Papers

This paper studies survival of certain bird species (Zebra Finches) under different food availability conditions. A competing risks model is studied under improved adaptive type-II progressive censoring scheme (IAT-II PCS) referring to this phenomena. Two independent competing causes of failure are considered where lifetime of these failures are assumed to follow Gompertz distribution with unknown scale and shape parameters. Maximum likelihood estimators (MLEs) of the unknown parameters are derived. It is established that they exist uniquely. Asymptotic confidence intervals (ACIs) are also constructed using asymptotic normality property of the MLE. Bayes estimates are obtained with respect to both non-informative and informative priors under different loss functions. Highest posterior density (HPD) credible intervals are calculated. A Monte Carlo simulation study is conducted to compare the performance of the proposed estimates. Three optimality criteria are studied to obtain the optimal censoring scheme. Finally, a real life data set is analyzed for further illustrations.

Expected Shortfall (ES) has become a core coherent risk measure in finance and statistics, and high-dimensional ES regression is crucial for characterizing heterogeneous tail risk with massive covariates. Existing offline methods for high-dimensional ES regression rely on access to full data, which fails under streaming data scenarios with sequential batch arrival and limited storage. To address this issue, this paper proposes a renewable estimation and inference framework for high-dimensional ES regression tailored to streaming data. By optimizing a surrogate loss function determined only by current data and historical information, the proposed procedure updates the estimator of ES regression coefficients without storing full raw data. Based on the online estimator, we design an online debiased estimator and further construct valid Wald-type confidence intervals using consistent variance estimation. Theoretically, we establish non-asymptotic error bounds for the online high-dimensional ES estimator and verify the asymptotic normality of the online debiased estimator. Extensive simulations show that the proposed method achieves estimation accuracy and inference performance comparable to the offline benchmark. Moreover, an application on the car insurance claim dataset demonstrates strong practical value in insurance risk management.

The focused information criterion is used to make a choice among several statistical models, or among several variables to include in a model. Different from other such information criteria, the focused information criterion is constructed to select the best model for a given interest quantity, the focus of the research question. Different such focus parameters may lead to different selected models, each one best for the corresponding focus. What is `best' is defined by a risk function, often the mean squared error. Other risks can be considered too for focused selection. Selections by the focused information criterion include using parametric (generalized) linear models, non- and semiparametric models, quantile regression models, graphical models, models for survival data, for longitudinal data, time series models, and many more. Extensions of the basic version include versions for high-dimensional data, regularized estimation, and Bayesian methods.

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.

Bayesian group sequential designs (GSDs) combine the efficiency of frequentist GSDs with clinically interpretable probability statements and principled external evidence incorporation. However, their uptake in confirmatory trials has been held back by the cost of evaluating frequentist operating characteristics at the design stage, which often nests Markov chain Monte Carlo or another approximate posterior inference within a Monte Carlo trial-simulation loop. We introduce adabay, an open-source R package implementing a semi-simulation framework for the rapid evaluation and calibration of Bayesian GSDs. Trial data paths are simulated by Monte Carlo, while per-look posteriors and posterior tail probabilities are computed analytically or by low-dimensional deterministic quadrature. Flexible prior specification is achieved by approximating any user-specified prior with a finite mixture of conjugate components, with tail-probability diagnostics that flag inadequate approximations at the decision thresholds. The package offers a unified application programming interface for continuous, binary, count and time-to-event endpoints, supports posterior-probability decision rules with one or more efficacy and futility criteria under binding or non-binding regimes, and includes a precomputation strategy that decouples threshold calibration and look-time selection from the simulation pass. adabay reproduces the operating characteristics of BATSS and adaptr in the continuous and binary case studies, the analytic gsbDesign values in the continuous case, and the BATSS results in the count case, all within Monte Carlo error, while running approximately four to five orders of magnitude faster than BATSS and one to over two orders of magnitude faster than adaptr per virtual trial on eight cores. The package is distributed under an MIT licence.

