Statistical inference for infinite‐dimensional parameters via asymptotically pivotal estimating functions

Abstract

Suppose that a consistent estimator for an infinite‐dimensional parameter can be readily obtained via a set of estimating functions which has a ‘good’ local linear approximation around the true value of the parameter. However, it may be difficult to estimate the variance function of this estimator well. We show that, if the set of estimating functions evaluated at the true parameter value is ‘asymptotically pivotal’, then the ‘fiducial’ distribution of the parameter can be used to approximate the distribution of this consistent estimator. We present three examples to illustrate that the corresponding inference for the parameter can be made via a simple simulation technique without involving complex, high‐dimensional nonparametric density estimates.