Higher-order asymptotic refinements in the multivariate Dirichlet regression model

Abstract

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.

KEYWORDS:

• Bartlett correction
• Bootstrap
• Compositional data
• Likelihood ratio test

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F03
• 62F05
• 62J05

Acknowledgments

We are very grateful to the anonymous referees for useful comments and suggestions on an early version of this paper. Tatiane Melo and Tiago Vargas gratefully acknowledge the financial support of the Brazilian agency FAPEG (grant 201410267001779). Artur Lemonte gratefully acknowledges the financial support of the Brazilian agency CNPq (grant 301808/2016–3). Germán Moreno-Arenas gratefully acknowledges the Mobility Program of the Universidad Industrial de Santander (UIS), Bucaramanga, Colombia.