Information Geometry of Estimating Functions in Semi-Parametric Statistical Models

For semi-parametric statistical estimation, when an estimating function exists, it often provides an efficient or a good consistent estimator of the parameter of interest against nuisance parameters of infinite dimensions. The present paper elucidates the structure of estimating functions, based on the dual differential geometry of statistical inference and its extension to fibre bundles. The paper studies the following problems. First, when does an estimating function exist and what is the set of all the estimating functions? Second, how are the asymptotic variances of the estimators derived from estimating functions and when are the estimators efficient? Third, how do we adaptively choose a practically good (quasi-)estimating function from the observed data? The concept of m-curvature freeness plays a fundamental role in solving the above problems.