A Bernstein-von Mises Theorem for Generalized Fiducial Distributions
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| Format: | Preprint |
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2024
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| _version_ | 1866910037135327232 |
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| author | Borgert, J. E. Hannig, Jan |
| author_facet | Borgert, J. E. Hannig, Jan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Bernstein-von Mises Theorem for Generalized Fiducial Distributions Borgert, J. E. Hannig, Jan Statistics Theory 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. |
| title | A Bernstein-von Mises Theorem for Generalized Fiducial Distributions |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2401.17961 |