A smoothed semiparametric likelihood for estimation of nonparametric finite mixture models with a copula-based dependence structure

Abstract

In this manuscript, we consider a finite multivariate nonparametric mixture model where the dependence between the marginal densities is modeled using the copula device. Pseudo expectation–maximization (EM) stochastic algorithms were recently proposed to estimate all of the components of this model under a location-scale constraint on the marginals. Here, we introduce a deterministic algorithm that seeks to maximize a smoothed semiparametric likelihood. No location-scale assumption is made about the marginals. The algorithm is monotonic in one special case, and, in another, leads to “approximate monotonicity”—whereby the difference between successive values of the objective function becomes non-negative up to an additive term that becomes negligible after a sufficiently large number of iterations. The behavior of this algorithm is illustrated on several simulated and real datasets. The results suggest that, under suitable conditions, the proposed algorithm may indeed be monotonic in general. A discussion of the results and some possible future research directions round out our presentation.