Testes de normalidade multivariada baseados em amostras betas independentes

Abstract

In inference, multivariate normality tests are very important, since many methods are based on assumptions that the data come from a multivariate normal distribution. Gnanadesikan and Kettenring (1972) proven that it is possible to obtain beta samples from normal samples using a transformation in the Mahalanobis quadratic distance. Checking the fit of the sample obtained by transformation to the beta distribution is an indication that the original sample is from a multivariate normal distribution. Embrechts, Frey and McNeil (2005) proposed a test based on Kolmogorov-Smirnov test using these concepts. However, this test is influenced by the sample dependence present in the quadratic distance. Liang, Pan and Yang (2004) presented a way to obtain univariate beta samples, each independent and identically distributed, through transformations in a p-variate normal sample. This work aimed to propose two tests for multivariate normality: a goodness-of-fit test based on Kolmogorov-Smirnov test and an intensive test based on parametric bootstrap. The R program (R CORE TEAM, 2015) was used to implement the algorithms of both proposed tests and Monte Carlo simulations were used in order to estimate type I error rates and the power of the tests. Comparisons were conducted between the proposed tests and the multivariate normality test that was presented by Embrechts, Frey and McNeil (2005) and the Shapiro-Wilk multivariate normality test proposed by Royston (1983). Although the proposed tests have obtained good control of the type I error rates, the use of these tests was not recommended due to the poor performance of power presented by them.