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GEOMETRY OF *f*-DIVERGENCE

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PAUL W. VOS

*Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.*
(Received December 26, 1989; revised August 7, 1990)

**Abstract.**
Amari's ±1-divergences and geometries provide an
important description of statistical inference. The
±1-divergences are constructed so that they are compatible
with a metric that is defined by the Fisher information.
In many cases, the ±1-divergences are but two in a family
of divergences, called the *f*-divergences, that are compatible
with the metric. We study the geometries induced by these
divergences. Minimizing the *f*-divergence provides geometric
estimators that are naturally described using certain
curvatures. These curvatures are related to asymptotic bias
and efficiency loss. Under special but important
restrictions, the geometry of *f*-divergence is closely
related to the *alpha*-geometry, Amari's extension of the
±1-geometries. One application of these results is
illustrated in an example.

*Key words and phrases*:
Divergence, contrast functional, yoke,
minimum divergence estimator, geometric estimator, curvature,
dual geometries, statistical manifold.

**Source**
( TeX ,
DVI ,
PS )