Most FDR-controlled feature selection methods are designed for coordinate-wise hypotheses, where each feature has a single weight or importance score. This abstraction fails in sequential and grouped models, where one original feature is represented by a block of sub-features, such as lags, recurrent states, or attention-based interactions. We propose a grouped-feature FDR control framework for such settings. For grouped linear models, we construct null-symmetric block-level mirror statistics with matrix-valued perturbations. For neural sequential models, we combine Permutation SHAP derivatives as model-agnostic block-level importance scores with kernel-based dependence measure. The framework is model-agnostic across network architectures, does not require specifying the covariate distribution, and reduces to Gaussian Mirror or Neural Gaussian Mirror when the block size is one. We prove FDR control for low- and high-dimensional grouped linear models and asymptotic symmetry of smoothed Permutation SHAP derivatives under fixed fitted nonlinear models. Experiments on simulated and real-world datasets show reliable FDR control and improved power under correlated grouped-feature signals.

The pinball-loss support vector machine is robust, but its asymmetry parameter is usually fixed in advance. We propose a data-driven elastic-net support vector machine that learns simplex-constrained weights over candidate pinball losses while retaining one classifier. The weighted loss is equivalent to a pinball loss with a data-dependent effective parameter. An empirical oracle inequality shows that, when weight regularization and simplex truncation vanish, the classifier objective at a global minimizer does not exceed that of the best fixed candidate; otherwise, the excess is explicitly bounded. For high-dimensional data, we develop a column-partitioned variable-splitting solver. It converges with a best-iterate $O(1/T)$ squared-step residual rate. Under common initialization and global parameters, any column partition produces, in exact arithmetic, the same iterates and solution as centralized training. Experiments assess predictive behavior, numerical equivalence, and multi-process scalability.

As global populations age, enhancing neighborhood walkability through inclusive urban design is important for mitigating built environment (BE) barriers that discourage physical activity and social participation among older adults. This study investigates the utility of in-context learning (ICL), using the transformer-based foundation model TabPFN, to determine how BE features influence perceived walkability, as measured by the Neighborhood Environment Walkability Scale (NEWS-A) survey. Using a small-scale dataset (N = 257) comprising a unique demographic of older adults with knee osteoarthritis or a history of falls, TabPFN achieved a macro F1 score of 54.89% for walkability perceptions categorized as Low, Neutral, and High using equal-width binning. This result outperformed optimized, grid-searched baseline models, including Random Forest (45.85%) and XGBoost (50.56%). To interpret these results, we employed Shapley Interaction Quantification (SHAP-IQ) to identify the hierarchical importance of feature interactions. Preliminary results revealed that the model's predictive logic was primarily driven by higher-order interactions. For example, the interaction between average street circuity and the ratio of drivable roads emerged as the primary discriminator of perceived walkability. Neighborhood greenery was found to have substantial predictive importance only when combined with an individual's fear of falling or perception of age-friendliness. Overall, ICL using TabPFN demonstrates superior performance on small-scale datasets, enhancing the fidelity of the resulting interpretive insights. Furthermore, SHAP-IQ provides a synergistic perspective on how higher-order feature interactions drive the model's predictions.

The Study of Women's Health Across the Nation (SWAN) has followed women for over 30 years, from midlife premenopause until later life. The study has 16 surveys at approximately 2 years intervals that cover a wide range of physical and psychological symptoms. These multivariate categorical survey responses potentially contain rich health-related information. Temporal trajectories of the survey responses can be characterized by both the responses profiles and the evolving dynamics of the responses over time. To capture those two features and investigate how they inform subsequent health outcomes, we propose a joint multi-layer latent transition model. We combine a latent transition model that classifies individuals based on their response profiles over time with an additional layer of clustering of these latent class transition sequences, with the goal of connecting these cluster profiles with health outcomes: in this application, self-reported falls. In addition, we evaluate the operating characteristics of the method through simulation studies.

To meet growing demand for granular demographic and socioeconomic indicators under tighter budgets, national statistical offices must continually develop new methods. These include using big data, satellite imagery, and transactional sources to improve or redesign data collection. Artificial intelligence can support this work, but algorithms generated with AI should not be trusted for production without rigorous verification. This paper focuses on two foundations of trust in the use of AI in official statistics: independent statistical verification before production use, and disciplined protection of respondent confidentiality during development and testing. The approach is illustrated through the author's experience directing AI to construct and implement a Mini Max Hierarchical Bayes sampling algorithm. Applied to a synthetic labour force population, the method met all specified precision targets while reducing the required sample size by 80 percent, as confirmed by a Monte Carlo study with 1000 replications. Applied to 2021 Australian Census microdata, it achieved a 90 percent reduction while producing national point estimates accurate to well below 1 percent. The paper concludes with a practical evaluation checklist aligned with the UN Fundamental Principles of Official Statistics and the HLG MOS Quality Framework for Statistical Algorithms.

