The covariance structure of cml-estimates in the Rasch model

Abstract

Abstract In this paper we consider conditional maximum likelihood (cml) estimates for item parameters in the Rasch model under random subject parameters. We give a simple approximation for the asymptotic covariance matrix of the cml-estimates. The approximation is stated as a limit theorem when the number of item parameters goes to infinity. The results contain precise mathematical information on the order of approximation. The results enable the analysis of the covariance structure of cml-estimates when the number of items is large. Let us give a rough picture. The covariance matrix has a dominating main diagonal containing the asymptotic variances of the estimators. These variances are almost equal to the efficient variances under ml-estimation when the distribution of the subject parameter is known. Apart from very small numbers n of item parameters the variances are almost not affected by the number n. The covariances are more or less negligible when the number of item parameters is large. Although this picture intuitively is not surprising it has to be established in precise mathematical terms. This has been done in the present paper. The paper is based on previous results [6] of the author concerning conditional distributions of non-identical replications of Bernoulli trials. The mathematical background are Edgeworth expansions for the central limit theorem. These previous results are the basis of approximations for the Fisher information matrices of cml-estimates. The main results of the present paper are concerned with the approximation of the covariance matrices. Numerical illustrations of the results and numerical experiments based on the results are presented in Strasser [5].