Estimation of a Mixture via the Empirical Characteristic Function

In many circumstances, the likelihood function does not have a simple tractable expression. The main examples discussed in the paper are the convolution and the mixture of distributions. Finite mixture models are commonly used to model data from a population composed of a finite number of homogeneous subpopulations. An example of application is the estimation of a cost function in presence of multiple technologies of production (see Beard, Caudill, and Gropper, 1991). Ignoring heterogeneity may lead to seriously misleading results. Convolution appears in models that include person-specific hetorogeneity that is not observable (see Lancaster, 1990). For such models, estimation using the characteristic function offers a nice alternative to maximum likelihood method. It has been shown by Feuerverger and McDunnough (1981) that the empirical characteristic function yields an efficient estimator when used with a specific weighting function. However, this weighting function depends on the likelihood which is of course unknown. This poses the problem of the implementation of this method. Here we show that the empirical characteristic function yields a continuum of moment conditions that can be handled by the method developed by Carrasco and Florens (1999). We simply estimate the parameters of the model by GMM based on this continuum of moment conditions. We show that this method delivers asymptotically efficient estimators while being relatively easy to implement. A close investigation shows that Carrasco-Florens' results gives a rationale to Feuerverger and McDunnough's approach and is much more general since it applies to any continuum of moments. Using our continuous GMM method avoids the explicit derivation of the optimal weighting function as in Feuerverger and McDunnough. We give a general method to estimate it from the data. Next, we allow for the presence of covariates in the model. We discuss the efficient estimation based on the conditional characteristic function conditionally on exogenous variables. As long as identifiability holds, our estimators reach the Cramer Rao efficiency bound for any choice of instruments. The issue on optimal instruments can be completely ignored here. The way we choose the weight in our GMM objective function guarantees efficiency. Finally, we intend to complete the paper by a Monte Carlo experiment in order to assess the small sample properties of our estimators.