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

TL;DR

This paper introduces a deterministic algorithm for estimating nonparametric finite mixture models with copula-based dependence, removing the need for location-scale assumptions and demonstrating promising monotonicity properties.

Contribution

The authors develop a new deterministic algorithm that maximizes a smoothed semiparametric likelihood for copula-based mixture models without relying on location-scale constraints.

Findings

Algorithm shows monotonic or approximately monotonic behavior in simulations.

Effective in estimating dependence structures in multivariate mixtures.

Potential for broader application in nonparametric dependence modeling.

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 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 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.