Geometry of f-divergence

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.

[1]  Timothy R. C. Read,et al.  Goodness-Of-Fit Statistics for Discrete Multivariate Data , 1988 .

[2]  Frederick Mosteller,et al.  Understanding robust and exploratory data analysis , 1983 .

[3]  P. Vos.  Fundamental equations for statistical submanifolds with applications to the Bartlett correction , 1989 .

[4]  R. E. Kass,et al.  Chapter 5: Differential Metrics in Probability Spaces , 1987 .

[5]  S. Eguchi.  A differential geometric approach to statistical inference on the basis of contrast functionals , 1985 .

[6]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[7]  J. Pfanzagl.  Asymptotic Expansions Related to Minimum Contrast Estimators , 1973 .

[8]  C. R. Rao,et al.  Differential metrics in probability spaces , 1984 .

[9]  F. Mosteller,et al.  Understanding robust and exploratory data analysis , 1985 .

[10]  I. Csiszar,et al.  Topological property of f-divergence , 1967 .

[11]  R. Kass.  Nonlinear Regression Analysis and its Applications , 1990 .

[12]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[13]  P. Vos,et al.  Minimum f-divergence estimators and quasi-likelihood functions , 1992 .

[14]  Calyampudi R. Rao,et al.  Chapter 4: Statistical Manifolds , 1987 .

[15]  David Ruppert,et al.  Transformations to symmetry and homoscedasticity , 1989 .

[16]  R. E. Kass,et al.  Chapter 2: Differential Geometrical Theory of Statistics , 1987 .

[17]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[18]  S. Eguchi.  Second Order Efficiency of Minimum Contrast Estimators in a Curved Exponential Family , 1983 .

[19]  Douglas M. Bates,et al.  Nonlinear Regression Analysis and Its Applications , 1988 .