Teste da razão de verossimilhanças para a variância generalizada normal multivariada

Abstract

An interesting measure of variability in the multivariate population is the determinant of the covariance matrix Sp p, jSj, of a population, known as generalized variance. This is a measure that summarizes the dispersion of a multivariate population in a single value, considering the dependence between the variables involved. Because of this, it has applications in several areas that aim to evaluate the dispersion existing in some multivariate population of interest. In industries, for example, there are several situations in which the simultaneous monitoring or control of two or more characteristics related to the quality process is necessary. So to say, assessing whether the process is, statistically, under control, consists of jointly analyzing all the variables related to the quality process, considering the dependence between them, as well as their variability. In addition to the industry, the study of multivariate variability through generalized variance is present in signal processing, cluster analysis, optimal designs and many other fields. In this way, the construction of hypothesis tests that evaluate the dispersion in multivariate populations is necessary given its wide field of action. This work is divided into two parts. The first is a bibliographic review that encompasses all the theory necessary to understand the construction of a hypothesis test for the generalized variance of the multivariate normal distribution, which consists of the second part and was presented in article format. The article deals with the proposition of two new hypothesis tests, one built via the likelihood ratio - the LRT test - and the other, it is also built via the likelihood ratio, however it is added Bartlett’s Correction for likelihood ratio tests, called BCLRT. Such hypothesis tests are designed to test the generalized variance of a normal multivariate population. For the evaluation of the type I error rate and the power of the tests, Monte Carlo simulations are performed for different scenarios in which are varied the sample size n, the number of variables p and the level of significance a for the proposed tests and for other tests already in the literature. The performance of the tests proposed in the evaluations of the type I error rate and power led us to recommend the use of the BCLRT test only in scenarios where we have p = 2, especially when n > 30. While for the LRT test, we recommend its use in situations where p = 2 and p = 3 for n > 30 and for p = 5 when n > 50.