Narrow Results

Geometry for q-exponential families is studied in this paper. A q-exponential family is a set of probability distributions, which is a natural generalization of the standard exponential family. A q-exponential family has information geometric structure and a dually flat structure. To describe these relations, generalized conformal structures for statistical manifolds are studied in this paper. As an application of geometry for q-exponential families, a geometric generalization of statistical inference is also studied.

We give a survey on geometry of statistical manifolds in terms of estimating functions. A statistical model naturally has a statistical manifold structure. In particular, a q-exponential family which is a generalization of an exponential family admits several statistical manifold structures. An estimating function can be regarded as a tangent vector of a statistical model, and it gives rise to dualistic structures on a statistical manifold. In this paper, we construct statistical manifold structures on statistical models and divergence functions from the viewpoint of estimating functions. We also study geometry of nonintegrable estimating functions.