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Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal
Estimación y pruebas de hipótesis en ANOVA a una vía cuando los errores se distribuyen como normal sesgados
DOI:
https://doi.org/10.15446/rce.v38n1.48802Palabras clave:
ANOVA, Modified Likelihood, Iteratively Reweighting Algorithm, Skew-Normal, Monte Carlo Simulation, Robustness (en)ANOVA, Estimación, Normal sesgada, Pruebas de hipótesis, Robustez. (es)
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We consider one-way analysis of variance (ANOVA) model when the error terms have skew- normal distribution. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies (see, Tiku 1967). In the ML method, iteratively reweighting algorithm (IRA) is used to solve the likelihood equations. The MML approach is a non-iterative method used to obtain the explicit estimators of model parameters. We also propose new test statistics based on these estimators for testing the equality of treatment effects. Simulation results show that the proposed estimators and the tests based on them are more efficient and robust than the corresponding normal theory solutions. Also, real data is analysed to show the performance of the proposed estimators and the tests.
Se considera el modelo de análisis de varianza a una vía (ANOVA) cuando los términos de error siguen una distribución normal sesgada. Se obtienen estimadores de los parámetros desconocidos mediante el uso de la metodología de máxima verosimilitud (ML). Se proponen nuevos estadísticos de prueba basados en estos estimadores. Los resultados de la simulación muestran que los estimadores propuestos y los tests basados en ellos son más eficientes y robustos que los correspondientes a las soluciones de la teoría normal. Un conjunto de datos real es analizado con el fin de mostrar el desempeño de los estimadores propuestos y sus tests relacionados.
https://doi.org/10.15446/rce.v38n1.488020
1Bartin University, Science Faculty, Department of Statistics, Bartin, Turkey. Professor. Email: ncelik@bartin.edu.tr
2Ankara University, Science Faculty, Department of Statistics, Ankara, Turkey. Professor. Email: senoglu@science.ankara.edu.tr
3Ankara University, Science Faculty, Department of Statistics, Ankara, Turkey. Professor. Email: oarslan@ankara.edu.tr
We consider one-way analysis of variance (ANOVA) model when the error terms have skew- normal distribution. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies (see, Tiku 1967). In the ML method, iteratively reweighting algorithm (IRA) is used to solve the likelihood equations. The MML approach is a non-iterative method used to obtain the explicit estimators of model parameters. We also propose new test statistics based on these estimators for testing the equality of treatment effects. Simulation results show that the proposed estimators and the tests based on them are more efficient and robust than the corresponding normal theory solutions. Also, real data is analysed to show the performance of the proposed estimators and the tests.
Key words: ANOVA, Modified Likelihood, Iteratively Reweighting Algorithm, Skew-Normal, Monte Carlo Simulation, Robustness.
Se considera el modelo de análisis de varianza a una vía (ANOVA) cuando los términos de error siguen una distribución normal sesgada. Se obtienen estimadores de los parámetros desconocidos mediante el uso de la metodología de máxima verosimilitud (ML). Se proponen nuevos estadísticos de prueba basados en estos estimadores. Los resultados de la simulación muestran que los estimadores propuestos y los tests basados en ellos son más eficientes y robustos que los correspondientes a las soluciones de la teoría normal. Un conjunto de datos real es analizado con el fin de mostrar el desempeño de los estimadores propuestos y sus tests relacionados.
Palabras clave: ANOVA, estimación, normal sesgada, pruebas de hipótesis, robustez.
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References
1. Acitas, S., Kasap, S., Senoglu, B. & Arslan, O. (2013), 'Robust estimation with the skew t_2 distribution', Pakistan Journal of Statistics 29(4), 409-430.
2. Arrellano-Valle, R., Bolfarine, H. & Lachos, V. (2005), 'Skew-normal linear mixed models', Journal of Data Science 3, 415-438.
3. Arslan, O. & Genc, .. (2009), 'The skew generalized t (sgt) distribution as the scale mixture of a skew exponential distribution and its application in robust estimation', Statistics 43, 481-498.
4. Azzalini, A. (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12, 171-178.
5. Azzalini, A. (1986), 'Further results on a class of distributions which includes the normal ones', Statistica 46, 199-208.
6. Azzalini, A. (2005), 'The skew-normal distribution and related multivariate families (with discussion)', Scandinavian Journal of Statistics 32, 159-188.
7. Bowman, K. & Shenton, L. (2001), 'Weibull distributions when the shape parameter is defined', Computational Statistics and Data Analysis 36, 299-310.
8. Box, G. (1953), 'Non-normality and test of variances', Biometrika 40, 336-346.
9. Box, G. & Tiao, G. (1964), 'A Bayesian approach to the importance of assumptions applied to the comparison of variances', Biometrika 51, 153-167.
