Proposta de um teste exato para avaliar a normalidade multivariada baseado em uma transformação t de Student

Abstract

The normal distribution is one of the most important continuous probability distribution. This distribution describe several phenomena and has great hole in inferential statistics. It is noteworthy that the normality directly influences the quality and reliability of scientific research since violations of assumption can lead to incorrect results and conclusions. The same is expected for multivariate inferences. A simple manner, however subjective, to verify the univariate or multivariate normality is through quantile-quantile plots (Q-Q plots). Furthermore, the Q-Q plots are efficient tools for the visualization of outliers. A disadvantage of the classical Q-Q plot is that the quantiles are only asymptotically identically distributed, but they are not independent. This fact compromises the efficiency of the Q-Q plot or any test based on the use of the observed distance quantiles. The objective of this study is to propose an accurate test and validate its performance by Monte Carlo simulation and also provide a Q-Q plot to detect further evidence of violation of multivariate normality in $ p $ dimensions. This Q-Q plot originates from a characterization of the multivariate normal distribution made by Yang et al. (1996) based on the spherical distribution properties (Fang et al., 1990). The R program version 3.1.0 was used to build this Q-Q plot normality test and to perform the validation of its performance by Monte Carlo simulations. The Monte Carlo simulation results showed that the proposed test successful controls the type I error rates being accurate, but shows lower power than any other multivariate normality test.