We study long-memory continuous-time moving-average processes driven by a Levy process and observed at random renewal times. The sampling scheme introduces an additional source of randomness through irregular observation times. We establish the asymptotic behaviour of normalized partial sums under both finite- and infinite-mean renewal sampling.

The dispersion of reactive solutes in shear flows is governed by the interplay between advective stretching, transverse diffusion, and boundary exchange kinetics. While classical analytical methods and grid-based numerical solvers have extensively characterised these transport mechanisms, accurately resolving the spatiotemporal evolution of solute plumes in asymmetric reactive environments remains computationally demanding. In this study, we introduce a physics-informed neural network (PINN) framework to simulate two-dimensional wall-reactive solute dispersion in canonical shear flows (Couette, Poiseuille, and Couette-Poiseuille) bounded by absorbing walls. By embedding the governing convection-diffusion equation and Robin boundary conditions into a unified loss function, the mesh-free PINN reconstructs the spatiotemporal concentration field. The network predictions are validated against an alternating-direction implicit (ADI) finite-difference benchmark, showing close agreement across non-reactive, symmetric, and asymmetric reactive regimes. The computations are carried out at $\mathrm{Pe}=10$ for impermeable walls, symmetric absorption $(\beta_1,\beta_2)=(1,1)$, and tenfold asymmetric wall-reactivity contrasts $(\beta_1,\beta_2)=(0.2,2)$ and (2,0.2). Leveraging the differentiable nature of the trained PINN, we extract wall-resolved transport diagnostics, including the apparent axial dispersion coefficient, cumulative wall-removal dynamics, and localised uptake fluxes. The results show that the imposed shear profile governs the streamwise organisation of reactive uptake, while unequal wall reactivities induce transverse asymmetry that modifies the macroscopic spreading rate. Overall, this framework establishes PINNs as an interpretable mesh-free tool for analysing boundary-coupled reactive transport in shear flows.

We develop a Bayesian framework for model comparison of second-order Langevin dynamics from position-only trajectories. While approximate increment likelihoods for nonlinear position-only inference have been formulated previously, a unified evidence-based framework for comparing multiple second-order models under positional observation has remained lacking. Here we address this problem by combining exact increment likelihoods for linear Gaussian models with a previously proposed approximate likelihood for nonlinear dynamics. Synthetic-data benchmarks show reliable recovery of the generating model at fine sampling intervals and progressive loss of identifiability under coarse temporal sampling. Application to Dictyostelium discoideum trajectories demonstrates that the statistically supported model depends strongly on temporal resolution. Moreover, the selected models reproduce key statistical properties of the experimental trajectories, providing additional support for the model-comparison results. Our framework therefore offers a practical approach to evidence-based comparison of partially observed stochastic dynamics.

Bayesian inference is difficult when likelihood evaluations are expensive and budgets are limited. We propose sampling-based adaptive active learning (SALE), a Gaussian-process (GP) framework for surrogate-based Bayesian inference. SALE uses the expected posterior (EP) induced by normalised GP sample paths as a common sequential-design measure: it defines a posterior-guided search region and weights uncertainty reduction (UR). A state-dependent rule allocates evaluations between Bayesian optimisation (BO) for localisation and UR for calibration. For BO, an annealed objective interpolates between the EP and Thompson sampling while regularising the query law against surrogate-path perturbations. For UR, we introduce an ideal EP-weighted rule and a computationally feasible proxy. Under a Bayesian GP framework, we characterise the annealed objective's stability–bias trade-off through perturbation and Bayesian regret bounds, derive explicit budget-dependent expected total-variation control for the ideal EP-weighted UR, and establish an expected total-variation rate for the implemented proxy. Across analytic benchmarks and simulated likelihoods, SALE reduces total-variation error across all considered settings while avoiding severe failures seen under several external baselines. Econometric and astrophysical examples demonstrate its practical value.

