Density estimation for toroidal data using semiparametric mixtures

Abstract

AbstractToroidal data is an extension of circular data on a torus and plays a critical part in various scientific fields. This article studies the density estimation of multivariate toroidal data based on semiparametric mixtures. One of the major challenges of semiparametric mixture modelling in a multi-dimensional space is that one can not directly maximize the likelihood over the unrestricted component density as it will result in a degenerate estimate with an unbounded likelihood. To overcome this problem, we propose to fix the maximum of the component density, which subsequently bounds the maximum of the mixture and its likelihood function, hence providing a satisfactory density estimate. The product of univariate circular distributions are utilized to form multivariate toroidal densities as candidates for mixture components. Numerical studies show that the mixture-based density estimator is superior in general to the kernel density estimator.