That Gaussian Markov random fields can be used to approximate Gaussian random fields with Mat & eacute;rn covariances functions gained widespread attention due to the seminal work of Lindgren, Rue, and Lindstr & ouml;m (J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 423-498). This was the culmination of a rich history within the statistical literature with pivotal contributions by Whittle (Biometrika 41 (1954) 434-449), Moran (J. Appl. Probab. 10 (1973) 54-62), and Besag (J. Roy. Statist. Soc. Ser. B (1974) 192-236). There developments often relied on analogies to time series analysis where the Markov properties of autoregressive processes were well understood. We revisit this literature while simultaneously offering an alternative demonstration of the result by Lindgren, Rue, and Lindstr & ouml;m (J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 423-498) that also relies on time series techniques, namely, spectral analysis. New insight is gained into understanding the role of the curse of dimensionality: as dimension increases, neighbourhoods of increasing size are required in order to exploit the GMRF-GRF approximation. Here the rigour of the differential equation argument in Lindgren, Rue, and Lindstr & ouml;m (J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 423-498) is sacrificed for the sake of clarity and simplicity. The result is accessible to students enrolled in their first course in time series analysis.
A family of exponential tilting density functions (ETD) is presented and compared with energy functions. This ETD family is shown to be associated with the normal density and the log-gamma density by the minimum cross entropy in information theory. In this paper, we show that ETD minimizes the Kulback-Leiber divergence under some moments constraints. Two examples are provided to illustrate how to approximate a baseline density using ETD. In addition, the normalizing constant of the ETD is approximated by three commonly used approximation methods: Gaussian variational approximation (GVA), Laplace approximation (LA) and saddlepoint approximation (SA). It is shown that the normalizing constant obtained by GVA and SA are asymptotically equivalent, and theoretically, both are more accurate than the one obtained by LA. With the availability of the normalizing constant, likelihood-based asymptotic inference can be obtained. To demonstrate the applicability of the proposed method, it is applied to parameter estimation of the Poisson mixed model and Bayesian inference.
Several indices of accuracy have been proposed to summarize ROC curves. Based on these indices, statistical tests have been developed to compare the accuracy of diagnostic tests. We briefly review three of them, AUC, Youden index, and the length of the ROC curve. Besides, we explore overlap measures as alternative measures for assessing the effectiveness of a diagnostic marker. Both parametric and nonparametric estimators as well as different methods for constructing confidence intervals are proposed for these measures. We show that they are related to distances and divergences and provide some properties. We then identify situations in which the overlap measures outperform the ROC summary indices through a simulation study. Furthermore, we compare the methods for constructing confidence intervals in terms of coverage and width. Our approaches are illustrated using a real data set.
Byron Morgan is an Emeritus Professor of Statistics, a Fellow of the Learned Society of Wales and represents a true pillar of the applied statistics community, contributing appreciable understanding to a number of fields, most recently statistical ecology. Byron has offered sustained service to the community, chairing the last Research Excellence Framework statistics panel in 2001 (after which statistics was merged with mathematics) and has reviewed and advised several external agencies at home and abroad. He has been editor of the Journal of Agricultural, Biological and Environmental Statistics, co-editor of Biometrics and of Applied Statistics and Associate Editor and Guest Editor for several other journals. He also served in multiple learned societies, most notably the International Biometric Society, (of which he was the president in 1996/1997 and Elected Honorary Life Member in 2014) and in the Royal Statistical Society, where he was a member of council and a vice-president during 1997-2001. His contributions to the organisation of scientific conferences are many and include his serving as Chair of Scientific Program Committee of the XVI-Ith International Biometric Conference in Hamilton, Ontario, 1994, and as Chair of the local organising committee for the International Statistical Ecology Conference, held in Canterbury in 2010. As well as leading many initiatives and grants, he has mentored and inspired generations of researchers, many of whom have gone on to research careers in both academia and NGOs. Anyone who has met Byron will not be surprised that it is a great pleasure to have an excuse to talk to him about his career and for him to relay some of his many entertaining stories.
We review some early contributions by Corrado Gini to modified binomial models and predictive probability and highlight their role, once extended to the multivariate context, for modelling voting behaviour in Ecological Inference. A collection of overdispersed multinomial models are described, their properties investigated and a connection to Gini's results for the corresponding binomial model, when available, is provided. After a concise introduction to Ecological Inference, we discuss recent developments aiming at more realistic models of voting behaviour and some connections to Gini's work.
In the field of approximate Bayesian inference, expectation propagation (EP) is an often overlooked counterpart to its older Laplace approximation and variational Bayes cousins, perhaps owing to a lack of theory (especially convergence guarantees) and a higher implementation overhead, where derivations need to be hand-crafted to the specific distributions being approximated. However, when EP is carefully implemented, it is often more accurate than these alternative approaches. With this in mind, the purpose of this review paper is to describe and to consolidate the current state of research on EP at a high level focusing on examples and applications. Our aim is for this broad-based, practical EP guide to encourage its novel application to both existing and new Bayesian problems, where it has the opportunity to dramatically extend the cutting edge in performance.
