Proposta de um teste de esfericidade usando estimadores robustos do parâmetro de dispersão multivariado

Abstract

For the hypothesis of sphericity with homogeneous variances and equal to one, we propose the study of eight tests to verify the robustness of the high dimensionality of the data, the robustness of the asymmetry of the distribution and also the robustness of the presence of outliers in the distribution. The likelihood ratio test degenerates when p ≥ n because the covariance matrix of the sample is unique and, for this reason, we introduce the test study based on the test statistic W , proposed by Ledoit e Wolf (2002), which is robust when p ≥ n, and W ’s modifications, in which the covariance matrix is replaced by the robust comedian estimator ( WAsR), the Monte Carlo version ( WMC) and the Monte Carlo version change ( WMCR). The modified likelihood ratio test statistic is also studied, following the same criteria of the W statistic: LRTAsR, LRT MC and LRT MCR, in that order. The distributions used are normal, log-normal, t -Student with n = 5 degrees of freedom, t -Student with n = 30 degrees of freedom and the contaminated normal, to evaluate the robustness of the presence of outliers and also the asymmetry of the distribution. Four robust tests for the presence of outliers are found, and they are the LRT MC, the LRT MCR, the WMC and the WMCR. The WMC and the WMCR are robust statistics also in terms of the high dimensionality of the data ( p ≥ n). The WMC and the WMCR are more powerful tests than the test based on the original W statistic.