MINIMUM f -DIVERGENCE ESTIMATORS AND QUASI-LIKELIHOOD FUNCTIONS

PAUL W. VOS

Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.

(Received September 4, 1989; revised September 14, 1990)

Abstract. Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari’s dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.

Key words and phrases: Quasi-likelihood, f-divergence, minimum divergence estimator, minimum chi-square estimator, dual geometries, generalized linear models.