10. Donaldson, T. (1968), 'Robustness of the f-test to errors of both kinds and the correlation between the numerator and denominator of the F ratio', Journal of American Statistical Association 63(322), 660- 667.
11. Garay, A., Lachos, V. & Abanto-Valle, C. (2011), 'Nonlinear regression models based on scale mixtures of skew-normal distributions', Journal of Korean Statistical Society 40, 115-124.
12. Garay, A., Lachos, V., Labra, F. & Ortega, E. (2013), 'Statistical diagnostics for nonlinear regression models based on scale mixtures of skew normal distributions', Journal of Statistical Computation and Simulation 84, 1761-1778.
13. Geary, R. (1947), 'Testing for normality', Biometrika 34, 209-242.
14. Gupta, A. & Huang, W. (2002), 'Quadratic forms in skew normal variates', Journal of Mathematical Analysis and Applications 273, 558-564.
15. Huber, P. (1981), Robust Statistics, Jonh Wiley, New York.
16. Islam, M. & Tiku, M. (2004), 'Multiple linear regression model under nonnormality', Communication Statistics -Theory Methods 33, 2443-2467.
17. Kantar, Y. & Senoglu, B. (2008), 'A comparative study for the location and scale parameters of the weibull distribution with given shape parameter', Computers and Geosciences 34, 1900-1909.
18. Lachos, V., Bandyopadhyay, D. & Garay, A. (2011), 'Heteroscedastic non linear regression models based on scale mixtures of skew-normal distributions', Statistics and Probability Letters 81, 1208-1217.
19. Lachos, V., Bolfarine, H., Arellano-Valle, R. B. & Montenegro, L. (2007), 'Likelihood based inference for multivariate skew normal regression models', Communications in Statistics 36(9), 1769-1786.
20. Lachos, V., Ghosh, P. & Arellano-Valle, R. (2010), 'Likelihood based inference for skew-normal independent linear mixed models', Statistica Sinica 20, 303-322.
21. Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), 'The family of Log-Skew-Normal Alpha-power distributions using precipitation data', Revista Colombiana de Estadística 36(1), 43-57.
22. Montgomery, D. (2005), Design and Analysis of Experiments, John Wiley & Sons Inc., United States of America.
23. Pearson, E. (1932), 'The analysis of variance in cases of nonnormal variation', Biometrika 23, 114-133.
24. Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), 'An empirical comparison of EM initialization methods and model choice criteria for mixtures of Skew-Normal distributions', Revista Colombiana de Estadística 35(3), 457-478.
25. Senoglu, B. & Tiku, M. (2001), 'Analysis of variance in experimental design with nonnormal error distributions', Communication Statistical Theory Methods 30, 1335-1352.
26. Senoglu, B. & Tiku, M. (2002), 'Linear contrasts in experimental design with non-identical error distributions', Biometrical Journal 44(3), 359-374.
27. Spjotvoll, E. & Aastveit, H. (1980), 'Comparison of robust estimators on some data from field experiments', Scandinavian Journal of Statistics 7, 1-13.
28. Srivastava, A. (1959), 'Effect of nonnormality on the power of the analysis of variance test', Biometrika 46, 114-122.
29. Tan, W. & Tiku, M. (1999), Sampling Distributions in Terms of Laguerre Polynomials with Applications, New Age International (formerly, Wiley Eastern), New Delhi.
30. Tiku, M. (1967), 'Estimating the mean and standard deviation from censored normal samples', Biometrika 54, 155-165.
31. Tiku, M. & Akkaya, A. (2004), Robust Estimation and Hypothesis Testing, New Age International, New Delhi.
32. Tiku, M., Tan, W. & Balakrishnan, N. (1986), Robust Inference, Marcel Dekker, New York.
33. Tukey, J. (1960), A survey of sampling from contaminated distributions, 'Contributions to Probability and Statistics', Stanford University Press, Stanford.
34. Xie, F., Wei, B. & Lin, J. (2009), 'Homogeneity diagnostics for skew-normal nonlinear regression models', Statistics and Probability Letters 79, 821-827.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv38n1a04,
AUTHOR = {Celik, Nuri and Senoglu, Birdal and Arslan, Olcay},
TITLE = {{Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2015},
volume = {38},
number = {1},
pages = {75-91}
}
Referencias
Acitas, S., Kasap, S., Senoglu, B. & Arslan, O. (2013), ‘Robust estimation with the skew t2 distribution’, Pakistan Journal of Statistics 29(4), 409–430.
Arrellano-Valle, R., Bolfarine, H. & Lachos, V. (2005), ‘Skew-normal linear mixed models’, Journal of Data Science 3, 415–438.
Arslan, O. & Genc, A. (2009), ‘The skew generalized t (sgt) distribution as the scale mixture of a skew exponential distribution and its application in robust estimation’, Statistics 43, 481–498.
