Papers

Kernel-based methods are used in the context of Granger Causality to enable the identification of nonlinear causal relationships between time series variables. In this paper, we show that two state of the art kernel-based Granger Causality (GC) approaches can be theoretically unified under the framework of Kernel Principal Component Regression (KPCR), and introduce a method based on this unification, demonstrating that this approach can improve causal identification. Additionally, we introduce a Gaussian Process score-based model with Smooth Information Criterion penalisation on the marginal likelihood, and demonstrate improved performance over existing state of the art time-series nonlinear causal discovery methods. Furthermore, we propose a contemporaneous causal identification algorithm, fully based on GC, using the proposed score-based $GP_{SIC}$ method, and compare its performance to a state of the art contemporaneous time series causal discovery algorithm.

In Astronomy, Survival Analysis and Epidemiology, among many other fields, doubly truncated data often appear. Double truncation generally induces a sampling bias, so ordinary estimators may be inconsistent. In this paper, smoothing spline density estimation from doubly truncated data is investigated. For this purpose, an appropriate correction of the penalized likelihood that accounts for the sampling bias is considered. The theoretical properties of the estimator are discussed, and its practical performance is evaluated through simulations. Two real datasets are analyzed using the proposed method for illustrative purposes. Comparison to kernel density smoothing is included.

The hazard ratio, typically estimated using Cox's famous proportional hazards model, is the most common effect measure used to describe the association or effect of a covariate on a time-to-event outcome. In recent years the hazard ratio has been argued by some to lack a causal interpretation, even in randomised trials, and even if the proportional hazards assumption holds. This is concerning, not least due to the ubiquity of hazard ratios in analyses of time-to-event data. We review these criticisms, describe how we think hazard ratios should be interpreted, and argue that they retain a valid causal interpretation. Nevertheless, alternative measures may be preferable to describe effects of exposures or treatments on time-to-event outcomes.

Large-scale neuroimaging studies often collect data from multiple scanners across different sites, where variations in scanners, scanning procedures, and other conditions across sites can introduce artificial site effects. These effects may bias brain connectivity measures, such as functional connectivity (FC), which quantify functional network organization derived from functional magnetic resonance imaging (fMRI). How to leverage high-dimensional network structures to effectively mitigate site effects has yet to be addressed. In this paper, we propose SLACC (Sparse LAtent Covariate-driven Connectome) factorization, a multivariate method that explicitly parameterizes covariate effects in latent subject scores corresponding to sparse rank-1 latent patterns derived from brain connectivity. The proposed method identifies localized site-driven variability within and across brain networks, enabling targeted correction. We develop a penalized Expectation-Maximization (EM) algorithm for parameter estimation, incorporating the Bayesian Information Criterion (BIC) to guide optimization. Extensive simulations validate SLACC's robustness in recovering the true parameters and underlying connectivity patterns. Applied to the Autism Brain Imaging Data Exchange (ABIDE) dataset, SLACC demonstrates its ability to reduce site effects.

Estimands can help to clarify the research questions being addressed in randomised trials. Because the choice of estimand can affect how relevant trial results are to patients and other stakeholders, such as clinicians or policymakers, it is important for them to be involved in these decisions. However, there are barriers to having these conversations. For instance, discussions around how intercurrent events should be addressed in the estimand definition typically involve complex concepts as well as technical language. We three tools to facilitate conversations between researchers and patients and other stakeholders about the choice of estimand and intercurrent event strategy: (i) a video explaining the concept of an estimand and the five different ways that intercurrent events can be incorporated into the estimand definition; (ii) an infographic outlining these five strategies; and (iii) an editable PowerPoint slide which can be completed with trial-specific details to facilitate conversations around choice of estimand for a particular trial. These resources can help to start conversations between the trial team and patients and other stakeholders about the best choice of estimand and intercurrent event strategies for a randomised trial.

We introduce a novel class of nonlinear tests for serial dependence in functional time series, grounded in the functional quantile autocorrelation framework. Unlike traditional approaches based on the classical autocovariance kernel, the functional quantile autocorrelation framework leverages quantile-based excursion sets to robustly capture temporal dependence within infinite-dimensional functional data, accommodating potential outliers and complex nonlinear dependencies. We propose omnibus test statistics and study their asymptotic properties under both known and estimated quantile curves, establishing their asymptotic distribution and consistency under mild assumptions. In particular, no moment conditions are required for the validity of the tests. Extensive simulations and an application to high-frequency financial functional time series demonstrate the methodology's effectiveness, reliably detecting complex serial dependence with superior power relative to several existing tests. This work expands the toolkit for functional time series, providing a robust framework for inference in settings where traditional methods may fail.

