Consistent Empirical Bayes Estimation of the Mean of a Mixing Distribution with Applications to Treatment of Nonresponse
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911276515459072 |
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| author | Greenshtein, Eitan |
| author_facet | Greenshtein, Eitan |
| contents | We consider a Nonparametric Empirical Bayes (NPEB) framework. Let $Y_i$ be random variables, $Y_i \sim f(y|θ_i)$, $i=1,...,n$, where $θ_i \sim G$, and $θ_i \in Θ$ are independent. The variables $Y_i $ are conditionally independent given $θ_i, \; i=1,...,n$. The mixing distribution $G$ is unknown and assumed to belong to a nonparametric class
$\{G \}$.
Let $η(θ)$ be a function of $θ$. We address the problem of consistently estimating $E_G η(θ) \equiv η_G$. This problem becomes particularly challenging when $G$ cannot be consistently estimated from the observed data.
We motivate this problem, especially in contexts involving nonresponse and missing data. For such cases, a consistent estimation method is suggested and its performance is demonstrated through simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15373 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Consistent Empirical Bayes Estimation of the Mean of a Mixing Distribution with Applications to Treatment of Nonresponse Greenshtein, Eitan Statistics Theory We consider a Nonparametric Empirical Bayes (NPEB) framework. Let $Y_i$ be random variables, $Y_i \sim f(y|θ_i)$, $i=1,...,n$, where $θ_i \sim G$, and $θ_i \in Θ$ are independent. The variables $Y_i $ are conditionally independent given $θ_i, \; i=1,...,n$. The mixing distribution $G$ is unknown and assumed to belong to a nonparametric class $\{G \}$. Let $η(θ)$ be a function of $θ$. We address the problem of consistently estimating $E_G η(θ) \equiv η_G$. This problem becomes particularly challenging when $G$ cannot be consistently estimated from the observed data. We motivate this problem, especially in contexts involving nonresponse and missing data. For such cases, a consistent estimation method is suggested and its performance is demonstrated through simulations. |
| title | Consistent Empirical Bayes Estimation of the Mean of a Mixing Distribution with Applications to Treatment of Nonresponse |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2511.15373 |