A refreshing take on the inverted Dirichlet via a mode parameterization with some statistical illustrations

Abstract

The inverted Dirichlet (IDir) distribution is a popular choice for modeling multivariate data with positive support; however, its conventional parameterization can be challenging to interpret. In this paper, we propose a refreshing take on the IDir distribution through a convenient mode-based parameterization, resulting in the mode-reparameterized IDir (mIDir). This new parameterization aims to enhance the use of the IDir in various contexts. We provide relevant statistical illustrations in robust and nonparametric statistics, model-based clustering, and semiparametric density estimation, all benefiting from this novel perspective on the IDir for computation and implementation. First, we define finite mIDir mixtures for clustering and semiparametric density estimation. Secondly, we introduce a smoother based on mIDir kernels, which, by design, avoids allocating probability mass to unrealistic negative values, thereby addressing the boundary bias issue. Thirdly, we introduce a heavy-tailed generalization of the mIDir distribution, referred to as the contaminated mIDir (cmIDir), which effectively handles and detects mild outliers, making it suitable for robust statistics. Maximum likelihood estimates of the parameters for the parametric models are obtained using a developed EM algorithm as well as direct numerical optimization. A parameter recovery analysis demonstrates the successful application of the estimation method, while a sensitivity analysis examines the impact of mild outliers on both the mIDir and cmIDir models. The flexibility and advantages of the proposed mIDir-based models are showcased through several real data analyses and illustrations.