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Empirical Bayes Estimates of Domain Scores Under Binomial and Hypergeometric Distributions for Test Scores

Published online by Cambridge University Press:  01 January 2025

Miao-Hsiang Lin  and
Chao A. Hsiung
Miao-Hsiang Lin*
Affiliation:
Institute of Statistical Science, Academia Sinica
Chao A. Hsiung
Affiliation:
Institute of Statistical Science, Academia Sinica
*
Requests for reprints should be sent to Miao-Hsiang Lin, Institute of Statistical Science, Academia Sinica, Taipei, 11529, Taiwan, R.O.C.
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Abstract

We introduce two simple empirical approximate Bayes estimators (EABEs)— \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}_N (x)$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta _N (x)$$\end{document}—for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test score x without relying on knowledge of prior domain score distributions) have been proven to hold Δ-asymptotic optimality in Robbins' sense of convergence in mean. We found that, where \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} are the monotonized versions of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}_N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta _N$$\end{document} under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} or \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} performs better than both \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} performs as well as \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.

Keywords

Bayes riskΔ-asymptotic optimalitybinomial and hypergeometric modelsprior-hyperprior approachsimple empirical Bayes estimateVan Houwelingen's monotonization method
Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 3 , September 1994 , pp. 331 - 359
Copyright
Copyright © 1994 The Psychometric Society

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Footnotes

The authors are indebted to both anonymous reviewers, especially Reviewer 2, and the Editor for their invaluable comments and suggestions. Thanks are also due to Yuan-Chin Chang and Chin-Fu Hsiao for their help with our simulation and programming work.

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