We study the problem of finding the minimum $\mathbb{L}\_2$-norm element in a transportation polytope. It corresponds to minimizing the index of coincidence or maximizing Rényi entropy over joint probability distributions with prescribed marginals. The solution is straightforward under a feasibility condition on the marginals. We make three contributions. First, we decompose the basic solution into a mixture of three interpretable couplings and derive a closed-form sampling procedure, revealing its role as the $\mathbb{L}\_2$-projection of the uniform distribution onto the transportation polytope. Second, we show that the feasibility condition becomes exponentially rare as the number of modalities grows, which highlights the practical necessity of a more general construction. Third, we provide a KKT-based iterative construction of the optimal coupling for arbitrary marginals with explicit Lagrange multipliers.

Surface anomaly detection using 3D point cloud data has gained increasing attention in industrial inspection. However, most existing methods rely on deep learning techniques that are highly dependent on large-scale datasets for training, which are difficult and expensive to acquire in real-world applications. To address this challenge, we propose a novel method based on self-organizing network for 3D anomaly ranking, also named 3D-SONAR. The core idea is to model the 3D point cloud as a dynamic system, where the points are represented as an undirected graph and interact via attractive and repulsive forces. The energy distribution induced by these forces can reveal surface anomalies. Experimental results show that our method achieves superior anomaly detection performance in both open surface and closed surface without training. This work provides a new perspective on unsupervised inspection and highlights the potential of physics-inspired models in industrial anomaly detection tasks with limited data.

Marked Temporal Point Processes (MTPPs) arise naturally in medical, social, commercial, and financial domains. However, existing Transformer-based methods mostly inject temporal information only via positional encodings, relying on shared or parametric decay structures, which limits their ability to capture heterogeneous and type-specific temporal effects. Inspired by this observation, we derive a novel attention operator called Hawkes Attention from the multivariate Hawkes process theory for MTPP, using learnable per-type neural kernels to modulate query, key and value projections, thereby replacing the corresponding parts in the traditional attention. Benefited from the design, Hawkes Attention unifies event timing and content interaction, learning both the time-relevant behavior and type-specific excitation patterns from the data. The experimental results show that our method achieves better performance compared to the baselines. In addition to the general MTPP, our attention mechanism can also be easily applied to specific temporal structures, such as time series forecasting.

Representational similarity methods compare the geometries of neural representations, but they do not measure how consistently the geometry of a single representation is recovered from subsets of its feature coordinates. We call this property geometric stability and introduce Shesha, which estimates it by correlating representational dissimilarity matrices from complementary random feature subsets. Shesha is not invariant to orthogonal rotations: representations with identical Gram matrices, and therefore identical linear CKA, can have different geometric stability. Controlled transformations further separate the quantities. Across $2{,}463$ encoder configurations spanning seven domains, similarity and stability are positively associated across non-PCA transformations ($\rho=+0.75$) but negatively associated under PCA-coordinate compression ($\rho=-0.47$). We further evaluate 170 pretrained vision models across six datasets. DINOv2 combines strong transfer performance with bottom-quartile stability on five of six datasets, showing that transferability and feature-split stability need not coincide. Across random feature subsets, the marginal relationship between Shesha and linear-probe variability is dataset-dependent; after controlling for task alignment with LogME, higher Shesha is associated with lower variability on five of six datasets. These results identify geometric stability as a basis-dependent property that complements representational similarity and task alignment.

We study the Bandit Clustering (BC) problem under the fixed confidence setting, where the objective is to group a collection of data sequences (arms) into clusters through sequential sampling from adaptively selected arms at each time step while ensuring a fixed error probability at the stopping time. We consider a setting where arms in a cluster may have different distributions. Unlike existing results in this setting, which assume Gaussian-distributed arms, we study a broader class of vector-parametric distributions that satisfy mild regularity conditions. Existing asymptotically optimal BC algorithms require solving an optimization problem as part of their sampling rule at each step, which is computationally costly. We propose an Efficient Bandit Clustering algorithm (EBC), which, instead of solving the full optimization problem, takes a single step toward the optimal value at each time step, making it computationally efficient while remaining asymptotically optimal. We also propose a heuristic variant of EBC, called EBC-H, which further simplifies the sampling rule, with arm selection based on quantities computed as part of the stopping rule. We highlight the computational efficiency of EBC and EBC-H by comparing their per-sample run time with that of existing algorithms. The asymptotic optimality of EBC is supported through simulations on the synthetic datasets. Through simulations on both synthetic and real-world datasets, we show the performance gain of EBC and EBC-H over existing approaches.

