Multivariate data exhibit unique characteristics that may challenge traditional statistical methods, particularly in model-based clustering and classification. In this context, the Dirichlet and inverted Dirichlet models are often an initial parametric choice when considering either data on the multivariate simplex or multivariate data with positive support; their parameterization, which relies on shape parameters, lacks intuitive and more insightful application. In this context, we (re)develop these Dirichlet-based distributions to address these challenges, inspired by a Gaussian philosophy of a model being parametrized in terms of a location and scale parameter.

Focusing on the multivariate simplex, we first introduce a unimodal Dirichlet (UD) distribution parameterized in terms of its mode and a dispersion parameter; subsequently, we study finite mixtures of UD distributions for clustering and classification. We also propose the contaminated UD distribution to handle atypical observations, a heavy-tailed generalization that allows for flexible tail behaviour. Parameter estimation is achieved through maximum likelihood or an expectation-maximization algorithm, with analyses conducted on simulated data highlighting the impact of atypical observations on parameter estimation and classification.

Secondly, for multivariate-yet-positive contexts, we propose a mode-based parameterization of the inverted Dirichlet (IDir) distribution, resulting in the mode-reparametrized IDir (mIDir), which enhances interpretability and applicability in various contexts, including robust and nonparametric statistics. We define finite mixtures of mIDir for clustering and semiparametric density estimation and introduce a smoother based on mIDir kernels to address boundary bias. Furthermore, we propose the contaminated mIDir distribution, a heavy-tailed extension that robustly handles mild outliers. We demonstrate the flexibility of these models through parameter recovery analysis, sensitivity analysis for mild outliers, and real data applications. These contributions provide a robust framework for modeling multivariate data with complex behaviors such as atypical observations and outliers.