Strong Gaussian approximation of the mixture Rasch model

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.

Nonparametric estimation of the ability density in the Mixed-Effect Rasch Model

“…In the following we introduce some of the frequently used terms where the terminology is partially adopted from [22]. These terms are required to understand the asymptotically equivalent Gaussian models in Subsection 2.2.…”

“…In order to provide some insight about this function, we mention that, in [22], some statistics containing the sums of the rows and the columns in the MRM have been detected to be sufficient; and their asymptotic distribution has been studied. Therein the conditional expectation of one statistic given the other equals minus the grandient of Ψ N and the corresponding conditional covariance matrix equals the Hessian matrix of Ψ N , see [22], p. 1337. Hence this function plays a major role in the asymptotically equivalent Gaussian models, on which our estimation strategy will be based.…”

“…Let us summarize the conditions on the parameters sets Θ and F which are assumed in [22] in order to establish asymptotic equivalence of MRM and the two-layer Gaussian experiment. As our procedure is based on this result we also impose these constraints to deduce all theoretical properties of the current paper while, in practice, the procedure may still be useful when these assumptions are violated.…”

“…Moreover this nonparametric estimator can be applied to treat parametric submodels and to construct parametric goodness-of-fit tests as well where the investigation of the asymptotic distribution of our estimator remains a challenging task. Our method significantly differs from the existing approaches as we do not use MLbased techniques but, in fact, we exploit asymptotic equivalence of the MRM to a two-layer Gaussian experiment (more precisely, a Gaussian observation andconditionally on that -another Gaussian observation) in Le Cam's sense, which has recently been proved in [22]. In that paper the authors use some components of the observation in order to construct an asymptotic confidence region for the difficulty parameters and leave the problem of nonparametric estimation of the ability density open for future research.…”

“…There is no straight forward extension of our procedures to the estimation of the ability distribution in the corresponding mixed-effect model since the sums of the rows and the columns do apparently not form a sufficient statistic in general. Thus the basic arguments from [22] cannot be taken over. Nonparametric estimation of the ability distribution in 2-PL and 3-PL IRT remains an interesting topic for future research.…”