Proposição de testes de normalidade multivariada baseados em distâncias robustas

Abstract

Multivariate normality is one of the most important assumptions for the realization of many inferential methods. The non-verification of this assumption can influence the reliability of the results. There are many tests in the specialized literature to verify normality. In the case of multivariate normality, the tests, in general, are based on correlation coefficients, asymmetry and kurtosis coefficients and distances. Despite the large number of tests, there is no test in the literature that is uniformly more powerful in all evaluated situations. The tests, in general, presents some restrictions, both in relation to size and to dimension. The presence of outliers in the data can result in bad parameter estimation and even distortions in the distribution fitting, making the tests fail. Therefore, the aim of this work is to propose and evaluate four outlier robust tests: multivariate normality test based on Mahalanobis distance with robust measures of the scale and location parameters (TNMD2RKS), multivariate normality test based on robust beta distance (TNMDbRKS), parametric bootstrap multivariate normality test based on robust distances (TNMD2RBoot) and the parametric bootstrap multivariate normality test based on robust beta distances (TNMDbRBoot). For the four tests it was used the robust scale and location estimators calculated via function CovOgk from the R software. Type I error rates and power of the tests were evaluated by comparing then to the parametric bootstrap multivariate normality test based on the correlation between the order statistics and the expected values and the Royston’s Shapiro-Wilk test, via Monte Carlo simulation. The tests TNMD2RKS, TNMD2RBoot and TNMDbRBoot were successfully proposed, obtaining excellent type I error rate control, especially in samples with the presence of outliers, in which the other tests did not perform well. In terms of power, the four tests performed well in large samples, however, they did not outperform the tests used as references.