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Distinguishing two classes of candidate models is a fundamental and practically important problem in statistical inference. Error rate control is crucial to the logic but, in complex nonparametric settings, such guarantees can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. In this paper we develop a novel e-value construction that leverages the so-called predictive recursion (PR) algorithm designed to recursively fit nonparametric mixture models. The resulting PRe-value affords anytime valid inference uniformly over stopping rules and is shown to be efficient in the sense that it achieves the maximal growth rate under the alternative relative to the mixture model being fit by PR. In the special case of testing the density for log-concavity, the PRe-value test is shown empirically to be significantly more efficient than a recently proposed anytime valid test based on universal inference.

In this paper, we consider objective Bayesian inference of the generalized exponential distribution using the independence Jeffreys prior and validate the propriety of the posterior distribution under a family of structured priors. We propose an efficient sampling algorithm via the generalized ratio-of-uniforms method to draw samples for making posterior inference. We carry out simulation studies to assess the finite-sample performance of the proposed Bayesian approach. Finally, a real-data application is provided for illustrative purposes.

We propose a new class of models for variable clustering called Asymptotic Independent block (AI-block) models, which defines population-level clusters based on the independence of the maxima of a multivariate stationary mixing random process among clusters. This class of models is identifiable, meaning that there exists a maximal element with a partial order between partitions, allowing for statistical inference. We also present an algorithm for recovering the clusters of variables without specifying the number of clusters \emph{a priori}. Our work provides some theoretical insights into the consistency of our algorithm, demonstrating that under certain conditions it can effectively identify clusters in the data with a computational complexity that is polynomial in the dimension. This implies that groups can be learned nonparametrically in which block maxima of a dependent process are only sub-asymptotic. To further illustrate the significance of our work, we applied our method to neuroscience and environmental real-datasets. These applications highlight the potential and versatility of the proposed approach.

Principal variables analysis (PVA) is a technique for selecting a subset of variables that capture as much of the information in a dataset as possible. Existing approaches for PVA are based on the Pearson correlation matrix, which is not well-suited to describing the relationships between non-Gaussian variables. We propose a generalized approach to PVA enabling the use of different types of correlation, and we explore using Spearman, Gaussian copula, and polychoric correlations as alternatives to Pearson correlation when performing PVA. We compare performance in simulation studies varying the form of the true multivariate distribution over a wide range of possibilities. Our results show that on continuous non-Gaussian data, using generalized PVA with Gaussian copula or Spearman correlations provides a major improvement in performance compared to Pearson. Meanwhile, on ordinal data, generalized PVA with polychoric correlations outperforms the rest by a wide margin. We apply generalized PVA to a dataset of 102 clinical variables measured on individuals with X-linked dystonia parkinsonism (XDP), a rare neurodegenerative disorder, and we find that using different types of correlation yields substantively different sets of principal variables.

A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.

We present here a new splitting method to solve Lyapunov equations in a Kronecker product form. Although this resulting matrix is of order $n^2$, each iteration demands two operations with the matrix $A$: a multiplication of the form $(A-\sigma I) \tilde{B}$ and a inversion of the form $(A-\sigma I)^{-1}\tilde{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1. Moreover we present a theorem that enables us to get a good starting vector for the method.

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.

A sequential pattern with negation, or negative sequential pattern, takes the form of a sequential pattern for which the negation symbol may be used in front of some of the pattern's itemsets. Intuitively, such a pattern occurs in a sequence if negated itemsets are absent in the sequence. Recent work has shown that different semantics can be attributed to these pattern forms, and that state-of-the-art algorithms do not extract the same sets of patterns. This raises the important question of the interpretability of sequential pattern with negation. In this study, our focus is on exploring how potential users perceive negation in sequential patterns. Our aim is to determine whether specific semantics are more "intuitive" than others and whether these align with the semantics employed by one or more state-of-the-art algorithms. To achieve this, we designed a questionnaire to reveal the semantics' intuition of each user. This article presents both the design of the questionnaire and an in-depth analysis of the 124 responses obtained. The outcomes indicate that two of the semantics are predominantly intuitive; however, neither of them aligns with the semantics of the primary state-of-the-art algorithms. As a result, we provide recommendations to account for this disparity in the conclusions drawn.

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.