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The likelihood ratio test statistic provides the basis for testing inference on the regression parameters in the class of multivariate Dirichlet regression models, which is very useful in modeling multivariate positive observations summing to one. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of the likelihood ratio statistic. Our simulation results suggest that the likelihood ratio test tends to be extremely liberal when the sample size is small. We derive a general Bartlett correction factor in matrix notation for the likelihood ratio test statistic, which reduces the size distortion of the test, and also consider a bootstrap-based Bartlett correction. We also employ the Skovgaard's adjustment to the likelihood ratio statistic. We numerically compare the proposed tests with the usual likelihood ratio test. Our simulation results suggest that the proposed corrected tests can be interesting alternatives to usual likelihood ratio test since they lead to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes. [ABSTRACT FROM AUTHOR]

In this paper we consider the multivariate Dirichlet regression model proposed by Melo et al. (2009), which is tailored to situations where the multivariate response consists of multivariate positive observations summing to one and the regression structure involves regressors and unknown parameters. We discuss maximum likelihood estimation for the model parameters and derive modified maximum likelihood estimators that are bias-free to second order. Monte Carlo simulation experiments are conducted in order to investigate the performance of the corrected estimators. The numerical results reveal that the bias correction scheme yields nearly unbiased estimates without increasing the mean squared errors. An empirical application is considered for illustrative purposes. [ABSTRACT FROM AUTHOR]