Estimation in the Presence of Infinitely many Nuisance Parameters--Geometry of Estimating Functions

Abstract

When there exist nuisance parameters whose number increases in proportion to that of independent observations, it is in general difficult to get a consistent or efficient estimator of a common structural parameter. The present paper proposes a new theory based on a vector bundle consisting of certain random variables over the statistical model. Structures and properties of estimating functions are elucidated in the class of consistent estimators. A necessary and sufficient condition is obtained for the existence of a consistent estimator given by an estimating function. A necessary and sufficient condition is then given for the existence of the optimal estimator in this class, which is further obtained when it exists. In their derivations, the concept of dual connections and parallel transports plays an essential role. The results are applied to a special type of exponential family, and the optimal estimators are explicitly obtained in some examples. This explains the reason why the conditional score plays an important role.