Consistent Estimation of the Influence Function of Locally Asymptotically Linear Estimators

Consider estimators which behave locally asymptotically like an average of some function taken at the observations. This function is called the influence function and one calls such estimators locally asymptotically linear. It is shown that the influence function of a locally asymptotically linear estimator can be estimated consistently and conversely, that, given a consistent estimator of the influence function, estimators can be constructed which are locally asymptotically linear in that influence function. With the help of these results an adaptive estimator is constructed for a partially irregular model.