We consider the famous Rasch model, which is applied to psychometric surveys when $n$ persons under test answer $m$ questions. The score is given by a realization of a random binary $n\times m$-matrix. Its $(j,k)$th component indicates whether or not the answer of the $j$th person to the $k$th question is correct. In the mixture, Rasch model one assumes that the persons are chosen randomly from a population. We prove that the mixture Rasch model is asymptotically equivalent to a Gaussian observation scheme in Le Cam’s sense as $n$ tends to infinity and $m$ is allowed to increase slowly in $n$. For that purpose, we show a general result on strong Gaussian approximation of the sum of independent high-dimensional binary random vectors. As a first application, we construct an asymptotic confidence region for the difficulty parameters of the questions.
"Strong Gaussian approximation of the mixture Rasch model." Bernoulli 25 (2) 1326 - 1354, May 2019. https://doi.org/10.3150/18-BEJ1022