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Asymptotic expansions for the ability estimator in item response theory

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  • Haruhiko Ogasawara

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    Abstract

    Asymptotic approximations to the distributions of the ability estimator and its transformations in item response theory are derived beyond the usual normal one when associated item parameters are given as in tailored testing. For the approximations, the asymptotic cumulants of the estimators up to the fourth order with the higher-order asymptotic variances are obtained under possible model misspecification. For testing and interval estimation of abilities, the asymptotic cumulants of the pivots studentized in four ways are derived. Numerical examples with simulations including those for confidence intervals for abilities are given using the three-parameter logistic model. Copyright Springer-Verlag 2012

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    File URL: http://hdl.handle.net/10.1007/s00180-011-0282-0
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    Bibliographic Info

    Article provided by Springer in its journal Computational Statistics.

    Volume (Year): 27 (2012)
    Issue (Month): 4 (December)
    Pages: 661-683

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    Handle: RePEc:spr:compst:v:27:y:2012:i:4:p:661-683

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    Related research

    Keywords: Ability; Asymptotic expansion; Cumulants; Studentization; Sandwich estimator; Model misspecification; Pivots;

    References

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    1. Ogasawara, Haruhiko, 2010. "Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2149-2167, October.
    2. Karl Klauer, 1991. "Exact and best confidence intervals for the ability parameter of the Rasch model," Psychometrika, Springer, vol. 56(3), pages 535-547, September.
    3. Frederic Lord, 1983. "Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability," Psychometrika, Springer, vol. 48(2), pages 233-245, June.
    4. R. Bock & Murray Aitkin, 1981. "Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm," Psychometrika, Springer, vol. 46(4), pages 443-459, December.
    5. Taniguchi, M. & Watanabe, Y., 1994. "Statistical Analysis of Curved Probability Densities," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 228-248, February.
    6. Haruhiko Ogasawara, 2009. "Asymptotic cumulants of the parameter estimators in item response theory," Computational Statistics, Springer, vol. 24(2), pages 313-331, May.
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    Cited by:
    1. Ogasawara, Haruhiko, 2013. "Asymptotic cumulants of ability estimators using fallible item parameters," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 144-162.

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