In multidimensional item response theory (MIRT), it is possible for the estimate of a subject’s ability in some dimension to decrease after they have answered a question correctly. This paper investigates how and when this type of paradoxical result can occur. We demonstrate that many response models and statistical estimates can produce paradoxical results and that in the popular class of linearly compensatory models, maximum likelihood estimates are guaranteed to do so. In light of these findings, the appropriateness of multidimensional item response methods for assigning scores in high-stakes testing is called into question.

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.

Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.

A necessary and sufficient condition is given in this paper for the existence and uniqueness of the maximum likelihood (the so-called joint maximum likelihood) estimate of the parameters of the Partial Credit Model. This condition is stated in terms of a structural property of the pattern of the data matrix that can be easily verified on the basis of a simple iterative procedure. The result is proved by using an argument of Haberman (1977).

General algorithms for computing the likelihood of any sequence generated by an absorbing Markov-chain are described. These algorithms enable an investigator to compute maximum likelihood estimates of parameters using unconstrained optimization techniques. The problem of parameter identifiability is reformulated into questions concerning the behavior of the likelihood function in the neighborhood of an extremum. An alternative characterization of the concept of identifiability is proposed. A procedure is developed for deciding whether or not this definition is satisfied.

The test-negative design (TND) has become a standard approach for vaccine effectiveness (VE) studies. However, previous studies suggested that it may be more vulnerable than other designs to misclassification of disease outcome caused by imperfect diagnostic tests. This could be a particular limitation in VE studies where simple tests (e.g. rapid influenza diagnostic tests) are used for logistical convenience. To address this issue, we derived a mathematical representation of the TND with imperfect tests, then developed a bias correction framework for possible misclassification. TND studies usually include multiple covariates other than vaccine history to adjust for potential confounders; our methods can also address multivariate analyses and be easily coupled with existing estimation tools. We validated the performance of these methods using simulations of common scenarios for vaccine efficacy and were able to obtain unbiased estimates in a variety of parameter settings.

The cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

Product yield reflects the potential product quality and reliability, which means thathigh yield corresponds to good quality and high reliability. Yet consumers usuallycouldn’t know the actual yield of the products they purchase. Generally, the products thatconsumers get from suppliers are all eligible. Since the quality characteristic of theeligible products is covered by the specifications, then the observations of qualitycharacteristic follow truncated normal distribution. In the light of maximum likelihoodestimation, this paper proposes an algorithm for calculating the parameters of fullGaussian distribution before truncation based on truncated data and estimating productyield. The confidence interval of the yield result is derived, and the effect of samplesize on the precision of the calculation result is also analyzed. Finally, theeffectiveness of this algorithm is verified by an actual instance.

This paper considers an M/M/R/N queue with heterogeneousservers in which customers balk (do not enter) with a constantprobability (1 - b). We develop the maximum likelihoodestimates of the parameters for the M/M/R/N queue with balking andheterogeneous servers. This is a generalization of the M/M/2queue with heterogeneous servers (without balking), and theM/M/2/N queue with balking and heterogeneous servers in theliterature. We also develop the confidence interval formula forthe parameter ρ, the probability of empty system P 0, andthe expected number of customers in the system E[N], of anM/M/R/N queue with balking and heterogeneous servers. The effectsof varying b, N, and R on the confidence intervals of P 0and E[N] are also investigated.

Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet–multinomial urn model for describing the above phenomena from a stochastic point of view.

We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.