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Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):745-754; doi:10.1093/biomet/asm045
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Copyright © 2007 Biometrika Trust

Miscellanea

On the approximation of the quadratic exponential distribution in a latent variable context

Francesco Bartolucci

Department of Economics, Finance and Statistics, University of Perugia, 06123 Perugia, Italy

Fulvia Pennoni

Department of Statistics, University of Milano-Bicocca, 20126 Milano, Italy

bart{at}stat.unipg.it

fulvia.pennoni{at}unimib.it

Received for publication 1 July 2006. Revision received 1 November 2006.
  Abstract

Following Cox & Wermuth (1994, 2002), we show that the distribution of a set of binary observable variables, induced by a certain discrete latent variable model, may be approximated by a quadratic exponential distribution. This discrete latent variable model is equivalent to the latent-class version of the two-parameter logistic model of Birnbaum (1968), which may be seen as a generalized version of the Rasch model (Rasch, 1960, 196). On the basis of this result, we develop an approximate maximum likelihood estimator of the item parameters of the two-parameter logistic model which is very simply implemented. The proposed approach is illustrated through an example based on a dataset on educational assessment.

Key Words: Approximate maximum likelihood • Estimation • Item response theory • Rasch model • Two-parameter logistic model


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