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The Asymptotic Distribution of Ability Estimates

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  • Sandip Sinharay

    (Pacific Metrics Corporation)

Abstract

The maximum likelihood estimate (MLE) of the ability parameter of an item response theory model with known item parameters was proved to be asymptotically normally distributed under a set of regularity conditions for tests involving dichotomous items and a unidimensional ability parameter (Klauer, 1990; Lord, 1983). This article first considers the more general case of tests that include a mix of dichotomous and polytomous items. A proof is given of the asymptotic normality of the MLE of the ability parameter for such tests under a set of regularity conditions. Then, it is proved that a similar result holds for the weighted likelihood estimate and the posterior mode of the ability parameter. Multidimensional ability parameters are considered next. Numerical illustrations are provided to demonstrate the asymptotic results.

Suggested Citation

  • Sandip Sinharay, 2015. "The Asymptotic Distribution of Ability Estimates," Journal of Educational and Behavioral Statistics, , vol. 40(5), pages 511-528, October.
    • Handle: RePEc:sae:jedbes:v:40:y:2015:i:5:p:511-528
    DOI: 10.3102/1076998615606115
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    References listed on IDEAS

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    1. Ogasawara, Haruhiko, 2012. "Supplement to the paper“ Asymptotic expansions for the ability estimator in item response theory”," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 63(2/3), pages 329-336.
    2. Chun Wang, 2015. "On Latent Trait Estimation in Multidimensional Compensatory Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 428-449, June.
    3. Daniel Segall, 1996. "Multidimensional adaptive testing," Psychometrika, Springer;The Psychometric Society, vol. 61(2), pages 331-354, June.
    4. Frederic Lord, 1983. "Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability," Psychometrika, Springer;The Psychometric Society, vol. 48(2), pages 233-245, June.
    5. Hua-Hua Chang, 1996. "The asymptotic posterior normality of the latent trait for polytomous IRT models," Psychometrika, Springer;The Psychometric Society, vol. 61(3), pages 445-463, September.
    6. Hua-Hua Chang & William Stout, 1993. "The asymptotic posterior normality of the latent trait in an IRT model," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 37-52, March.
    7. C. Glas & Anna Dagohoy, 2007. "A Person Fit Test For Irt Models For Polytomous Items," Psychometrika, Springer;The Psychometric Society, vol. 72(2), pages 159-180, June.
    8. David Magis, 2015. "A Note on Weighted Likelihood and Jeffreys Modal Estimation of Proficiency Levels in Polytomous Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 200-204, March.
    9. Ogasawara, Haruhiko, 2013. "Asymptotic properties of the Bayes and pseudo Bayes estimators of ability in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 359-377.
    10. A. Philippou & G. Roussas, 1975. "Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 27(1), pages 45-55, December.
    11. Haruhiko Ogasawara, 2012. "Asymptotic expansions for the ability estimator in item response theory," Computational Statistics, Springer, vol. 27(4), pages 661-683, December.
    12. Thomas Warm, 1989. "Weighted likelihood estimation of ability in item response theory," Psychometrika, Springer;The Psychometric Society, vol. 54(3), pages 427-450, September.
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