Semiparametric Two-Component Mixture Models When One Component Is Defined Through Linear Constraints

Abstract

We propose a structure of a semiparametric two-component mixture model when one component is parametric and the other is defined through linear constraints on its distribution function. Estimation of a two-component mixture model with an unknown component is very difficult when no particular assumption is made on the structure of the unknown component. A symmetry assumption was used in the literature to simplify the estimation. Such method has the advantage of producing consistent and asymptotically normal estimators, and identifiability of the semiparametric mixture model becomes tractable. Still, existing methods which estimate a semiparametric mixture model have their limits when the parametric component has unknown parameters or the proportion of the parametric part is either very high or very low. We propose in this paper a method to incorporate a prior linear information about the distribution of the unknown component in order to better estimate the model when existing estimation methods fail. The new method is based on ϕ−divergences and has an original form since the minimization is carried over both arguments of the divergence. The resulting estimators are proved to be consistent and asymptotically normal under standard assumptions. We show that using the Pearson's χ 2 divergence our algorithm has a linear complexity when the constraints are moment-type. Simulations on univariate and multivariate mixtures demonstrate the viability and the interest of our novel approach.

An Overview of Semiparametric Extensions of Finite Mixture Models

“…Al Mohamad and Boumahdaf (2018) considered a semiparametric two-component mixture model when one component is parametric and the other is defined based on linear constraints on its distribution function. A new estimation method is proposed, which incorporates a prior linear information about the distribution of the unknown component and is based on φ-divergences.…”

Research on Control Method of Pollutant Total Amount of Water Quality based on Fuzzy Mathematics

“…The adopted technology is the chance constraint plan, where the certainty equivalent replaces the probability restriction in optimization problems of water quality. First-order uncertainty analysis is used to represent the intrinsic uncertainty of random factors (Mohamad & Boumahdaf, 2017). The total amount of the polluted water was controlled via a linear programming model.…”