First, we consider multiple comparisons for the differences among parameters in k Poisson populations. We may give the Tukey-Kramer type multiple test procedure based on estimators of k means. However, the degree of conservativeness for the multiple tests depends on unknown rate parameters. Therefore, multiple tests based on estimators of k standard deviations are proposed. It is found that the degree of conservativeness for the proposed tests is controlled by the sample sizes. Furthermore, the closed testing procedure, more powerful than the REGW (Ryan⁄Einot-Gabriel⁄Welsch) tests, is proposed. Simultaneous confidence intervals for the differences of the square roots of the rates are discussed. Next, for the multiple comparisons with a control, we propose the multiple test procedure based on the estimators of k standard deviations. It is shown that the proposed multiple test is superior to the tests based on the Bonferroni inequality asymptotically. A sequentially rejective procedure is derived under unequal sample sizes.
Probability testing (PT) is a way to respond to multiple-choice test items. In PT the examinee gives to each response option his/her subjective probability of its being correct as an expression of partial knowledge. By using PT more item information can be drawn from the subjects than the other scoring methods that can be used for multiple-choice items. In this research, a multi-dimensional continuous item response model for PT is proposed. Moreover, the matrix of information function, a method of estimating item parameter, a method of estimating the subject’s vector of latent traits are introduced.
Estimation of the covariance between two of the variables in a set of observed variables is investigated when the factor analysis model holds. Under the assumption of multivariate normality, explicit formulae for the asymptotic sampling variances and covariances of the maximum likelihood estimators are derived. Asymptotic sampling variances of functions of the covariances are also investigated. Examples suggest that while the usual unbiased estimators can perform poorly in estimating the covariances, they may perform to the same extent as the maximum likelihood estimators in estimating functions of the covariances.
Asymptotic cumulants of functions of multinomial sample proportions with and without studentization up to the fourth order are derived, where observed proportions are possibly added by some quantities. Some of the asymptotic cumulants of non-studentized estimators are invariant with respect to the added quantities used. On the other hand, most of the asymptotic cumulants for studentized estimators are the same as those for the estimators without the added quantities when the estimator of the asymptotic variance of the non-studentized estimator is appropriately constructed to avoid the problem of sampling zeroes or empty cells. Especially, when the quantities of order O(1⁄n) are used, all the asymptotic cumulants of the studentized estimators up to the fourth order are the same as those for the estimators without the added quantities. A numerical example using the log odds-ratio and Yule’s coefficients is illustrated.