Semiparametric Mixtures of Symmetric Distributions

type="main" xml:id="sjos12015-abs-0001"> <title type="main">ABSTRACT</title>We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M-estimators of these parameters based on a Fourier approach and prove that they are <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" display="block" altimg="urn:x-wiley:03036898:media:sjos12015:sjos12015-math-0001" wiley:location="equation/sjos12015-math-0001.gif"><msqrt><mrow><mi>n</mi>< /mrow></msqrt></math>-consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.