Likelihood ratio tests in the Rasch model for item response data when the number of persons and items goes to infinity

When the number of persons and items goes to infinity simultaneously, the maximum likelihood estimator in the Rasch model for dichotomous item response data has been shown to be consistency and asymptotic normality. However, the limiting distributions of the likelihood ratio tests in the past thirty years are still unknown. In this paper, we establish the Wilks type of results for the likelihood ratio tests under some simple and composite null hypotheses. Our proof crucially depends on the approximated inverse of the Fisher information matrix with small approximation errors. Simulation studies are provided to illustrate the asymptotic results.