, 37, 341-365.

ABNT

CASTRO-KURISS, C.; LEIVA, V.; ATHAYDE, E. Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions. Revista Colombiana de Estadística, [S. l.], v. 37, n. 2Spe, p. 341–365, 2014. DOI: 10.15446/rce.v37n2spe.47941. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/47941. Acesso em: 16 jan. 2025.

Chicago

Castro-Kuriss, Claudia, Víctor Leiva, and Emilia Athayde. 2014. “Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions”. Revista Colombiana De Estadística 37 (2Spe):341-65. https://doi.org/10.15446/rce.v37n2spe.47941.

Harvard

Castro-Kuriss, C., Leiva, V. and Athayde, E. (2014) “Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions”, Revista Colombiana de Estadística, 37(2Spe), pp. 341–365. doi: 10.15446/rce.v37n2spe.47941.

IEEE

[1]
C. Castro-Kuriss, V. Leiva, and E. Athayde, “Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions”, Rev. colomb. estad., vol. 37, no. 2Spe, pp. 341–365, Jul. 2014.

MLA

Castro-Kuriss, C., V. Leiva, and E. Athayde. “Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions”. Revista Colombiana de Estadística, vol. 37, no. 2Spe, July 2014, pp. 341-65, doi:10.15446/rce.v37n2spe.47941.

Turabian

Castro-Kuriss, Claudia, Víctor Leiva, and Emilia Athayde. “Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions”. Revista Colombiana de Estadística 37, no. 2Spe (July 1, 2014): 341–365. Accessed January 16, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/47941.

Vancouver

1.
Castro-Kuriss C, Leiva V, Athayde E. Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions. Rev. colomb. estad. [Internet]. 2014 Jul. 1 [cited 2025 Jan. 16];37(2Spe):341-65. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/47941

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CrossRef citations5

1. Víctor Leiva, Fabrizio Ruggeri, Helton Saulo, Juan F. Vivanco. (2017). A methodology based on the Birnbaum–Saunders distribution for reliability analysis applied to nano-materials. Reliability Engineering & System Safety, 157, p.192. https://doi.org/10.1016/j.ress.2016.08.024.

2. Helton Saulo, N. Balakrishnan, Xiaojun Zhu, Jhon F. B. Gonzales, Jeremias Leão. (2017). Estimation in generalized bivariate Birnbaum–Saunders models. Metrika, 80(4), p.427. https://doi.org/10.1007/s00184-017-0612-5.

3. Camilo Lillo, Víctor Leiva, Orietta Nicolis, Robert G. Aykroyd. (2018). L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation, goodness of fit and application to earthquake data. Journal of Applied Statistics, 45(2), p.187. https://doi.org/10.1080/02664763.2016.1269729.

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