The Bernstein-von Mises theorem for Semiparametric Mixtures
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912138823467008 |
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| author | Franssen, Stefan Nguyen, Jeanne van der Vaart, Aad |
| author_facet | Franssen, Stefan Nguyen, Jeanne van der Vaart, Aad |
| contents | Semiparametric mixture models are parametric models with latent variables. They are defined kernel, $p_θ(x | z)$, where z is the unknown latent variable, and $θ$ is the parameter of interest. We assume that the latent variables are an i.i.d. sample from some mixing distribution $F$. A Bayesian would put a prior on the pair $(θ, F)$. We prove consistency for these models in fair generality and then study efficiency. We first prove an abstract Semiparametric Bernstein-von Mises theorem, and then provide tools to verify the assumptions. We use these tools to study the efficiency for estimating $θ$ in the frailty model and the errors in variables model in the case were we put a generic prior on $θ$ and a species sampling process prior on $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Bernstein-von Mises theorem for Semiparametric Mixtures Franssen, Stefan Nguyen, Jeanne van der Vaart, Aad Statistics Theory Semiparametric mixture models are parametric models with latent variables. They are defined kernel, $p_θ(x | z)$, where z is the unknown latent variable, and $θ$ is the parameter of interest. We assume that the latent variables are an i.i.d. sample from some mixing distribution $F$. A Bayesian would put a prior on the pair $(θ, F)$. We prove consistency for these models in fair generality and then study efficiency. We first prove an abstract Semiparametric Bernstein-von Mises theorem, and then provide tools to verify the assumptions. We use these tools to study the efficiency for estimating $θ$ in the frailty model and the errors in variables model in the case were we put a generic prior on $θ$ and a species sampling process prior on $F$. |
| title | The Bernstein-von Mises theorem for Semiparametric Mixtures |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2412.00219 |