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

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, 1961). 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.