A fundamental assumption of classical hypothesis testing is that the significance threshold alpha is chosen independently from the data. The validity of confidence intervals likewise relies on choosing alpha beforehand. We point out that the independence of alpha is guaranteed in practice, because in most fields there exists one standard alpha that everyone uses-so that alpha is automatically independent of everything. However, there have been recent calls to decrease alpha from 0.05 to 0.005. We note that this may lead to multiple accepted standard thresholds within one scientific field. For example, different journals may require different significance thresholds. As a consequence, some researchers may be tempted to conveniently choose their alpha based on their p-value. We use examples to illustrate that this severely invalidates hypothesis tests, and mention some potential solutions.
In the analyses of cluster-randomized trials, mixed-model analysis of covariance (ANCOVA) is a standard approach for covariate adjustment and handling within-cluster correlations. However, when the normality, linearity, or the random-intercept assumption is violated, the validity and efficiency of the mixed-model ANCOVA estimators for estimating the average treatment effect remain unclear. Under the potential outcomes framework, we prove that the mixed-model ANCOVA estimators for the average treatment effect are consistent and asymptotically normal under arbitrary misspecification of its working model. If the probability of receiving treatment is 0.5 for each cluster, we further show that the model-based variance estimator under mixed-model ANCOVA1 (ANCOVA without treatment-covariate interactions) remains consistent, clarifying that the confidence interval given by standard software is asymptotically valid even under model misspecification. Beyond robustness, we discuss several insights on precision among classical methods for analyzing cluster-randomized trials, including the mixed-model ANCOVA, individual-level ANCOVA, and cluster-level ANCOVA estimators. These insights may inform the choice of methods in practice. Our analytical results and insights are illustrated via simulation studies and analyses of three cluster-randomized trials.
Nonparametric tests for functional data are a challenging class of tests to work with because of the potentially high dimensional nature of the data. One of the main challenges for considering rank-based tests, like the Mann-Whitney or Wilcoxon Rank Sum tests (MWW), is that the unit of observation is typically a curve. Thus any rank-based test must consider ways of ranking curves. While several procedures, including depth-based methods, have recently been used to create scores for rank-based tests, these scores are not constructed under the null and often introduce additional, uncontrolled for variability. We therefore reconsider the problem of rank-based tests for functional data and develop an alternative approach that incorporates the null hypothesis throughout. Our approach first ranks realizations from the curves at each measurement occurrence, then calculates a summary statistic for the ranks of each subject, and finally re-ranks the summary statistic in a procedure we refer to as a doubly ranked test. We propose two summaries for the middle step: a sufficient statistic and the average rank. As we demonstrate, doubly rank tests are more powerful while maintaining ideal type I error in the two sample, MWW setting. We also extend our framework to more than two samples, developing a Kruskal-Wallis test for functional data which exhibits good test characteristics as well. Finally, we illustrate the use of doubly ranked tests in functional data contexts from material science, climatology, and public health policy.
In many applications, researchers seek to identify overlapping entities across multiple data files. Record linkage algorithms facilitate this task, in the absence of unique identifiers. As these algorithms rely on semi-identifying information, they may miss records that represent the same entity, or incorrectly link records that do not represent the same entity. Analysis of linked files commonly ignores such linkage errors, resulting in biased, or overly precise estimates of the associations of interest. We view record linkage as a missing data problem, and delineate the linkage mechanisms that underpin analysis methods with linked files. Following the missing data literature, we group these methods under three categories: likelihood and Bayesian methods, imputation methods, and weighting methods. We summarize the assumptions and limitations of the methods, and evaluate their performance in a wide range of simulation scenarios.
This study investigates the extension of distance variance, a validated spread metric for continuous and binary variables [Edelmann et al., 2020, Ann. Stat., 48(6)], to quantify the spread of general categorical variables. We provide both geometric and algebraic characterizations of distance variance, revealing its connections to some commonly used entropy measures, and the variance-covariance matrix of the one-hot encoded representation. However, we demonstrate that distance variance fails to satisfy the Schur-concavity axiom for categorical variables with more than two categories, leading to counterintuitive results. This limitation hinders its applicability as a universal measure of spread.