Retrieval-augmented large language models (LLMs) promise source-linked clinical support, but their value depends on whether displayed evidence guides rather than distorts physician reliance. We developed CORA, an agentic retrieval-augmented LLM, to investigate how source-linked assistance affects physician decision-making. CORA maintained benchmark performance and achieved larger gains on cases published after the models' training-data cutoffs. In a study of 46 physicians, accuracy increased from 70.8% unaided to 82.6% with CORA. Supporting citations predicted correct answers (87.7% vs 65.5%), but citations created an important asymmetry: perceived support increased adoption of correct advice from 34% to 76.9% but when an incorrect LLM answer appeared citation-supported, physician resistance to it fell from 92% to 34.8%. These findings show that source-linked LLM assistance can improve physician accuracy while introducing a grounding-dependent safety risk.

Causal estimation of treatment effects within prespecified subgroups, such as biomarker-defined strata, disease phenotypes, or demographic groups are often of clinical interest. Bayesian approaches are attractive for subgroup effect estimation because flexible priors can represent complex treatment heterogeneity and propagate posterior uncertainty. Frequentist methods for prespecified causal subgroup analysis with time-to-event outcomes are also available, including propensity-score weighting approaches that emphasize subgroup-level covariate balance. In the posterior g-formula, subgroup survival estimands also depend on the subgroup-specific distribution of baseline covariates, which may be unstable when some subgroups are small or unevenly represented. Right censoring separately reduces information for the outcome-model component of the estimand, increasing overall uncertainty in subgroup causal survival contrasts. We extend the hierarchical Bayesian bootstrap (HBB) to subgroup causal inference with right-censored time-to-event outcomes. The HBB places a nonparametric hierarchical prior on subgroup-specific baseline covariate distributions, enabling principled borrowing of information across related subgroups while preserving subgroup-specific structure. We combine this distributional regularization with a Bayesian accelerated failure time model and right censoring to perform a posterior g-formula that propagates uncertainty from both the survival model and the subgroup covariate distribution. The resulting framework stabilizes subgroup causal survival estimands in sparse strata without imposing parametric assumptions on the covariate distribution. Simulation studies examine performance across varying degrees of subgroup sparsity and censoring and compare the proposed approach to the popular Bayesian additive regression tree for heterogeneous survival effects.

Estimation of quantile inequality curves and measures is considered in a parametric model based on grouped data. The unknown parameters of the distribution are estimated using the minimum divergence method, using various $\phi$-divergences. The consistency of the plug-in estimators of the inequality curves and measures and the asymptotic normality of the indices estimators are proved. In a simulation study, the methods are verified and compared in terms of the accuracy of the estimation. The practical applications of the proposed methods are illustrated by the analysis of two real data sets.

Gradient-based Bayesian inference methods require efficient access to Jacobian and adjoint-Jacobian operators of high-dimensional forward models. While multigrid solvers provide near-optimal complexity for elliptic partial differential equations, they are rarely available in forms compatible with automatic differentiation (AD). We develop a differentiable multigrid solver for steady-state diffusion-absorption problems and derive its adjoint operations analytically through the full multigrid hierarchy. The resulting solver, DMGS, is implemented in C++ and interfaced with JAX to provide efficient Jacobian-vector and vector-Jacobian products for variational inference in the NIFTy framework. We validate the approach on a 3D inverse problem involving diffuse radiative transfer in tissue, reconstructing an effective radiative source from Monte Carlo-simulated data. The reconstruction reproduces the data at a reduced chi-squared of 1.1 and generalizes to 32 independent validation datasets. Benchmarks against a JAX-native multigrid implementation show comparable runtimes and consistently lower peak memory for the hand-derived adjoint, with modest reverse-mode overhead. These results establish differentiable multigrid solvers as practical building blocks for variational inference in PDE-constrained problems.

Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source–target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a distributional distance that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.

