Bernoulli

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

Strong Gaussian approximation of the mixture Rasch model

Friedrich Liese, Alexander Meister, and Johanna Kappus

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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
Received: February 2017
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

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

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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References

  • [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.