Estimation and Testing in One-Way ANOVA when the Errors are Skew-Normal

• NURI CELIK ^[1] ; BIRDAL SENOGLU ^[2] ; OLCAY ARSLAN ^[2]
  1. [1] Bartin University

     Bartin University

     Turquía

     
  2. [2] Ankara University

     Ankara University

     Turquía

     
• Localización: Revista Colombiana de Estadística, ISSN-e 2389-8976, ISSN 0120-1751, Vol. 38, Nº. 1, 2015, págs. 75-91
• Idioma: inglés
• DOI: 10.15446/rce.v38n1.48802
• Títulos paralelos:
  • Estimación y pruebas de hipótesis en ANOVA a una vía cuando los errores se distribuyen como normal sesgados
• Enlaces
  • Texto completo
• Resumen
  • español

    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.

  • English

    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.

• Referencias bibliográficas
  • Acitas, S.,Kasap, S.,Senoglu, B.,Arslan, O.. (2013). 'Robust estimation with the skew t_2 distribution'. Pakistan Journal of Statistics....
  • Arrellano-Valle, R.,Bolfarine, H.,Lachos, V.. (2005). 'Skew-normal linear mixed models'. Journal of Data Science. 3. 415-438
  • Arslan, O.,Genc, ... (2009). 'The skew generalized t (sgt) distribution as the scale mixture of a skew exponential distribution and its...
  • 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...
  • Bowman, K.,Shenton, L.. (2001). 'Weibull distributions when the shape parameter is defined'. Computational Statistics and Data Analysis....
  • 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....
  • Donaldson, T.. (1968). 'Robustness of the f-test to errors of both kinds and the correlation between the numerator and denominator of...
  • Garay, A.,Lachos, V.,Abanto-Valle, C.. (2011). 'Nonlinear regression models based on scale mixtures of skew-normal distributions'....
  • Garay, A.,Lachos, V.,Labra, F.,Ortega, E.. (2013). 'Statistical diagnostics for nonlinear regression models based on scale mixtures of...
  • 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.
  • 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...
  • Lachos, V.,Bandyopadhyay, D.,Garay, A.. (2011). 'Heteroscedastic non linear regression models based on scale mixtures of skew-normal distributions'....
  • Lachos, V.,Bolfarine, H.,Arellano-Valle, R. B.,Montenegro, L.. (2007). 'Likelihood based inference for multivariate skew normal regression...
  • Lachos, V.,Ghosh, P.,Arellano-Valle, R.. (2010). 'Likelihood based inference for skew-normal independent linear mixed models'. Statistica...
  • Martínez-Flórez, G.,Vergara-Cardozo, S.,González, L. M.. (2013). 'The family of Log-Skew-Normal Alpha-power distributions using precipitation...
  • Montgomery, D.. (2005). Design and Analysis of Experiments. John Wiley & Sons Inc..
  • 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...
  • Senoglu, B.,Tiku, M.. (2001). 'Analysis of variance in experimental design with nonnormal error distributions'. Communication Statistical...
  • Senoglu, B.,Tiku, M.. (2002). 'Linear contrasts in experimental design with non-identical error distributions'. Biometrical Journal....
  • Spjotvoll, E.,Aastveit, H.. (1980). 'Comparison of robust estimators on some data from field experiments'. Scandinavian Journal of...
  • 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...
  • 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.
  • Tiku, M.,Tan, W.,Balakrishnan, N.. (1986). Robust Inference. Marcel Dekker.
  • Tukey, J.. (1960). 'Contributions to Probability and Statistics'. Stanford University Press.
  • Xie, F.,Wei, B.,Lin, J.. (2009). 'Homogeneity diagnostics for skew-normal nonlinear regression models'. Statistics and Probability...