Robust multivariate analysis of variability

A general testing procedure is proposed to multivariately test for equality of p variances among k groups. The procedure applies a multivariate analysis of variance on an appropriate measure of spread for the uncensored original observations. Three such measures of spread are compared in a simulation experiment which considered two and three variables with equal and unequal sample sizes for the null and alternative hypotheses for Gaussian, Student's t (8, 12, and 20 degrees of freedom) and gamma (α=2,4,6 and 10) distributions . The likelihood ratio test (Box, 1949) was included in the above simulations. The results suggest that if one chooses a measure of spread appropriate for the distribution of the original observations, the proposed MANOVA-based testing procedure is robust and reasonably powerful. Using this procedure for the normal distribution, similar power was observed to that of the likelihood ratio test when the variables were uncorrelated or had little positive correlation.     

CONCEPTS,THEORY, AND TECHNIQUES

The beta distribution is becoming more widely used in business applications and in economic model-building. A simple way to estimate the parameters of the univariate distribution and the general multivariate distribution would be useful but there appears to be no simple way of accomplishing this task. For example, the method of maximum likelihood does not yield tractable results. The method of moments presented in this paper is a practical way to obtain parameter estimates for the beta distribution. Explicit estimators are derived for the univariate and bivariate case and estimators are then inferred for the n-variate case. Empirical tests suggest that the method of moments may be used to establish useful estimates for these parameters. Also, estimates obtained by the method of moments may be used to develop bounds on the parameters so that computational techniques can be utilized with other estimation methods possessing more desirable properties.