**Sankhya:
The Indian Journal of Statistics**

2004, Volume 66, Pt. 2, 213--233

**Neighbourhoods of Randomness and
Geometry of McKay Bivariate Gamma 3-Manifold**By

Khadiga Arwini and C.T.J. Dodson, University of Manchester, UK

SUMMARY. We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. Moreover, every neighbourhood of the uniform distribution contains a neighbourhood of log-gamma distributions. We derive also the information geometry of the 3-manifold of McKay bivariate gamma distributions, which can provide a metrization of departures from randomness and departures from independence for bivariate processes. The curvature objects are derived, including those on three submanifolds. As in the case of bivariate normal manifolds, we have negative scalar curvature but here it is not constant and we show how it depends on correlation. These results have applications, for example, in the characterization of stochastic materials..

*AMS (1991) subject classification}. *53B10, 60A10.

*Key words and phrases. *Statistical manifold, information geometry,
exponential family, affine immersion, McKay bivariate gamma distribution, near-random
processes.