GEOMETRY OF f -DIVERGENCE

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 α-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.