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Occasionally, investigators collect auxiliary marks at the time of failure in a clinical study. Because the failure event may be censored at the end of the follow-up period, these marked endpoints are subject to induced censoring. We propose two new families of two-sample tests for the null hypothesis of no difference in mark-scale distribution that allows for arbitrary associations between mark and time. One family of proposed tests is a nonparametric extension of an existing semi-parametric linear test of the same null hypothesis while a second family of tests is based on novel marked rank processes. Simulation studies indicate that the proposed tests have the desired size and possess adequate statistical power to reject the null hypothesis under a simple change of location in the marginal mark distribution. When the marginal mark distribution has heavy tails, the proposed rank-based tests can be nearly twice as powerful as linear tests.

Modelling for marked point processes is an important problem, but has received remarkably little attention in the statistical literature. The authors developed a marked point process model that incorporates the use of functional data analysis in a joint estimation of the frequency function of the point process and the intensity of the mark, with application to data from 22 lupus patients consisting of times of flares in symptom severity combined with a quantitative assessment of the severity. The data indicate that a rapid decrease in drug dose is significantly associated with a decrease in flare frequency. Experiments with simulated data designed to model the actual data further support this conclusion. La modélisation des processus ponctuels marqués est un problème important, mais il a reçu très peu d'attention en statistique. Les auteurs développent un modèle de processus ponctuel marqué qui incorpore l'utilisation de l'analyse de données fonctionnelles dans l'estimation jointe de la fonction de fréquence du processus ponctuel et de l'intensité des marqueurs. Une application est faite à l'aide des données sur les périodes de poussées de lupus combinées à une évaluation quantitative de la sévérité de ces poussées provenant de 22 patients. Les données indiquent qu'une croissance rapide dans le dosage des médicaments est associée de façon significative avec la décroissance dans la fréquence des périodes de poussées. Des expériences de simulation basées sur un modèle construit à l'aide des données supportent cette conclusion.

1. This article reviews the application of some summary statistics from current theory of spatial point processes for extracting information from spatial patterns of plants. Theoretical measures and issues connected with their estimation are described. Results are illustrated in the context of specific ecological questions about spatial patterns of trees in two forests. 2. The pair correlation function, related to Ripley's K function, provides a formal measure of the density of neighbouring plants and makes precise the general notion of a 'plant's-eye' view of a community. The pair correlation function can also be used to describe spatial relationships of neighbouring plants with different qualitative properties, such as species identity and size class. 3. The mark correlation function can be used to describe the spatial relationships of quantitative measures (e.g. biomass). We discuss two types of correlation function for quantitative marks. Applying these functions to the distribution of biomass in a temperate forest, it is shown that the spatial pattern of biomass is uncoupled from the spatial pattern of plant locations. 4. The inhomogeneous pair correlation function enables first-order heterogeneity in the environment to be removed from second-order spatial statistics. We illustrate this for a tree species in a forest of high topographic heterogeneity and show that spatial aggregation remains after allowing for spatial variation in density. An alternative method, the master function, takes a weighted average of homogeneous pair correlation functions computed in subareas; when applied to the same data and compared with the former method, the spatial aggregations are smaller in size. 5. Synthesis. These spatial statistics, especially those derived from pair densities, will help ecologists to extract important ecological information from intricate spatially correlated plants in populations and communities.

We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

Geostatistics involves the fitting of spatially continuous models to spatially discrete data. Preferential sampling arises when the process that determines the data locations and the process being modelled are stochastically dependent. Conventional geostatistical methods assume, if only implicitly, that sampling is non-preferential. However, these methods are often used in situations where sampling is likely to be preferential. For example, in mineral exploration, samples may be concentrated in areas that are thought likely to yield high grade ore. We give a general expression for the likelihood function of preferentially sampled geostatistical data and describe how this can be evaluated approximately by using Monte Carlo methods. We present a model for preferential sampling and demonstrate through simulated examples that ignoring preferential sampling can lead to misleading inferences. We describe an application of the model to a set of biomonitoring data from Galicia, northern Spain, in which making allowance for preferential sampling materially changes the results of the analysis.

Problem solving has been recognized as a central skill that today’s students need to thrive and shape their world. As a result, the measurement of problem-solving competency has received much attention in education in recent years. A popular tool for the measurement of problem solving is simulated interactive tasks, which require students to uncover some of the information needed to solve the problem through interactions with a computer-simulated environment. A computer log file records a student’s problem-solving process in details, including his/her actions and the time stamps of these actions. It thus provides rich information for the measurement of students’ problem-solving competency. On the other hand, extracting useful information from log files is a challenging task, due to its complex data structure. In this paper, we show how log file process data can be viewed as a marked point process, based on which we propose a continuous-time dynamic choice model. The proposed model can serve as a measurement model for scaling students along the latent traits of problem-solving competency and action speed, based on data from one or multiple tasks. A real data example is given based on data from Program for International Student Assessment 2012.

The potential for statistical complexity in species distribution models (SDMs) has greatly increased with advances in computational power. Structurally complex models provide the flexibility to analyse intricate ecological systems and realistically messy data, but can be difficult to interpret, reducing their practical impact. Founding model complexity in ecological theory can improve insight gained from SDMs. Here, we evaluate a marked point process approach, which uses multiple Gaussian random fields to represent population dynamics of the Eurasian crane Grus grus in a spatio‐temporal species distribution model. We discuss the role of model components and their impacts on predictions, in comparison with a simpler binomial presence/absence approach. Inference is carried out using Integrated Nested Laplace Approximation (INLA) with inlabru, an accessible and computationally efficient approach for Bayesian hierarchical modelling, which is not yet widely used in SDMs. Using the marked point process approach, crane distribution was predicted to be dependent on the density of suitable habitat patches, as well as close to observations of the existing population. This demonstrates the advantage of complex model components in accounting for spatio‐temporal population dynamics (such as habitat preferences and dispersal limitations) that are not explained by environmental variables. However, including an AR1 temporal correlation structure in the models resulted in unrealistic predictions of species distribution; highlighting the need for careful consideration when determining the level of model complexity. Increasing model complexity, with careful evaluation of the effects of additional model components, can provide a more realistic representation of a system, which is of particular importance for a practical and impact‐focused discipline such as ecology (though these methods extend to applications for a wide range of systems). Founding complexity in contextual theory is not only fundamental to maintaining model interpretability but can be a useful approach to improving insight gained from model outputs.

Subordinating a multivariate Lévy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Lévy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Lévy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs (T, X) of Lévy processes which creates a Lévy process X ☉ T. Here, T is a subordinator, but X is an arbitrary Lévy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-α-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Lévy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.

In this paper, we consider the problem of computing a median of marked point process data under an edit distance. We formulate this problem as a binary linear program, and propose to solve it to optimality by software. We show results of numerical experiments to demonstrate the effectiveness of the proposed method and its application in earthquake prediction.

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.