Community detection in multilayer networks, which aims to identify groups of nodes exhibiting similar connectivity patterns across multiple network layers, has attracted considerable attention in recent years. Most existing methods are based on the assumption that different layers are either independent or follow specific dependence structures, and edges within the same layer are independent. In this article, we propose a novel method for community detection in multilayer networks that accounts for a broad range of inter-layer and intra-layer dependence structures. The proposed method integrates the multilayer stochastic block model for community detection with a multivariate probit model to capture the structures of inter-layer dependence, which also allows intra-layer dependence. To facilitate parameter estimation, we develop a constrained pairwise likelihood method coupled with an efficient alternating updating algorithm. The asymptotic properties of the proposed method are also established, with a focus on examining the influence of inter-layer and intra-layer dependences on the accuracy of both parameter estimation and community detection. The theoretical results are supported by extensive numerical experiments on both simulated networks and a real-world multilayer trade network.

Distributional representations of data collected using digital health technologies have been shown to outperform scalar summaries for clinical prediction, with carefully quantified tail-behavior often driving the gains. Motivated by these findings, we propose a unified generalized odds (GO) framework that represents subject-specific distributions through ratios of probabilities over arbitrary regions of the sample space, subsuming hazard, survival, and residual life representations as special cases. We develop a scale-on-odds regression model using spline-based functional representations with penalization for efficient estimation. Applied to wrist-worn accelerometry data from the HEAL-MS study, generalized odds models yield improved prediction of Expanded Disability Status Scale (EDSS) scores compared to classical scalar and survival-based approaches, demonstrating the value of odds-based distributional covariates for modeling DHT data.

Posterior tempering reduces the influence of the likelihood in the calculation of the posterior by raising the likelihood to a fractional power $\alpha$. The resulting power posterior - also known as an $\alpha$-posterior or fractional posterior - has been shown to exhibit appealing properties, including robustness to model misspecification and asymptotic normality (Bernstein-von Mises theorem). However, practical recommendations for selecting the tempering parameter and statistical guarantees for the resulting power posterior remain open questions. Cross-validation-based approaches to tuning this parameter suggest interesting asymptotic regimes for the selected $\alpha$, which can either vanish or behave like a mixture distribution with a point mass at infinity and the remaining mass converging to zero. We formalize the asymptotic properties of the power posterior in these regimes. In particular, we provide sufficient conditions for (i) consistency of the power posterior moments and (ii) asymptotic normality of the power posterior mean. Our analysis required us to establish a new Laplace approximation that is interesting in its own right and is the key technical tool for showing a critical threshold $\alpha\asymp 1/\sqrt{n}$ where the asymptotic normality of the posterior mean breaks. Our results allow for the power to depend on the data in an arbitrary way.

Developing safe, effective, and practically useful clinical prediction models (CPMs) traditionally requires iterative collaboration between clinical experts, data scientists, and informaticists. This process refines the often small but critical details of the model building process, such as which features/patients to include and how clinical categories should be defined. However, this traditional collaboration process is extremely time- and resource-intensive, resulting in only a small fraction of CPMs reaching clinical practice. This challenge intensifies when teams attempt to incorporate unstructured clinical notes, which can contain an enormous number of concepts. To address this challenge, we introduce HACHI, an iterative human-in-the-loop framework that uses AI agents to accelerate the development of fully interpretable CPMs by enabling the exploration of concepts in clinical notes. HACHI alternates between (i) an AI agent rapidly exploring and evaluating candidate concepts in clinical notes and (ii) clinical and domain experts providing feedback to improve the CPM learning process. HACHI defines concepts as simple yes-no questions that are used in linear models, allowing the clinical AI team to transparently review, refine, and validate the CPM learned in each round. In two real-world prediction tasks (acute kidney injury and traumatic brain injury), HACHI outperforms existing approaches, surfaces new clinically relevant concepts not included in commonly-used CPMs, and improves model generalizability across clinical sites and time periods. Furthermore, HACHI reveals the critical role of the clinical AI team, such as directing the AI agent to explore concepts that it had not previously considered, adjusting the granularity of concepts it considers, changing the objective function to better align with the clinical objectives, and identifying issues of data bias and leakage.

