In contrast to dichotomous item response theory (IRT) models, most well-known polytomous IRT models do not imply stochastic ordering of the latent trait by the total test score (SOL). This has been thought to make the ordering of respondents on the latent trait using the total test score questionable and throws doubt on the justifiability of using nonparametric polytomous IRT models for ordinal measurement. We show that a broad class of polytomous IRT models has a weaker form of SOL, denoted weak SOL, and argue that weak SOL justifies ordering respondents on the latent trait using the total test score and, therefore, the use of nonparametric polytomous IRT models for ordinal measurement.

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p-values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.

Three classes of polytomous IRT models are distinguished. These classes are the adjacent category models, the cumulative probability models, and the continuation ratio models. So far, the latter class has received relatively little attention. The class of continuation ratio models includes logistic models, such as the sequential model (Tutz, 1990), and nonlogistic models, such as the acceleration model (Samejima, 1995) and the nonparametric sequential model (Hemker, 1996). Four measurement properties are discussed. These are monotone likelihood ratio of the total score, stochastic ordering of the latent trait by the total score, stochastic ordering of the total score by the latent trait, and invariant item ordering. These properties have been investigated previously for the adjacent category models and the cumulative probability models, and for the continuation ratio models this is done here. It is shown that stochastic ordering of the total score by the latent trait is implied by all continuation ratio models, while monotone likelihood ratio of the total score and stochastic ordering on the latent trait by the total score are not implied by any of the continuation ratio models. Only the sequential rating scale model implies the property of invariant item ordering. Also, we present a Venn-diagram showing the relationships between all known polytomous IRT models from all three classes.

In a restricted class of item response theory (IRT) models for polytomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait ϑ. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold.

In this study, first a taxonomy is given for nonparametric and parametric models for polytomous items based on the hierarchical relationship between the models. Next, it is investigated which models have the MLR property and which have the SO properties. It is shown that all models in the taxonomy possess the SOM property. However, counterexamples illustrate that many models do not, in general, possess the even more useful SOL property.

This paper concerns ordinal responses. An ordered Dirichlet distribution describes prior and posterior beliefs about the cumulative probabilities of response categories. Associating the response categories with intervals of a latent random variable then induces a distribution on the order statistics of that variable. The psychometrician can use the asymptotic theory of order statistics to learn how distributional assumptions about the latent variable effect inference. An example relates the skewness of a latent variable to the proportional odds and proportional hazards models of McCullagh [1980].

Scheiblechner (1995) proposes a probabilistic axiomatization of measurement called ISOP (isotonic ordinal probabilistic models) that replaces Rasch's (1980) specific objectivity assumptions with two interesting ordinal assumptions. Special cases of Scheiblechner's model include standard unidimensional factor analysis models in which the loadings are held constant, and the Rasch model for binary item responses. Closely related are the doubly-monotone item response models of Mokken (1971; see also Mokken & Lewis, 1982; Sijtsma, 1988; Molenaar, 1991; Sijtsma & Junker, 1996; and Sijtsma & Hemker, in press). More generally, strictly unidimensional latent variable models have been considered in some detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg (1993), and Junker (1990, 1993). The purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from Scheiblechner's (1995) paper, and to point out important related work by Hemker, Sijtsma, Molenaar, & Junker (1996, 1997), Ellis and Junker (in press) and Junker and Ellis (1997). We also discuss counterexamples to three major theorems in the paper. By carrying out these three tasks, we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the ISOP approach the careful attention it deserves.

Consider an experiment in which a subject guesses repeatedly at a randomly chosen target on a continuum. To guarantee a positive probability of success, the continuum is partitioned into a finite but large number of segments. The subject is given directional feedback. General guessing strategies are characterized, and an optimal strategy is identified. The hypothesis that the subject's performance can be explained by “chance alone” is of interest in such experiments. A test is developed based on comparing the subject's performance to expected performance using the optimal strategy. A “skill-scoring” procedure is developed for assessing a subject's performance in light of the strategy used, and a test based on skill-scoring is advanced.

The sum score is often used to order respondents on the latent trait measured by the test. Therefore, it is desirable that under the chosen model the sum score stochastically orders the latent trait. It is known that unlike dichotomous item response theory (IRT) models, most polytomous IRT models do not imply stochastic ordering. It is unknown, however, (1) whether stochastic ordering is often or rarely violated and (2) whether violations yield a serious problem for practical data analysis. These are the central issues of this paper. First, some unanswered questions that pertain to polytomous IRT models implying stochastic ordering were investigated. Second, simulation studies were conducted to evaluate stochastic ordering in practical situations. It was found that for most polytomous IRT models that do not imply stochastic ordering, the sum score can be used safely to order respondents on the latent trait.

The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

We prove existence and uniqueness for the inverse-first-passage time problem for soft-killed Brownian motion using rather elementary methods relying on basic results from probability theory only. We completely avoid the relation to a suitable partial differential equation via a suitable Feynman–Kac representation, which was previously one of the main tools.

The notion of ordered system signature, originally defined for independent and identical coherent systems, is first extended to the case of independent and non-identical coherent systems, and then some key properties that help simplify its computation are established. Through its use, a dynamic ordered system signature is defined next, which facilitates a systematic study of dynamic properties of several coherent systems under a life test. The theoretical results established here are then illustrated through some specific examples. Finally, the usefulness in the evaluation of aging used systems of the concepts introduced is demonstrated.

In this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.

The signature of a coherent system has been studied extensively in the recent literature. Signatures are particularly useful in the comparison of coherent or mixed systems under a variety of stochastic orderings. Also, certain signature-based closure and preservation theorems have been established. For example, it is now well known that certain stochastic orderings are preserved from signatures to system lifetimes when components have independent and identical distributions. This applies to the likelihood ratio order, the hazard rate order, and the stochastic order. The point of departure of the present paper is the question of whether or not a similar preservation result will hold for the mean residual life order. A counterexample is provided which shows that the answer is negative. Classes of distributions for the component lifetimes for which the latter implication holds are then derived. Connections to the theory of order statistics are also considered.

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.

In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

This paper starts with the Beta transform and discusses the stochastic ordering properties of this transform under different parameter settings. Later, the distribution of discounted aggregate claims in a compound renewal risk model with dependence between inter-claim times and claim sizes is studied. Recursive formulas for moments and joint moments are expressed in terms of the Beta transform of the inter-claim times and claim severities. Particularly, our moments formula is more explicit and computation-friendly than earlier ones in the references. Lastly, numerical examples are provided to illustrate our results.

Comparison results are given for time-inhomogeneous Markov processes with respect to function classes with induced stochastic orderings. The main result states the comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments, and to Lévy-driven diffusion processes.

We have found a mistake in the proofs of Navarro (2008, Theorem 2.3(b) and 2.3(c)) due to misapplication of properties of hazard rate and likelihood ratio orders. In this paper we show with an example that the stated results do not hold. This example is interesting since it proves some unexpected properties for these orderings under the formation of coherent systems. The result stated in Navarro (2008, Theorem 2.3(a)) for the usual stochastic order is correct.

In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/c queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/c queues started respectively from the empty state and the state of the M/G/c queue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.