SOME REMARKS ON THE INFORMATION GEOMETRY OF THE GAMMA DISTRIBUTION

ABSTRACT

Bounds for the Fisher information metric associated with the Gamma statistical model are found in terms of Poincaré type metric. This results in the determination of bounds for the Rao distance, that is the Riemannian distance induced by the information metric, between Gamma distributions, and of bounds for the Gaussian curvature of the Gamma model. The bounds seem to be sharp, where the lower and upper Rao distance bounds are Poincaré distances with Gaussian curvatures −1/4 and −1/2, respectively. In addition, the sign of the Gaussian curvature of the Gamma model is shown to be negative which means, in particular, that the geometry of the model is hyperbolic.

Keywords:

• Gamma distribution
• Information geometry
• Rao distance
• Gaussian curvature
• Hyperbolic geometry

ACKNOWLEDGMENTS

The authors wish to thank the referee for valuable comments and suggestions. This work is supported by DGICYT grant (Spain), PB96-1004-C02-01 and 1999SGR00059.