¹Hacettepe University, Faculty of Science, Department of Statistics, Ankara, Turkey. PhD. Email: deryacal@hacettepe.edu.tr
²Hacettepe University, Faculty of Science, Department of Statistics, Ankara, Turkey. Professor. Email: toker@hacettepe.edu.tr

In this study, robust Brown-Forsythe and robust Modified Brown-Forsythe ANOVA tests are proposed to take into consideration heteroscedastic and non-normality data sets with outliers. The non-normal data is assumed to be a two parameters Weibull distribution. Robust proposed tests are obtained by using robust mean and variance estimators based on median/MAD and median/Q_n methods instead of maximum likelihood. The behaviors of the robust proposed and classical ANOVA tests are examined by simulation study. The results shows that the proposed robust tests have good performance especially in the presence of heteroscedasticity and contamination.

Key words: Brown-Forsythe, Modified Brown-Forsythe, ANOVA, Weibull Distribution.

En este estudio se proponen tests Brown-Forsythe y robustos Brown-Forsythe ANOVA para tener en cuenta la no-normalidad en datos debida a la presencia de datos atípicos. Se asume que los datos no-normales tienen una distribución Weibull de dos parámetros. Estos tests se construyen en base a estimadores robustos de media y varianza obtenidos con métodos basados en la mediana en vez de métodos de máxima verosimilitud. Se examina en comportamiento de estos tests con datos simulados. Los resultados muestran que éstos tienen un buen desempeño, especialmente en presencia de atípicos y datos contaminados.

Palabras clave: ANOVA, Brown-Forsythe, Brown-Forsythe modificado, distribución Weibull.

Texto completo disponible en PDF

References

1. Anderson, T. W. & Darling, D. A. (1952), 'Asymptotic theory of certain 'goodness-of-fit' criteria based on stochastic processes', Annals of Mathematical Statistics 23, 193-212.         [ Links ]

2. Boudt, K., Caliskan, D. & Croux, C. (2011), 'Robust explicit estimators of weibull parameters', Metrika 73, 187-209.         [ Links ]

3. Brown, M. & Forsythe, A. (1974), 'Small sample behavior of some statistics which test the equality of several means', Technometrics 16, 129–132.         [ Links ]

4. Gamage, J. & Weerahandi, S. (1998), 'Size performance of some tests in one-way anova', Communications in Statistics - Simulation and Computation 27, 625-640.         [ Links ]

5. Gupta, R. & Kundu, D. (2001), 'Exponentiated exponential family: an alternative to gamma and weibull distributions', Biometrical Journal 43(1), 117-130.         [ Links ]

6. James, G. (1951), 'The comparison of several groups of observations when the ratios of the population variances are unkown', Biometrika 38, 324-329.         [ Links ]

7. Kulinskaya, E. & Dollinger, M. (2007), 'Robust weighted One-Way ANOVA: Improved approximation and efficiency', Journal of Statistical Planning and Inference 137, 462-472.         [ Links ]

8. Mehrotra, D. (1997), 'Improving the brown-forsythe solution to the generalized behrens fisher problem', Communications in Statistics Simulation and Computation 26, 1139-1145.         [ Links ]

9. Olive, D. (2006), 'Robust estimators for transformed location scale families', https://www.researchgate.net/profile/David_Olive2/publication/228390054_Robust_estimators_for_transformed_location_scale_families/links/02bfe51015be6ca4fa000000.pdf. Accessed: 2015-12-24.         [ Links ]

10. Rousseeuw, P. & Croux, C. (1993), 'Alternatives to the median absolute deviation', Journal of the American Statistical Association 88, 1273-1283.         [ Links ]

11. Senoglu, B. (2005), 'Robust 2^k factorial design with Weibull error distributions', Journal of Applied Statistics 32, 1051-1066.         [ Links ]

12. Senoglu, B. (2007), 'Robust estimation and hypothesis testing in 2^k factorial design', Communications de la Faculte des Sciences de l'Universite. Series A1 56, 39-50.         [ Links ]

13. Trujillo-Ortiz, A., Hernandez-Walls, R., Barba-Rojo, K., Castro-Perez, A. & Lavaniegos-Espejo, B. (2007), 'Andarwtest:anderson-darling test for assess- ing weibull distribution of a sample data'. http://www.mathworks.com/matlabcentral/fileexchange/15745-anderwtest         [ Links ]

14. Weerahandi, S. (1995), 'Anova under unequal error variances', Biometrics 51, 589-599.         [ Links ]

15. Welch, B. (1951), 'On the comparison of several mean values', Biometrika 38, 330-336.         [ Links ]

16. Wilcox, R. (1995), 'The practical importance of heteroscedastic methods, using trimmed means versus means, and designing simulation studies', British Journal of Mathematical and Statistical Psychology 48, 99-114.         [ Links ]

17. Wilcox, R. (1997), Introduction to Robust Estimation and Hypothesis Testing, Academic Press, New York.         [ Links ]

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