## Bernoulli

• Bernoulli
• Volume 25, Number 2 (2019), 1326-1354.

### 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 $n\times m$-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.

#### Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1326-1354.

Dates
Revised: August 2017
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862852

Digital Object Identifier
doi:10.3150/18-BEJ1022

Mathematical Reviews number (MathSciNet)
MR3920374

Zentralblatt MATH identifier
07049408

#### Citation

Liese, Friedrich; Meister, Alexander; Kappus, Johanna. Strong Gaussian approximation of the mixture Rasch model. Bernoulli 25 (2019), no. 2, 1326--1354. doi:10.3150/18-BEJ1022. https://projecteuclid.org/euclid.bj/1551862852

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