Proposição de teste de médias robustos e não robustos sob distribuições normais contaminadas e não normais

Abstract

In a statistical analysis, the inferential process for the mean of a population includes the socalled hypothesis test. Considering the univariate case and dealing with a normally distributedpopulation, when the population variance is not known, the t-student distribution is used tomake the decision to reject or not the null hypothesis that the population average is equal to agiven constant . For the multivariate case, under the same conditions, the T2 test proposed byHarold Hotelling is used to test the hypotheses about a population mean vector. Another wayto perform these hypothesis tests is through the likelihood ratio test, called the LRT (likelihoodratio test). In the presence of outliers, both the T2 and LRT tests should not be used, as theirestimators, which are the vector of means and the sample covariance matrix, are influenced byoutliers. In the literature, there are many studies on the construction of estimators for the vectorof averages and for the covariance matrix that are robust to the presence of outliers, possessinga high breaking point and maintaining the affine-equivalence property. However, many of theseestimators are conditioned to the existence of the first two moments. A special case of theseestimators is the comedian robust estimator, which is not conditioned to the existence of the firsttwo moments and, in summary, considers correlated medians. Therefore, in this thesis, we seekto build robust tests alternative to the traditional Hotelling T2 and the likelihood ratio test, usingthe comedian estimators of the mean vector and covariance matrix. These tests were generatedthrough adaptations made both in the T2 test and in the LRT test, with some of these tests beingasymptotic and others being constructed using the comedic parametric bootstrap estimators forthe mean vector and covariance matrix. Tests adapted to the LRT test have the advantage thatthey can be used for high-dimensional data. The performance of these tests was evaluated andcompared to the traditional T2 test considering normal contaminated and non-normal distributions, using Monte Carlo simulations. The power of the test and the type I error rate wereconsidered as evaluative measures. As a result, it was found that the performance of the parametric boostrap version with trace of the TLRPBT likelihood ratio test (Trace Likelihood RatioBootstrap Parametric test) had better performance among all the considered tests. Another factis that the asymptotic version of the ATLRT trace likelihood ratio test (Asymptotic Trace Likelihood Ratio Test) should not be discarded among all the proposed tests, as the same resistedcontrolled type I error and power in extreme situations asymmetry.