Variational autoencoders (VAEs) trained on multiple sequence alignments (MSAs) have emerged as powerful generative models for biological sequences, with applications ranging from disease variant prediction to functional RNA design. However, standard biological VAE training treats all sequences as exchangeable, ignoring the rich evolutionary structure that organizes homologous sequences from evolutionarily close to highly divergent. We propose Evolutionary Curriculum Learning (ECL), a training strategy that exploits this structure by progressively exposing the model to sequences of increasing evolutionary distance from sampled anchors, following a power-law expansion schedule. Applied to two architecturally distinct VAE models and two biological domains–protein variant effect prediction with EVE and RNA family sequence generation with RfamGen–ECL improves downstream task performance across five random seeds per configuration. Mean ClinVar classification AUROC rises from 0.981 to 0.989 for p53; for PTEN, ECL attains 1.000 in every seed whereas the baseline is unstable (mean 0.905, falling as low as 0.54). For RNA, ECL raises mean covariance-model bit scores on all three families tested and exceeds its seed-matched baseline in 12 of 15 training runs, though with only three families the effect cannot be established as significant at the family level. Ablation experiments show that progressively expanding the sampled sequences by evolutionary distance outperforms fixed-size neighborhood sampling in addition to uniform random sampling. Evolutionary distance is therefore a useful inductive bias for ordering the training curriculum in biological sequence modeling.

Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against. We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag, and show that this bidirectionality supplies a measurement-free test-time error signal: rolling forward $i$ steps and then backward $i$ steps must return the model to its start, so the round-trip discrepancy $\mathcal{C}_i$ is a self-supervised proxy for the unobservable rollout error: no ensembles, no held-out data, no governing equations, for one extra rollout. We validate on compressible magnetohydrodynamics (MHD), an astrophysical turbulent radiative mixing layer, and natural face videos (CelebV-HQ). On held-out MHD trajectories, $\mathcal{C}_i$ ranks rollout error (Spearman $0.91$-$0.98$ at fixed depth; $0.69 \pm 0.16$ within trajectories), and a simple calibrator fit on training rollouts predicts its magnitude to within $1.14\times$ ($68\%$) and $1.29\times$ ($95\%$) with near-nominal coverage - one nat beyond a depth-only predictor, transferring to all six decoded physical fields. The same signal flags the out-of-distribution Orszag-Tang vortex (AUROC $0.98$; $1.0$ by depth $10$) exactly where sampling-dispersion baselines invert, and it cuts incurred error by $15\%$ at $80\%$ coverage - three times the depth-only baseline. Bidirectional training comes at negative cost, beating direction specialists in both directions, and the backward direction doubles as a fast inverse solver. On LE-PDE-UQ's turbulent Navier-Stokes benchmark, a single bidirectional model reaches accuracy within $1.3\times$ of their ten-model ensemble at a tenth of the training cost, with the best training-free pixel-level calibration. Round-trip consistency turns reversibility into a practical trust signal for generative models.

Basket trials are clinical trials in which one treatment is investigated in multiple subpopulations within a single trial. The subpopulations are called strata in the following. Stratified cohorts are typical but not limited to early oncological trials, where targeted therapies can be investigated in different strata defined by tumor tissue. In terms of statistical methodology, the stratification of the trial can be leveraged using information borrowing. The idea is that the strata will be analyzed separately if they respond differently to treatment, but that a stratum may share information from another if their response rates are similar. This can increase power while keeping type-I error inflation moderate. The borrowing methods can be tuned with respect to possible outcome scenarios. In this paper, we provide a framework for tuning basket trials: The compromise between power and type-I error rate is defined by a utility function which is then optimized using optimization algorithms. We investigated this framework by performing a pre-specified comparison study. Part I of the study compared optimization algorithms in terms of reliability and efficiency, part II compared the statistical performance of utility functions in simulated scenarios. This study shows how targeting different utility functions based on measures such as local power in each stratum or the number of correct decisions leads to different borrowing behavior of the optimized design. Furthermore, we discuss a mathematical counterexample which shows that no uniformly most powerful (UMP) test exists for basket trials, thus showing the limitations of basket trials. While information borrowing may result in power gains for some scenarios, there is no design which is optimal for all possible scenarios. Hence, a transparent optimization procedure is crucial when planning basket trials.

