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"The MM, ME, ML, EL, EF and GMN Approaches to Estimation: A Synthesis"

Anil K. Bera and Yannis Bilias


First Author :

Anil K. Bera
University of Illinois at Urbana-Champaign
1206 S. Sixth Street, M/C 706
Champaign, IL 61820

Second Author :

Yannis Bilias
University of Cyprus
School of Economics and Management
CY-1678 Cyprus

Abstract :
The 20th century began on an auspicious statistical note with the publication of Karl Pearson’s (1900) goodness-of-fit test, which is regarded as one of the most important scientific breakthroughs. The basic motivation behind this test was to see whether an assumed probability model adequately described the data at hand. Pearson (1894) also introduced a formal approach to statistical estimation through his method of moments (MM) estimation. Ronald A. Fisher, while he was a third year undergraduate at the Gonville and Caius College, Cambridge, suggested the maximum likelihood estimation (MLE) procedure as an alternative to Pearson’s MM approach. In 1922 Fisher published a monumental paper that introduced such basic concepts as consistency, efficiency, sufficiency – and even the term “parameter” with its present meaning. Fisher (1922) provided the analytical foundation of MLE and studied its efficiency relative to the MM estimator. Fisher (1924a) established the asymptotic equivalence of minimum X² and ML estimators and wrote in favor of using minimum X² method rather than Pearson’s MM approach. Recently, econometricians have found working under assumed likelihood functions restrictive, and have suggested using a generalized version of Pearson’s MM approach, commonly known as the GMM estimation procedure as advocated in Hansen (1982). Earlier, Godambe (1960) and Durbin (1960) developed the estimating function (EF) approach to estimation that has been proven very useful for many statistical models. A fundamental result is that score is the optimum EF. Ferguson (1958) considered an approach very similar to GMM and showed that estimation based on the Pearson chi-squared statistic is equivalent to efficient GMM. Golan, Judge and Miller (1966) developed entropy-based formulation that allowed them to solve a wide range of estimation and inference problems in econometrics. More recently, Imbens, Spady and Johnson (1998), Kitamura and Stutzer (1997) and Mittlehammer, Judge and Miller (2000) put GMM within the framework of empirical likelihood (EL) and maximum entropy (ME) estimation. It can be shown that many of these estimation techniques can be obtained as special cases of minimizing Cressie and Read (1984) power divergence criterion that comes directly from the Pearson (1900) chi-squared statistic. In this way we are able to assimilate a number of seemingly unrelated estimation techniques into a unified framework.
Manuscript Received : 2001
Manuscript Published : 2001
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