Strong Gaussian approximation of the mixture Rasch model

TL;DR

This paper demonstrates that the mixture Rasch model, used in psychometric testing, can be approximated by a Gaussian model as the number of respondents grows, enabling new statistical inference methods.

Contribution

It proves the asymptotic Gaussian approximation of the mixture Rasch model and introduces a method for constructing confidence regions for question difficulty parameters.

Findings

Mixture Rasch model is asymptotically equivalent to a Gaussian scheme.

Established a strong Gaussian approximation for high-dimensional binary sums.

Constructed confidence regions for question difficulty parameters.

Abstract

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,m)-matrix. Its (j,k)th component indicates whether or not the answer of the jth person to the kth 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.