Causal inference usually concerns a scalar treatment, yet in many problems the treatment is unstructured: a text, an image, or a sequence of clinical decisions. Consider an instructor writing a course description to attract more students: the treatment is the course description, and the outcome is enrollment. The standard target, the average treatment effect of fixing the treatment to one exact value versus another, runs into two problems. It cannot be estimated, because almost no exact description recurs across courses, leaving no comparable group from which to measure its effect; and it would be of little use even if it could, since no one wants every course to carry the same description. What the instructor actually wants to know is which features of a description raise enrollment, and which of those features can be acted on across many courses. To this end, we propose a causal query for unstructured treatments: the maximally influential feature (MIF), the feature of the treatment that most strongly influences the outcome. We formalize the MIF as a binary feature of the treatment, defined by a feature-scoring function, constrained so that both of its values stay well populated, and chosen to maximize the causal effect it induces. Turning the feature on shifts the distribution of treatments toward those that display it, turning it off shifts away, and the MIF effect contrasts the two average potential outcomes. We study identification conditions for the MIF, develop algorithms to estimate it, and make it actionable through a nudging algorithm that revises a treatment along the MIF into an outcome-improving version. We illustrate the MIF algorithm across applications in text, image, and dynamic treatment sequences.

Denoising is often treated as a technical prelude to calcium transients analysis, but it can redefine the variables used for behavior decoding and causal structure learning. An inspectable motorneuron recording with a visible artefact motivated this study: removing artefact-linked blind source separation (BSS) components also changed trace dynamics outside the targeted frame. We therefore tested whether BSS denoising preserves behavior and causal evidence on already extracted calcium transients. In a synthetic benchmark with known lagged graphs, raw corrupted traces retained graph recovery, whereas component-removing BSS variants collapsed median graph F1 to 0 across estimator families. We then evaluated four BSS methods: FastICA, Infomax, SOBI, and JADE, on four larval zebrafish recordings of v2a reticulospinal neurons (v2a-RSNs) with tail behavior. BSS sometimes improved behavior decoding, but gains depended on fish, method and retained clusters; matched PCA and low-pass controls often matched or exceeded BSS in fold-audited comparisons. Connectivity effects were more consistent: c-GC and c-GC* graphs inferred from BSS-cleaned traces were much denser than raw-trace graphs. BSS should therefore be treated as an intervention on the measured process, not a neutral cleanup step.

Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselves act as location-dependent dual prices and sensitivity variables. We study the estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference distribution $\mu$ is transported to an unknown distribution $\nu$, accessed to via its empirical measure. Our main ingredient is a stability inequality that controls the $L^1(\mu)$ distance, modulo additive constants, between a strongly convex potential $\varphi$ and a convex potential $\widetilde\varphi$ by a weak dual norm of $(\nabla\widetilde\varphi)_\#\mu-(\nabla\varphi)_\#\mu,$ together with a second-order Wasserstein remainder of logarithmic type. This separation between the leading empirical-process term and the Wasserstein remainder yields faster convergence for potentials than for the corresponding transport maps. Under smoothness and uniform convexity assumptions, the exact semidiscrete Brenier potential converges in $L^1(\mu)$ at rate $n^{-1/2}$ for $d\leq3$, at rate $n^{-1/2}(\log n)^{5/2}$ for $d=4$, and at rate $n^{-2/d}(\log n)^{(d+2)/d}$ for $d\geq5$. The polynomial exponents are sharp. In dimensions $d\leq3$, we further establish a nondegenerate function-space central limit theorem and prove consistency of the nonparametric bootstrap. These results yield joint root-$n$ inference for every fixed finite collection of normalization-invariant weighted contrasts of the potential, including regional shadow premia in reference-based risk problems. Finally, we prove matching upper and lower bounds of order $\varepsilon\log(1/\varepsilon)$ for the normalization-invariant sum of the entropic dual potentials.

