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MINIMUM *f*-DIVERGENCE ESTIMATORS AND

QUASI-LIKELIHOOD FUNCTIONS}

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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.

**Source**
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