Bandwidth rules for Dirichlet kernel density estimation on the simplex

Abstract

Dirichlet kernels provide a natural tool for density estimation on the simplex, but data-driven bandwidth selection for them has been mostly limited to likelihood cross-validation (LCV). We develop bandwidth selectors for Dirichlet kernel estimators of densities and projected first derivatives for compositional data. Using a Dirichlet mixture as the reference model, we derive an exact representation of the mean integrated squared error (MISE) and define the exact MISE (EMI) selector. We also derive a closed-form least-squares cross-validation (LSCV) criterion and introduce the reference-smoothed least-squares (RSLS) selector, which replaces the unstable leave-one-out term of LSCV with the fitted Dirichlet mixture. For derivative estimation, we obtain projected derivatives of the Dirichlet density and their integrability conditions, extend the new bandwidth selectors, and carry out kernel mean shift clustering on the simplex. Simulations show that EMI is the most competitive selector for larger samples and Dirichlet-like targets, RSLS generally performs best for small samples and is more stable than LSCV for derivative estimation, and LCV remains competitive for non-Dirichlet targets in density estimation. Applications to wholesale customer expenditure and sectoral employment data illustrate density estimation and clustering, respectively.