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Saddlepoint Approximations of the Distribution of the Person Parameter in the Two Parameter Logistic Model

Published online by Cambridge University Press:  01 January 2025

Martin Biehler ,
Heinz Holling  and
Philipp Doebler
Martin Biehler*
Affiliation:
Westfälische Wilhelms-Universität Münster
Heinz Holling
Affiliation:
Westfälische Wilhelms-Universität Münster
Philipp Doebler
Affiliation:
Westfälische Wilhelms-Universität Münster
*
Requests for reprints should be sent to Martin Biehler, Westfälische Wilhelms-Universität Münster, Münster, Germany. E-mail: Martin.A.Biehler@uni-giessen.de
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Abstract

Large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic (2PL) model. In short tests, however, the assumption of normality can be grossly wrong. As a consequence, intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative. Methods belonging to the higher-order-theory, more specifically saddlepoint approximations, are a convenient way to deal with small-sample problems. Confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter. Moreover, an approximation to the exact distribution permits to compute median unbiased estimates (MUE) that are as likely to overestimate as to underestimate the true person parameter. Additionally, in small samples, these MUE are less mean-biased than the often-used maximum likelihood estimator.

Keywords

small-samplessaddlepoint approximationLugannani–Ricemodified signed likelihood ratio statistichigher-order-theoryparameter distribution2PLexponential familyperson parameterconfidence intervalsmedian unbiased estimatormean-bias
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 3 , September 2015 , pp. 665 - 688
Copyright
Copyright © 2014 The Psychometric Society

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Footnotes

Electronic Supplementary Material The online version of this article (doi:10.1007/s11336-014-9405-1) contains supplementary material, which is available to authorized users.

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