Azzalini, A. (1985), ‘A class of distributions which includes the normal ones’, Scandinavian Journal of Statistics 12, 171–178.
Azzalini, A. (1986), ‘Further results on a class of distributions which includes the normal ones’, Statistica 46, 199–208.
Azzalini, A. (2005), ‘The skew-normal distribution and related multivariate families (with discussion)’, Scandinavian Journal of Statistics 32, 159–188.
Bowman, K. & Shenton, L. (2001), ‘Weibull distributions when the shape parameter is defined’, Computational Statistics and Data Analysis 36, 299–310.
Box, G. (1953), ‘Non-normality and test of variances’, Biometrika 40, 336–346.
Box, G. & Tiao, G. (1964), ‘A Bayesian approach to the importance of assumptions applied to the comparison of variances’, Biometrika 51, 153–167.
Donaldson, T. (1968), ‘Robustness of the f-test to errors of both kinds and the correlation between the numerator and denominator of the F ratio’, Journal of American Statistical Association 63(322), 660– 667.
Garay, A., Lachos, V. & Abanto-Valle, C. (2011), ‘Nonlinear regression models based on scale mixtures of skew-normal distributions’, Journal of Korean Statistical Society 40, 115–124.
Garay, A., Lachos, V., Labra, F. & Ortega, E. (2013), ‘Statistical diagnostics for nonlinear regression models based on scale mixtures of skew normal distributions’, Journal of Statistical Computation and Simulation 84, 1761–1778.
Geary, R. (1947), ‘Testing for normality’, Biometrika 34, 209–242.
Gupta, A. & Huang, W. (2002), ‘Quadratic forms in skew normal variates’, Journal of Mathematical Analysis and Applications 273, 558–564.
Huber, P. (1981), Robust Statistics, Jonh Wiley, New York.
Islam, M. & Tiku, M. (2004), ‘Multiple linear regression model under nonnormality’, Communication Statistics -Theory Methods 33, 2443–2467.
Kantar, Y. & Senoglu, B. (2008), ‘A comparative study for the location and scale parameters of the weibull distribution with given shape parameter’, Computers and Geosciences 34, 1900–1909.
Lachos, V., Bandyopadhyay, D. & Garay, A. (2011), ‘Heteroscedastic non linear regression models based on scale mixtures of skew-normal distributions’, Statistics and Probability Letters 81, 1208–1217.
Lachos, V., Bolfarine, H., Arellano-Valle, R. B. & Montenegro, L. (2007), ‘Likelihood based inference for multivariate skew normal regression models’, Communications in Statistics 36(9), 1769–1786.
Lachos, V., Ghosh, P. & Arellano-Valle, R. (2010), ‘Likelihood based inference for skew-normal independent linear mixed models’, Statistica Sinica 20, 303–322.
Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), ‘The family of Log-Skew-Normal Alpha-power distributions using precipitation data’, Revista Colombiana de Estadística 36(1), 43–57.
Montgomery, D. (2005), Design and Analysis of Experiments, John Wiley & Sons Inc., United States of America.
Pearson, E. (1932), ‘The analysis of variance in cases of nonnormal variation’, Biometrika 23, 114–133.
Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), ‘An empirical comparison of EM initialization methods and model choice criteria for mixtures of Skew-Normal distributions’, Revista Colombiana de Estadística 35(3), 457–478.
Senoglu, B. & Tiku, M. (2001), ‘Analysis of variance in experimental design with nonnormal error distributions’, Communication Statistical Theory Methods 30, 1335–1352.
Senoglu, B. & Tiku, M. (2002), ‘Linear contrasts in experimental design with non-identical error distributions’, Biometrical Journal 44(3), 359–374.
Spjotvoll, E. & Aastveit, H. (1980), ‘Comparison of robust estimators on some data from field experiments’, Scandinavian Journal of Statistics 7, 1–13.
Srivastava, A. (1959), ‘Effect of nonnormality on the power of the analysis of variance test’, Biometrika 46, 114–122.
Tan, W. & Tiku, M. (1999), Sampling Distributions in Terms of Laguerre Polynomials with Applications, New Age International (formerly, Wiley Eastern), New Delhi.
Tiku, M. (1967), ‘Estimating the mean and standard deviation from censored normal samples’, Biometrika 54, 155–165.
Tiku, M. & Akkaya, A. (2004), Robust Estimation and Hypothesis Testing, New Age International, New Delhi.
Tiku, M., Tan, W. & Balakrishnan, N. (1986), Robust Inference, Marcel Dekker, New York.
Tukey, J. (1960), A survey of sampling from contaminated distributions, in I. Olkin, ed., ‘Contributions to Probability and Statistics’, Stanford University Press, Stanford.
Xie, F., Wei, B. & Lin, J. (2009), ‘Homogeneity diagnostics for skew-normal nonlinear regression models’, Statistics and Probability Letters 79, 821–827.
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