Higher order asymptotics of Generalized Fiducial Distribution

Generalized Fiducial Inference (GFI) is motivated by R.A. Fisher's approach of obtaining posterior-like distributions when there is no prior information available for the unknown parameter. Without the use of Bayes' theorem GFI proposes a distribution on the parameter space using a technique called increasing precision asymptotics \cite{hannig2013generalized}. In this article we analyzed the regularity conditions under which the Generalized Fiducial Distribution (GFD) will be first and second order exact in a frequentist sense. We used a modification of an ingenious technique named "Shrinkage method" \cite{bickel1990decomposition}, which has been extensively used in the probability matching prior contexts, to find the higher order expansion of the frequentist coverage of Fiducial quantile. We identified when the higher order terms of one-sided coverage of Fiducial quantile will vanish and derived a workable recipe for obtaining such GFDs. These ideas are demonstrated on several examples.