Horseshoe mixtures-of-experts (HS-MoE) models provide a Bayesian framework for sparse expert selection in mixture-of-experts architectures. We combine the horseshoe prior's adaptive global-local shrinkage with input-dependent gating, yielding data-adaptive sparsity in expert usage. Our primary methodological contribution is a particle learning algorithm for sequential inference, in which the filter is propagated forward in time while tracking only sufficient statistics. We also discuss how HS-MoE relates to modern mixture-of-experts layers in large language models, which are deployed under extreme sparsity constraints (e.g., activating a small number of experts per token out of a large pool).

This work addresses the fundamental problem of unbounded metric movement costs in bandit online convex optimization, by considering high-dimensional dynamic quadratic hitting costs and $\ell_2$-norm switching costs in a noisy bandit feedback model. For a general class of stochastic environments, we provide the first algorithm SCaLE that provably achieves a distribution-agnostic sub-linear dynamic regret, without the knowledge of hitting cost structure. En-route, we present a novel spectral regret analysis that separately quantifies eigenvalue-error driven regret and eigenbasis-perturbation driven regret. Extensive numerical experiments, against online-learning baselines, corroborate our claims, and highlight statistical consistency of our algorithm.

We propose graph canonical coherence analysis (gCChA), a novel framework that extends canonical correlation analysis to multivariate graph signals in the graph frequency domain. The proposed method addresses challenges posed by the inherent features of graphs: discreteness, finiteness, and irregularity. It identifies pairs of canonical graph signals that maximize their coherence, enabling the exploration of relationships between two sets of graph signals from a spectral perspective. This framework shows how these relationships change across different structural scales of the graph. We demonstrate the usefulness of this method through applications to economic and energy datasets of G20 countries and the USPS handwritten digit dataset.

Named Entity Recognition (NER) serves as a foundational component in many natural language processing (NLP) pipelines. However, current NER models typically output a single predicted label sequence without any accompanying measure of uncertainty, leaving downstream applications vulnerable to cascading errors. In this paper, we introduce a general framework for adapting sequence-labeling-based NER models to produce uncertainty-aware prediction sets. These prediction sets are collections of full-sentence labelings that are guaranteed to contain the correct labeling with a user-specified confidence level. This approach serves a role analogous to confidence intervals in classical statistics by providing formal guarantees about the reliability of model predictions. Our method builds on conformal prediction, which offers finite-sample coverage guarantees under minimal assumptions. We design efficient nonconformity scoring functions to construct efficient, well-calibrated prediction sets that support both unconditional and class-conditional coverage. This framework accounts for heterogeneity across sentence length, language, entity type, and number of entities within a sentence. Empirical experiments on four NER models across three benchmark datasets demonstrate the broad applicability, validity, and efficiency of the proposed methods.

Hamiltonian Monte Carlo (HMC) algorithms are among the most widely used sampling methods in high dimensional settings, yet their convergence properties are poorly understood in divergences that quantify relative density mismatch, such as Kullback-Leibler (KL) and Rényi divergences. These divergences naturally govern acceptance probabilities and warm-start requirements for Metropolis-adjusted Markov chains. In this work, we develop a framework for upgrading Wasserstein convergence guarantees for unadjusted Hamiltonian Monte Carlo (uHMC) to guarantees in tail-sensitive KL and Rényi divergences. Our approach is based on one-shot couplings, which we use to establish a regularization property of the uHMC transition kernel. This regularization allows Wasserstein-2 mixing-time and asymptotic bias bounds to be lifted to KL divergence, and analogous Orlicz-Wasserstein bounds to be lifted to Rényi divergence, paralleling earlier work of Bou-Rabee and Eberle (2023) that upgrade Wasserstein-1 bounds to total variation distance via kernel smoothing. As a consequence, our results provide quantitative control of relative density mismatch, clarify the role of discretization bias in strong divergences, and yield principled guarantees relevant both for unadjusted sampling and for generating warm starts for Metropolis-adjusted Markov chains.

