Distinguishing two candidate models is a fundamental and practically important statistical problem. Error rate control is crucial to the testing logic but, in complex nonparametric settings, can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. This paper proposes an e-process construction based on the predictive recursion (PR) algorithm originally designed to recursively fit nonparametric mixture models. The resulting PRe-process affords anytime valid inference and is asymptotically efficient in the sense that its growth rate is first-order optimal relative to PR's mixture model.
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Most of the scientific literature on causal modeling considers the structural framework of Pearl and the potential-outcome framework of Rubin to be formally equivalent, and therefore interchangeably uses do-interventions and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we agnostically superimpose the two causal models to specify under which mathematical conditions structural counterfactual outcomes and potential outcomes need to, do not need to, can, or cannot be equal (almost surely or law). Our comparison reminds that a structural causal model and a Rubin causal model compatible with the same observations do not have to coincide, and highlights real-world problems where they even cannot correspond. Then, we examine common claims and practices from the causal-inference literature in the light of these results. In doing so, we aim at clarifying the relationship between the two causal frameworks, and the interpretation of their respective counterfactuals.
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.
A discrete spatial lattice can be cast as a network structure over which spatially-correlated outcomes are observed. A second network structure may also capture similarities among measured features, when such information is available. Incorporating the network structures when analyzing such doubly-structured data can improve predictive power, and lead to better identification of important features in the data-generating process. Motivated by applications in spatial disease mapping, we develop a new doubly regularized regression framework to incorporate these network structures for analyzing high-dimensional datasets. Our estimators can be easily implemented with standard convex optimization algorithms. In addition, we describe a procedure to obtain asymptotically valid confidence intervals and hypothesis tests for our model parameters. We show empirically that our framework provides improved predictive accuracy and inferential power compared to existing high-dimensional spatial methods. These advantages hold given fully accurate network information, and also with networks which are partially misspecified or uninformative. The application of the proposed method to modeling COVID-19 mortality data suggests that it can improve prediction of deaths beyond standard spatial models, and that it selects relevant covariates more often.
For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.
Boundary-preserving weak approximations of some semilinear stochastic partial differential equations
We propose and analyse a boundary-preserving numerical scheme for the weak approximations of some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter splitting followed by exact simulation and exact integration in time. We prove weak convergence of optimal order 1/4 for globally Lipschitz continuous test functions of the scheme by proving strong convergence towards a strong solution driven by a different noise process. Boundary-preservation is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing methods for SPDEs.
Principal component analysis (PCA) is a widely used dimension reduction method, but its performance is known to be non-robust to outliers. Recently, product-PCA (PPCA) has been shown to possess the efficiency-loss free ordering-robustness property: (i) in the absence of outliers, PPCA and PCA share the same asymptotic distributions; (ii), in the presence of outliers, PPCA is more ordering-robust than PCA in estimating the leading eigenspace. PPCA is thus different from the conventional robust PCA methods, and may deserve further investigations. In this article, we study the high-dimensional statistical properties of the PPCA eigenvalues via the techniques of random matrix theory. In particular, we derive the critical value for being distant spiked eigenvalues, the limiting values of the sample spiked eigenvalues, and the limiting spectral distribution of PPCA. Similar to the case of PCA, the explicit forms of the asymptotic properties of PPCA become available under the special case of the simple spiked model. These results enable us to more clearly understand the superiorities of PPCA in comparison with PCA. Numerical studies are conducted to verify our results.
Multi-agent systems (MAS) have gained relevance in the field of artificial intelligence by offering tools for modelling complex environments where autonomous agents interact to achieve common or individual goals. In these systems, norms emerge as a fundamental component to regulate the behaviour of agents, promoting cooperation, coordination and conflict resolution. This article presents a systematic review, following the PRISMA method, on the emergence of norms in MAS, exploring the main mechanisms and factors that influence this process. Sociological, structural, emotional and cognitive aspects that facilitate the creation, propagation and reinforcement of norms are addressed. The findings highlight the crucial role of social network topology, as well as the importance of emotions and shared values in the adoption and maintenance of norms. Furthermore, opportunities are identified for future research that more explicitly integrates emotional and ethical dynamics in the design of adaptive normative systems. This work provides a comprehensive overview of the current state of research on norm emergence in MAS, serving as a basis for advancing the development of more efficient and flexible systems in artificial and real-world contexts.
Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.
Sequential positivity is often a necessary assumption for drawing causal inferences, such as through marginal structural modeling. Unfortunately, verification of this assumption can be challenging because it usually relies on multiple parametric propensity score models, unlikely all correctly specified. Therefore, we propose a new algorithm, called "sequential Positivity Regression Tree" (sPoRT), to check this assumption with greater ease under either static or dynamic treatment strategies. This algorithm also identifies the subgroups found to be violating this assumption, allowing for insights about the nature of the violations and potential solutions. We first present different versions of sPoRT based on either stratifying or pooling over time. Finally, we illustrate its use in a real-life application of HIV-positive children in Southern Africa with and without pooling over time. An R notebook showing how to use sPoRT is available at github.com/ArthurChatton/sPoRT-notebook.
In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.
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