Narrow Results

In order to construct estimating functions in some parametric models, this paper introduces two classes of information matrices. Some necessary and sufficient conditions for the information matrices achieving their upper bounds are given. For the problem of estimating the median, some optimum estimating functions based on the information matrices are acquired. Under some regularity conditions, an approach to carrying out the best basis function is introduced. In nonlinear regression models, an optimum estimating function based on the information matrices is obtained. Some examples are given to illustrate the results. Finally, the concept of optimum estimating function and the methods of constructing optimum estimating function are developed in more general statistical models.

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.