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

Strong Gaussian approximation of the mixture Rasch model

Friedrich Liese, Alexander Meister, and Johanna Kappus

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotic equivalence of statistical experiments high-dimensional central limit theorem item response model Le Cam distance psychometrics


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.

Export citation


  • [1] Alagumalai, S., Curtis, D.D. and Hungi, N. (2005). Applied Rasch Measurement: A Book of Exemplars. Berlin: Springer.
  • [2] Andersen, E.B. (1977). Sufficient statistics and latent trait models. Psychometrika 42 69–81.
  • [3] Andersen, E.B. (1980). Comparing latent distributions. Psychometrika 45 121–134.
  • [4] Andrich, D. (2010). Sufficiency and conditional estimation of person parameters in the polytomous Rasch model. Psychometrika 75 292–308.
  • [5] Bezruczko, N. (2005). Rasch Measurement in Health Sciences. Maple Grove, MN: JAM Press.
  • [6] Biehler, M., Holling, H. and Doebler, P. (2015). Saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model. Psychometrika 80 665–688.
  • [7] Brown, L.D., Carter, A.V., Low, M.G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074–2097.
  • [8] Brown, L.D. and Low, M.G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
  • [9] Cai, T.T. and Zhou, H.H. (2009). Asymptotic equivalence and adaptive estimation for robust nonparametric regression. Ann. Statist. 37 3204–3235.
  • [10] Carter, A.V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708–730.
  • [11] Carter, A.V. (2006). A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 143–156.
  • [12] DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. New York: Springer.
  • [13] de Leeuw, J. and Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. J. Educ. Behav. Stat. 11 183–196.
  • [14] Doebler, A., Doebler, P. and Holling, H. (2012). Optimal and most exact confidence intervals for person parameters in item response theory models. Psychometrika 77 98–115.
  • [15] Fischer, G.H. and Molenaar, I.W., eds. (1995). Rasch Models. New York: Springer.
  • [16] Genon-Catalot, V. and Larédo, C. (2014). Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments. Ann. Statist. 42 1145–1165.
  • [17] Haberman, S.J. (1977). Maximum likelihood estimates in exponential response models. Ann. Statist. 5 815–841.
  • [18] Hoderlein, S., Mammen, E. and Yu, K. (2011). Non-parametric models in binary choice fixed effects panel data. Econom. J. 14 351–367.
  • [19] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. New York: Springer.
  • [20] Le Cam, L. and Yang, G.L. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [21] Liese, F. and Miescke, K.-J. (2008). Statistical Decision Theory. Springer Series in Statistics. New York: Springer.
  • [22] Lindsay, B., Clogg, C.C. and Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. J. Amer. Statist. Assoc. 86 96–107.
  • [23] Mariucci, E. (2016). Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise. Stochastic Process. Appl. 126 503–541.
  • [24] Meister, A. (2011). Asymptotic equivalence of functional linear regression and a white noise inverse problem. Ann. Statist. 39 1471–1495.
  • [25] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields 155 201–229.
  • [26] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • [27] Pfanzagl, J. (1993). On the consistency of conditional maximum likelihood estimators. Ann. Inst. Statist. Math. 45 703–719.
  • [28] Pfanzagl, J. (1994). On the identifiability of structural parameters in mixtures: Applications to psychological tests. J. Statist. Plann. Inference 38 309–326.
  • [29] Rasch, G. (1960/1980). Probabilistic Models for Some Intelligence and Attainment Tests (Expanded Edition). Chicago: Univ. Chicago Press.
  • [30] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772–802.
  • [31] Rice, K.M. (2004). Equivalence between conditional and mixture approaches to the Rasch model and matched case-control studies, with applications. J. Amer. Statist. Assoc. 99 510–522.
  • [32] Rohde, A. (2004). On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions 22 235–243.
  • [33] Schmidt-Hieber, J. (2014). Asymptotic equivalence for regression under fractional noise. Ann. Statist. 42 2557–2585.
  • [34] Shiryaev, A.N. and Spokoiny, V.G. (2000). Statistical Experiments and Decisions: Asymptotic theory. Advanced Series on Statistical Science & Applied Probability 8. River Edge, NJ: World Scientific.
  • [35] Strasser, H. (1985). Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory. De Gruyter Studies in Mathematics 7. Berlin: de Gruyter.
  • [36] Strasser, H. (2012a). The covariance structure of cml-estimates in the Rasch model. Stat. Risk Model. 29 315–326.
  • [37] Strasser, H. (2012b). Asymptotic expansions for conditional moments of Bernoulli trials. Stat. Risk Model. 29 327–343.
  • [38] von Davier, M. and Carstensen, C.H. (2007). Multivariate and Mixture Distribution Rasch Models – Extensions and Applications. Statistics for Social and Behavioral Sciences. New York: Springer.