A Family of Minimum Discrepancy Estimators

In Part II, to avoid an explicit specification of both the joint probability distribution of the data sampling process underlying the observed data and the associated likelihood function, a possible sample distribution function is chosen from the multinomial family that assigns probability weight pi to observation yi. Under this concept, likelihood weights, p, supported on the sample data, are used to reduce a problem characterized by an infinite dimensional set of unknowns to a finite dimensional one. Based on this idea, alternative divergence measures, maximum empirical likelihood (MEL) and maximum empirical exponential likelihood (MEEL), were proposed and a basis for estimation and inference was developed. In Part III, we use goodness-of-fit measures proposed by Cressie and Read (CR) to specify a family of divergence measures that subsumes MEL and MEEL and that may be used to characterize data sampling process outcomes. In this context, the CR objective can be interpreted as a generalized divergence measure that leads to a generalized likelihood function concept and an associated empirical maximum likelihood method of estimation and inference.