Theoretical Grounding for Estimation in Conditional Independence Multivariate Finite Mixture Models

For the nonparametric estimation of multivariate finite mixture models with the conditional independence assumption, we propose a new formulation of the objective function in terms of penalised smoothed Kullback-Leibler distance. The nonlinearly smoothed majorisation-minimisation (NSMM) algorithm is derived from this perspective. An elegant representation of the NSMM algorithm is obtained using a novel projection-multiplication operator, a more precise monotonicity property of the algorithm is discovered, and the existence of a solution to the main optimisation problem is proved for the first time.