Statistical Inferences by Perturbing Estimating Function or Minimand

L.J. Wei

Harvard University

Suppose that inferences for θ₀ (finite- or infinite-dimensional parameter) to be made is based on an asymptotically pivotal estimating function S_X (θ), where X is the observable random quantity. That is, the asymptotic distribution of S_X (θ₀) can be approximated well by the distribution of a random quantity of Z, which is free of any unknown parameter. Let _X be the consistent estimator for θ₀ based on the estimating function S_X (θ). The controversial Fisher’s fiducial distribution of θ can be generated by the random quantity Ө, where S_x (Ө) = Z and x is the observed X. It turns out that the fiducial distribution of θ can be used to approximate the distribution of _X well. That is, the distribution of (_X – θ₀) can be approximated by the conditional (on the data) distribution of (Ө- _x ). In this talk, we will discuss how this works especially when the estimating function is not smooth in θ with several examples. We will also discuss other types of perturbation methods for estimating functions or minimands to make inferences about θ_0.