## Non-admissibility of the Rubin estimator of the variance in multiple imputation in the Bayesian Gaussian model

### Authors:

- Konstancja Bobecka-Wesołowska,
- Jacek Wesołowski

### Abstract

Multiple imputation is nowadays a generally accepted approach to statistical inference based on incomplete data sets. Within this methodology it is standard to assess the quality of the estimation by the Rubin estimator of the variance, which, when based on m imputations, has the form U¯m + (1+1/m)Bm. Here U¯m is the average of imputation estimators of variance and Bm is the empirical variance of imputation estimators. We consider the problem of estimation of variance of multiple imputation estimator in the Bayesian Gaussian model with the Gaussian mean. We show that the Rubin estimator is inadmissible in the class of estimators of the form ν2(α,β) = αU¯m + βBm, α,β ∈ R. We derive the optimal weights α∗ and β∗, i.e. such that ν2(α∗,β∗) has the smallest MSE in this class of estimators. Since α∗ and β∗ are defined through complicated expressions we also derive approximate optimal estimators with simple weights α∗∗ = 1 f , β∗∗ = − f n(1−f), where f is the response rate and n is the original size of the sample. These estimators outperform the Rubin estimator with respect to both the bias and the MSE. We also consider the case of a non-informative prior. Then the Rubin estimator is unbiased, though it remains inadmissible. Numerical experiments show that the performance of the optimal and the approximate optimal estimators is rather similar, therefore we recommend to use simplified approximate weights.

- Record ID
- WUTad3c476bd2f84805987f8ac2d2c863d5
- Author
- Pages
- 9-39
- Publication size in sheets
- 1.50
- Book
- Kotus Janina Janina Kotus (eds.): 20 Years of the Faculty of Mathematics and Information Science. A collection of research papers in mathematics, 2020, Oficyna Wydawnicza Politechniki Warszawskiej, 194 p., ISBN 978-83-8156-156-3
- Keywords in English
- multiple imputation, Rubin estimator, Bayesian Gaussian model
- URL
- https://ww2.mini.pw.edu.pl/wp-content/uploads/2020_monografia_minipw.pdf Opening in a new tab
- Language
- eng (en) English
- Score (nominal)
- 20
- Score source
- publisherList
- Score
- = 20.0, 05-05-2021, MonographChapterAuthor

- Uniform Resource Identifier
- https://repo.pw.edu.pl/info/article/WUTad3c476bd2f84805987f8ac2d2c863d5/

- URN
`urn:pw-repo:WUTad3c476bd2f84805987f8ac2d2c863d5`

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