Engineers and scientists have been collecting and analyzing fatigue data since the 1800s to ensure the reliability of life-critical structures. Applications include (but are not limited to) bridges, building structures, aircraft and spacecraft components, ships, ground-based vehicles, and medical devices. Engineers need to estimate S-N relationships (Stress or Strain versus Number of cycles to failure), typically with a focus on estimating small quantiles of the fatigue-life distribution. Estimates from this kind of model are used as input to models (e.g., cumulative damage models) that predict failure-time distributions under varying stress patterns. Also, design engineers need to estimate lower-tail quantiles of the closely related fatigue-strength distribution. The history of applying incorrect statistical methods is nearly as long and such practices continue to the present. Examples include treating the applied stress (or strain) as the response and the number of cycles to failure as the explanatory variable in regression analyses (because of the need to estimate strength distributions) and ignoring or otherwise mishandling censored observations (known as runouts in the fatigue literature). The first part of the paper reviews the traditional modeling approach where a fatigue-life model is specified. We then show how this specification induces a corresponding fatigue-strength model. The second part of the paper presents a novel alternative modeling approach where a fatigue-strength model is specified and a corresponding fatigue-life model is induced. We explain and illustrate the important advantages of this new modeling approach.
A very simple example demonstrates that Fisher's application of the conditionality principle to regression ("fixed x regression"), endorsed by Sprott and many other followers, makes prediction impossible in the context of statistical learning theory. On the other hand, relaxing the requirement of conditionality makes it possible via, e.g., conformal prediction.
The recent proliferation of computers and the internet have opened new opportunities for collecting and processing data. However, such data are often obtained without a well-planned probability survey design. Such non-probability based samples cannot be automatically regarded as representative of the population of interest. Several classes of methods for estimation and inferences from non-probability samples have been developed in recent years. The quasi-randomization methods assume that non-probability sample selection is governed by an underlying latent random mechanism. The basic idea is to use information collected from a probability ("reference") sample to uncover latent non-probability survey participation probabilities (also known as "propensity scores") and use them in estimation of target finite population parameters. In this paper, we review and compare theoretical properties of recently developed methods of estimation survey participation probabilities and study their relative performances in simulations.
Markov chain Monte Carlo significance tests were first introduced by Besag and Clifford in [4]. These methods produce statistical valid p-values in problems where sampling from the null hypotheses is intractable. We give an overview of the methods of Besag and Clifford and some recent developments. A range of examples and applications are discussed.
Many natural systems are observed as point patterns in time, space, or space and time. Examples include plant and cellular systems, animal colonies, earthquakes, and wildfires. In practice the locations of the points are not always observed correctly. However, in the point process literature, there has been relatively scant attention paid to the issue of errors in the location of points. In this paper, we discuss how the observed point pattern may deviate from the actual point pattern and review methods and models that exist to handle such deviations. The discussion is supplemented with several scientific illustrations.
Neural networks have achieved remarkable performance across various problem domains, but their widespread applicability is hindered by inherent limitations such as overconfidence in predictions, lack of interpretability, and vulnerability to adversarial attacks. To address these challenges, Bayesian neural networks (BNNs) have emerged as a compelling extension of conventional neural networks, integrating uncertainty estimation into their predictive capabilities. This comprehensive primer presents a systematic introduction to the fundamental concepts of neural networks and Bayesian inference, elucidating their synergistic integration for the development of BNNs. The target audience comprises statisticians with a potential background in Bayesian methods but lacking deep learning expertise, as well as machine learners proficient in deep neural networks but with limited exposure to Bayesian statistics. We provide an overview of commonly employed priors, examining their impact on model behavior and performance. Additionally, we delve into the practical considerations associated with training and inference in BNNs. Furthermore, we explore advanced topics within the realm of BNN research, acknowledging the existence of ongoing debates and controversies. By offering insights into cutting-edge developments, this primer not only equips researchers and practitioners with a solid foundation in BNNs, but also illuminates the potential applications of this dynamic field. As a valuable resource, it fosters an understanding of BNNs and their promising prospects, facilitating further advancements in the pursuit of knowledge and innovation.
The classical Mat\'ern model has been a staple in spatial statistics. Novel data-rich applications in environmental and physical sciences, however, call for new, flexible vector-valued spatial and space-time models. Therefore, the extension of the classical Mat\'ern model has been a problem of active theoretical and methodological interest. In this paper, we offer a new perspective to extending the Mat\'ern covariance model to the vector-valued setting. We adopt a spectral, stochastic integral approach, which allows us to address challenging issues on the validity of the covariance structure and at the same time to obtain new, flexible, and interpretable models. In particular, our multivariate extensions of the Mat\'ern model allow for time-irreversible or, more generally, asymmetric covariance structures. Moreover, the spectral approach provides an essentially complete flexibility in modeling the local structure of the process. We establish closed-form representations of the cross-covariances when available, compare them with existing models, simulate Gaussian instances of these new processes, and demonstrate estimation of the model's parameters through maximum likelihood. An application of the new class of multivariate Mat\'ern models to environmental data indicate their success in capturing inherent covariance-asymmetry phenomena.
This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful for different purposes, pertaining to estimation and testing, the characterization of cluster structure in community detection, and understanding latent space geometry. This paper concludes with a discussion of open problems and ongoing research activities addressing asymptotic normality and its implications for statistical network modeling.
