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A Bernstein-von Mises Theorem for Generalized Fiducial Distributions

J. E. Borgert, Jan Hannig

Abstract

An established and growing literature on generalized fiducial inference and related fiducial ideas points to the adoption of fiducial inference as a mainstream perspective among modern statisticians. Like Bayesian posteriors, generalized fiducial distributions (GFDs) are known to satisfy Bernstein-von Mises (BvM)-type results under classical regularity conditions. Existing fiducial BvM results, however, rely on relatively restrictive smoothness assumptions and are limited in scope. In this paper, we establish a Bernstein-von Mises theorem for generalized fiducial inference under the general framework of local asymptotic normality, which accommodates non-i.i.d. data settings and reduces to the familiar differentiability in quadratic mean condition in the i.i.d. case. We apply our result to extend existing fiducial theory for free-knot spline models first developed in Sonderegger and Hannig (2014), and further illustrate its generality in models where classical regularity conditions fail or i.i.d. assumptions are not met.

A Bernstein-von Mises Theorem for Generalized Fiducial Distributions

Abstract

An established and growing literature on generalized fiducial inference and related fiducial ideas points to the adoption of fiducial inference as a mainstream perspective among modern statisticians. Like Bayesian posteriors, generalized fiducial distributions (GFDs) are known to satisfy Bernstein-von Mises (BvM)-type results under classical regularity conditions. Existing fiducial BvM results, however, rely on relatively restrictive smoothness assumptions and are limited in scope. In this paper, we establish a Bernstein-von Mises theorem for generalized fiducial inference under the general framework of local asymptotic normality, which accommodates non-i.i.d. data settings and reduces to the familiar differentiability in quadratic mean condition in the i.i.d. case. We apply our result to extend existing fiducial theory for free-knot spline models first developed in Sonderegger and Hannig (2014), and further illustrate its generality in models where classical regularity conditions fail or i.i.d. assumptions are not met.
Paper Structure (15 sections, 3 theorems, 159 equations, 7 figures, 1 table)

This paper contains 15 sections, 3 theorems, 159 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Bernstein--von Mises results
  4. Mechanics of generalized fiducial inference
  5. A Bernstein--von Mises theorem for generalized fiducial distributions
  6. Motivation
  7. Problem setup and regularity conditions
  8. Main result
  9. Fiducial theory for free-knot spline models
  10. Examples and simulations
  11. Preliminaries
  12. Notation
  13. Proofs
  14. Example 4.1
  15. AR(2) model

Key Result

Theorem 3.1

Under Assumptions 1-5, the fiducial distribution satisfies where $R_{\sqrt{n}(\bar{\boldsymbol{\Theta}}-\boldsymbol{\theta}_0)|\mathbf{Y}}$ denotes the generalized fiducial distribution on the locally rescaled parameter space and the distance computed is the total variation distance.

Figures (7)

  • Figure 1: Kernel density estimates of the GFD (red) and Bayesian posteriors with Jeffreys (blue) and flat (black) priors, based on samples of size $n= 20, 200, 2,000$ (left to right) for a true parameter value of $\theta_0 = 0.3$. Densities are shown on the local scale, with corresponding normal approximations overlaid (black dashed). As $n$ increases, all distributions converge to the normal limit.
  • Figure 2: Coverage probabilities across sample sizes for various $\theta_0$ (blue: Jeffreys Bayesian; magenta: modified GFD; black: flat Bayesian; red: GFD), illustrating improved boundary performance of Jeffreys Bayesian and modified GFD methods, with all approaches converging for large samples.
  • Figure 3: Box-and-whisker plots of two-sided interval lengths for each method at various sample sizes, shown for a true parameter value near the boundary (top row) and in the interior (bottom row).
  • Figure 4: Kernel density estimates for the GFD (red) and Bayesian posterior with Jeffreys prior (blue) for the AR(2) model with true parameters $(\phi_1^0,\phi_2^0,\sigma^0) = (0.5, -0.3, 1)$. Black dashed curves show the corresponding normal approximations $N(I_{\boldsymbol{\theta}_0}^{-1}\Delta_{T,\boldsymbol{\theta}_0}, I_{\boldsymbol{\theta}_0}^{-1})$. Top and bottom rows correspond to sample sizes $n=50$ and $n=500$, respectively, with draws rescaled to the local scale; convergence to the normal approximation is evident as $n$ increases from $50$ to $500$.
  • Figure 5: Empirical coverage of two-sided GFD (red) and Bayesian intervals (blue) for the true AR parameter $\phi_1^0$ across nominal levels $1-\alpha$. Exact coverage indicated by a black dashed line. Top and bottom panels correspond to sample sizes $n=25$ and $n=100$, respectively, with data generated from true parameters $(\phi_1^0,\phi_2^0,\sigma^0)=(0.5,-0.3,1)$. Note that Bayesian intervals tend to under-cover compared to the fiducial.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Example 2.1
  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof