This paper first discusses the relationship between Kullback–Leibler information (KL) and Fisher information in the context of multi-dimensional item response theory and is further interpreted for the two-dimensional case, from a geometric perspective. This explication should allow for a better understanding of the various item selection methods in multi-dimensional adaptive tests (MAT) which are based on these two information measures. The KL information index (KI) method is then discussed and two theorems are derived to quantify the relationship between KI and item parameters. Due to the fact that most of the existing item selection algorithms for MAT bear severe computational complexity, which substantially lowers the applicability of MAT, two versions of simplified KL index (SKI), built from the analytical results, are proposed to mimic the behavior of KI, while reducing the overall computational intensity.

It has been widely reported that in computerized adaptive testing some examinees may get much lower scores than they would normally if an alternative paper-and-pencil version were given. The main purpose of this investigation is to quantitatively reveal the cause for the underestimation phenomenon. The logistic models, including the 1PL, 2PL, and 3PL models, are used to demonstrate our assertions. Our analytical derivation shows that, under the maximum information item selection strategy, if an examinee failed a few items at the beginning of the test, easy but more discriminating items are likely to be administered. Such items are ineffective to move the estimate close to the true θ, unless the test is sufficiently long or a variable-length test is used. Our results also indicate that a certain weighting mechanism is necessary to make the algorithm rely less on the items administered at the beginning of the test.

It is known that a family of fixed-effects item response models with equal discrimination and different guessing parameters has no model identifiability. For this family, some types of information including the Fisher information and a new one are maximized to have model identification. The conditions of monotonicity of these types of information with respect to a tuning parameter are given. In the case of the logistic model with guessing parameters, it is shown that maxima do not exist under some parametrization, where negative lower asymptote can be employed without changing the probabilities of correct responses by examinees.

Over the past thirty years, obtaining diagnostic information from examinees’ item responses has become an increasingly important feature of educational and psychological testing. The objective can be achieved by sequentially selecting multidimensional items to fit the class of latent traits being assessed, and therefore Multidimensional Computerized Adaptive Testing (MCAT) is one reasonable approach to such task. This study conducts a rigorous investigation on the relationships among four promising item selection methods: D-optimality, KL information index, continuous entropy, and mutual information. Some theoretical connections among the methods are demonstrated to show how information about the unknown vector θ can be gained from different perspectives. Two simulation studies were carried out to compare the performance of the four methods. The simulation results showed that mutual information not only improved the overall estimation accuracy but also yielded the smallest conditional mean squared error in most region of θ. In the end, the overlap rates were calculated to empirically show the similarity and difference among the four methods.

Chang and Stout (1993) presented a derivation of the asymptotic posterior normality of the latent trait given examinee responses under nonrestrictive nonparametric assumptions for dichotomous IRT models. This paper presents an extention of their results to polytomous IRT models in a fairly straightforward manner. In addition, a global information function is defined, and the relationship between the global information function and the currently used information functions is discussed. An information index that combines both the global and local information is proposed for adaptive testing applications.

An extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.

In insurance underwriting, misrepresentation represents the type of insurance fraud when an applicant purposely makes a false statement on a risk factor that may lower his or her cost of insurance. Under the insurance ratemaking context, we propose to use the expectation-maximization (EM) algorithm to perform maximum likelihood estimation of the regression effects and the prevalence of misrepresentation for the misrepresentation model proposed by Xia and Gustafson [(2016) The Canadian Journal of Statistics, 44, 198–218]. For applying the EM algorithm, the unobserved status of misrepresentation is treated as a latent variable in the complete-data likelihood function. We derive the iterative formulas for the EM algorithm and obtain the analytical form of the Fisher information matrix for frequentist inference on the parameters of interest for lognormal losses. We implement the algorithm and demonstrate that valid inference can be obtained on the risk effect despite the unobserved status of misrepresentation. Applying the proposed algorithm, we perform a loss severity analysis with the Medical Expenditure Panel Survey data. The analysis reveals not only the potential impact misrepresentation may have on the risk effect but also statistical evidence on the presence of misrepresentation in the self-reported insurance status.

De Bruijn's identity relates two important concepts in information theory: Fisher information and differential entropy. Unlike the common practice in the literature, in this paper we consider general additive non-Gaussian noise channels where more realistically, the input signal and additive noise are not independently distributed. It is shown that, for general dependent signal and noise, the first derivative of the differential entropy is directly related to the conditional mean estimate of the input. Then, by using Gaussian and Farlie–Gumbel–Morgenstern copulas, special versions of the result are given in the respective case of additive normally distributed noise. The previous result on independent Gaussian noise channels is included as a special case. Illustrative examples are also provided.

We consider statistical inference for a parametric cooperative sequential adsorption model for spatial time series data, based on maximum likelihood. We establish asymptotic normality of the maximum likelihood estimator in the thermodynamic limit. We also perform and discuss some numerical simulations of the model, which illustrate the procedure for creating confidence intervals for large samples.

This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.

Let Y(t) be a pure birth process. If a maximum likelihood estimator of the birth intensity is desired and the number n of observational points and the last observation T are given in advance, it is shown that equidistant sampling is not an optimal procedure. Properties of ‘optimal' designs and the corresponding maximum likelihood estimators are investigated and compared with equidistant and continuous sampling.