Volume 29, Issue 4 -

In this paper, we consider the sample estimators for the expected return, the variance, the value-at-risk (VaR), and the conditional VaR (CVaR) of the minimum VaR and the minimum CVaR portfolio. Their exact distributions are derived. These expressions are used for studying the distributional properties of the estimated characteristics. We prove that the expectation does not exist for the estimated variance, while the second moment does not exist for the estimated expected return. Moreover, expressions for the joint densities and the corresponding dependence measures between the estimators for the expected return and the variance as well as between the estimated expected return and the estimated VaR (CVaR) are derived. Finally, we present a confidence region for the minimum VaR portfolio and the minimum CVaR portfolio in the mean-variance space as well as in the mean-VaR (mean-CVaR) space. The obtained results are illustrated in an empirical study throughout the paper.

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].

In this paper we study conditional distributions of independent, but not identically distributed Bernoulli random variables. The conditioning variable is the sum of the Bernoulli variables. We obtain Edgeworth expansions for the conditional expectations and the conditional variances and covariances. The results are of basic interest for several applications, e.g. for the study of conditional maximum likelihood estimation in Rasch models with many item parameters.