Sarmanov copulas offer a simple and tractable way to build multivariate distributions by perturbing the independence copula. They admit closed-form expressions for densities and many functionals of interest, making them attractive for practical applications. However, the complex conditions on the dependence parameters to ensure that Sarmanov copulas are valid limit their application in high dimensions. Verifying the $d$-increasing property typically requires satisfying a combinatorial set of inequalities that makes direct construction difficult. To circumvent this issue, we develop a stochastic representation for bivariate Sarmanov copulas. We prove that every admissible Sarmanov can be realized as a mixture of independent univariate distributions indexed by a latent Bernoulli pair. The stochastic representation replaces the problem of verifying copula validity with the problem of ensuring nonnegativity of a Bernoulli probability mass function. The representation also recovers classical copula families, including Farlie–Gumbel–Morgenstern, Huang–Kotz, and Bairamov–Kotz–Bekçi as special cases. We further derive sharp global bounds for Spearman's rho and Kendall's tau. We then introduce a Bernoulli-mixing construction in higher dimensions, leading to a new class of multivariate Sarmanov copulas with easily verifiable parameter constraints and scalable simulation algorithms. Finally, we show that powered versions of bivariate Sarmanov copulas admit a similar stochastic representation through block-maximal order statistics.

Analysis often splits change into components. For example, how much of the observed variance is caused by genes or environment? In many cases, the split is ultimately made by the logic of the chain rule, which divides the difference of a product into two terms. Each term quantifies the partial difference associated with change in one component while holding the other component constant. The chain rule is of course widely known. However, this article argues that its deep fundamental role often goes unrecognized. The article shows how simply the basic chain rule unifies Fisher's fundamental theorem of natural selection, the Price equation description of evolutionary change, the Oaxaca-Blinder decomposition of wage differences in economics, the Kitagawa decomposition of mortality differences in demography, many expressions of thermodynamics, and most strikingly back propagation, the core optimization method of modern machine learning and artificial intelligence. The success in creating good designs and finding good solutions in both natural selection and artificial intelligence depends on how the chain rule propagates causes from instances of success or failure back to the underlying genes or parameters of the system. The mathematical analysis presented here shows that, for finite differences, the product rule form of the chain rule yields a basic decomposition of change into two components of a regression equation. That regression decomposition is purely a description of change with no explicit causal meaning. However, simple additional assumptions lead naturally to the modern counterfactual analysis of causality. From that perspective, we can easily understand the causal interpretation that Fisher gave to his fundamental theorem, and we can see the same causal structure in the Oaxaca-Blinder decomposition of economics and in causal analyses across many disciplines.

Non-linear statistical inverse problems pose major challenges both for statistical analysis and computation. Likelihood-based estimators typically lead to non-convex and possibly multimodal optimization landscapes, and Markov chain Monte Carlo (MCMC) methods may mix exponentially slowly. We propose a class of computationally tractable estimators–plug-in and PDE-penalized M-estimators–for inverse problems defined through operator equations of the form $L_f u = g$, where $f$ is the unknown parameter and $u$ is the observed solution. The key idea is to replace the exact PDE constraint by a weakly enforced relaxation, yielding conditionally convex and, in many PDE examples, nested quadratic optimization problems that avoid evaluating the forward map $G(f)$ and do not require PDE solvers. For prototypical non-linear inverse problems arising from elliptic PDEs, including the Darcy flow model $L_f u = \nabla\!\cdot(f\nabla u)$ and a steady-state Schrödinger model, we prove that these estimators attain the best currently known statistical convergence rates while being globally computable in polynomial time. In the Darcy model, we obtain an explicit sub-quadratic $o(N^2)$ arithmetic runtime bound for estimating $f$ from $N$ noisy samples. Our analysis is based on new generalized stability estimates, extending classical stability beyond the range of the forward operator, combined with tools from nonparametric M-estimation. We also derive adaptive rates for the Darcy problem, providing a blueprint for designing provably polynomial-time statistical algorithms for a broad class of non-linear inverse problems. Our estimators also provide principled warm-start initializations for polynomial-time Bayesian computation.

Interval-censored covariates are frequently encountered in biomedical studies, particularly in time-to-event data or when measurements are subject to detection or quantification limits. Yet, the estimation of regression models with interval-censored covariates remains methodologically underdeveloped. In this article, we address the estimation of generalized linear models when one covariate is subject to interval censoring. We propose a likelihood-based approach, GELc, that builds upon an augmented version of Turnbull's nonparametric estimator for interval-censored data. We prove that the GELc estimator is consistent and asymptotically normal under mild regularity conditions, with available standard errors. Simulation studies demonstrate favorable finite-sample performance of the estimator and satisfactory coverage of the confidence intervals. Finally, we illustrate the method using two real-world applications: the AIDS Clinical Trials Group Study 359 and an observational nutrition study on circulating carotenoids. The proposed methodology is available as an R package at github.com/atoloba/ICenCov.

In this paper I consider improving the KernelSHAP algorithm. I suggest to use the Wallenius' noncentral hypergeometric distribution for sampling the number of coalitions and perform sampling without replacement, so that the KernelSHAP estimation framework is improved further. I also introduce the Symmetric bootstrap to calculate the standard deviations and also use the Doubled half bootstrap method to compare the performance. The new bootstrap algorithm performs better or equally well in the two simulation studies performed in this paper. The new KernelSHAP algorithm performs similarly as the improved KernelSHAP method in the state-of-the-art R-package shapr, which samples coalitions with replacement in one of the options

Multilabel Classification (MLC) deals with the simultaneous classification of multiple binary labels. The task is challenging because, not only may there be arbitrarily different and complex relationships between predictor variables and each label, but associations among labels may exist even after accounting for effects of predictor variables. In this paper, we present a Bayesian additive regression tree (BART) framework to model the problem. BART is a nonparametric and flexible model structure capable of uncovering complex relationships within the data. Our adaptation, MLCBART, assumes that labels arise from thresholding an underlying numeric scale, where a multivariate normal model allows explicit estimation of the correlation structure among labels. This enables the discovery of complicated relationships in various forms and improves MLC predictive performance. Our Bayesian framework not only enables uncertainty quantification for each predicted label, but our MCMC draws produce an estimated conditional probability distribution of label combinations for any predictor values. Simulation experiments demonstrate the effectiveness of the proposed model by comparing its performance with a set of models, including the oracle model with the correct functional form. Results show that our model predicts vectors of labels more accurately than other contenders and its performance is close to the oracle model. An example highlights how the method's ability to produce measures of uncertainty on predictions provides nuanced understanding of classification results.

Machine Learning algorithms are ubiquitous in key decision-making contexts such as justice, healthcare and finance, which has spawned a great demand for fairness in these procedures. However, the theoretical properties of such models in relation with fairness are still poorly understood, and the intuition behind the relationship between group and individual fairness is still lacking. In this paper, we provide a theoretical framework based on Sheaf Diffusion to leverage tools based on dynamical systems and homology to model fairness. Concretely, the proposed method projects input data into a bias-free space that encodes fairness constrains, resulting in fair solutions. Furthermore, we present a collection of network topologies handling different fairness metrics, leading to a unified method capable of dealing with both individual and group bias. The resulting models have a layer of interpretability in the form of closed-form expressions for their SHAP values, consolidating their place in the responsible Artificial Intelligence landscape. Finally, these intuitions are tested on a simulation study and standard fairness benchmarks, where the proposed methods achieve satisfactory results. More concretely, the paper showcases the performance of the proposed models in terms of accuracy and fairness, studying available trade-offs on the Pareto frontier, checking the effects of changing the different hyper-parameters, and delving into the interpretation of its outputs.

Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in (Annales de l'Institut Henri Poincare Probability and Statistics, 56(2):1465-1483, 2020). We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.

We revisit the null distribution of the high-dimensional spatial-sign test of Wang et al. (2015) under mild structural assumptions on the scatter matrix. We show that the standardized test statistic converges to a non-Gaussian limit, characterized as a mixture of a normal component and a weighted chi-square component. To facilitate practical implementation, we propose a wild bootstrap procedure for computing critical values and establish its asymptotic validity. Numerical experiments demonstrate that the proposed bootstrap test delivers accurate size control across a wide range of dependence settings and dimension-sample-size regimes.

Integrating probability and non-probability samples is increasingly important, yet unknown non-probability inclusion mechanisms complicate identification and efficient estimation. We develop semiparametric theory for dual-frame data integration and propose two complementary estimators. The first models the conditional inclusion probability parametrically and, under correct specification, attains the semiparametric efficiency bound. The probability sample serves as an outcome-observed validation sample, identifying the conditional inclusion parameters without instrumental variables even under informative selection. We derive the efficient influence function while leaving the conditional distribution of the design inclusion probability unrestricted. The resulting estimating equations satisfy exact augmentation invariance, which supports both same-sample sieve estimation and pooled cross-fitting under weak nuisance-rate conditions. The second estimator, motivated by a two-stage sampling approximation, does not require a non-probability inclusion model; although not fully efficient, it is efficient within a restricted augmentation class, is robust to misspecification of the non-probability inclusion model, and its oracle version weakly dominates optimally augmented probability-sample-only inference. Simulations and a public-data repeated-sampling study show efficiency gains under correct specification and stable performance under misspecification and weak identification. Across the main simulations, the efficient procedures substantially reduce RMSE relative to probability-sample-only estimation, while the restricted estimator remains stable under inclusion-model misspecification and weak identification.