Modern machine learning pipelines increasingly rely on reusing pretrained and foundation models across downstream tasks. These pretrained models can differ not only in performance but also in how they can be used: some only provide black-box predictions, while others may permit white-box access to internal representations that can be probed or fine-tuned. When deployed to the target domain in the presence of distribution shift, no single strategy, including zero-shot application, fine-tuning, or directly training a target-specific model, is uniformly the best. In this work, we propose Frontier Learning, a framework that treats a library of candidate models spanning different training histories and access regimes as complementary sources of information rather than mutually exclusive alternatives. Frontier Learning constructs a unified target-domain feature by concatenating internal representations from white-box candidates as well as prediction outputs from black-box candidates, then fits a lightweight, regularized supervised learner on this concatenated representation using labeled target data. Because the resulting hypothesis class contains predictors obtained by zero-shot reuse, fine-tuning, and direct training as special cases, empirical risk minimization over the frontier learner is guaranteed to be no worse, on the training sample, than any individual baseline. We evaluate the framework in simulations spanning varying degrees of source-target compatibility and in two real-world distribution-shift settings: visual domain adaptation on DomainNet/VisDA and clinical mortality prediction across intensive care unit domains using MIMIC-IV-Notes. Across all settings, Frontier Learning matches or outperforms the strongest individual reuse strategy, with the largest gains arising precisely when no single baseline is reliable across the range of shift considered.

Integer-valued time series are ubiquitous in fields such as finance, economics, and epidemiology. As spatiotemporal data structures in these domains grow increasingly complex and high-dimensional, the matrix integer-valued autoregressive (MINAR) model efficiently captures row-column cross-correlations to reduce dimensionality. However, it fundamentally fails to accommodate negative values, which is a critical flaw for analyzing real-world differenced data or financial tick fluctuations. To bridge this theoretical and practical gap, this paper introduces the Z-MINAR model, a novel matrix autoregressive framework defined on the full integer domain (Z). By pioneering a signed matrix thinning operator and utilizing an extended poisson distribution for the innovations, the Z-MINAR model elegantly handles both positive and negative integers while strictly preserving the crucial topological interactions inherent in matrix data. Furthermore, we employ a projection-based conditional least squares estimation procedure and rigorously establish the model's stationarity, causality, and asymptotic normality. Extensive simulations demonstrate the superior estimation accuracy, robustness, and adaptability of Z-MINAR over existing benchmark models. Finally, an empirical application focusing on crime count variations across different urban regions confirms the model's practical efficacy in uncovering dynamic spatiotemporal dependence structures in Z-valued matrix time series.

We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.

Population censuses increasingly rely on imputation to assign usual-residence addresses for non-responding dwellings, yet no formal statistical framework has existed for comparing competing imputation methods on their combined coverage and address-accuracy performance. We develop such a framework within a Total Statistical Error (TSE) paradigm that is not restricted to survey-based data collection and applies equally to register-based and administrative data sources. The central quantity is a unit-level binary correctness indicator, equal to one if and only if a person is both enumerated and assigned to the correct usual-residence address; its complement is the unit TSE, and aggregating over the population yields a correctness rate that serves as the basis for head-to-head method comparison. We distinguish two assessment paradigms: Paradigm I, in which a post-enumeration survey (PES) provides reference values for a probability sample, and Paradigm II, in which a complete benchmark makes the correctness rate directly computable. We illustrate the framework using 2021 Australian census microdata, comparing k-nearest-neighbour and random forest imputation under a not-missing-at-random (NMAR) mechanism. A key finding is that dependent nonresponse between the census and a simulated PES causes naive Paradigm I subgroup estimates to be severely biased when subgroup deletion rates are small. Augmenting the PES response propensity model with the census nonresponse indicator reduces this bias by approximately 90%; simulation experiments show that the nonresponse indicator alone drives the correction, with the additionally included imputed set A values contributing negligible further reduction. This is the preferred strategy whenever dependent nonresponse between the census and PES is a concern.

Predictive coding offers a powerful framework for cortical computation, yet scalable implementations that respect both Bayesian exactness and neurobiological constraints remain scarce. We bridge this gap by formally connecting predictive coding to Recursive Gaussian Processes (RGPs). RGPs employ a single Gaussian process \( g(t, \cdot) \) indexed by layer index and input value, preventing the representational collapse of standard deep Gaussian processes while allowing learnable cross-layer dependence via \( r_{1g} \). We demonstrate that RGPs intrinsically implement hierarchical Bayesian inference, uncertainty propagation, and precision-weighted prediction error. Critically, we map RGP components—the shared GP, spike-and-slab variable selection, and MCMC dynamics—onto the canonical cortical microcircuit, providing a neurobiological substrate for these computations. Drawing on the free energy principle, we show that RGP inference minimizes variational free energy, formally linking Bayesian mechanics to neuronal dynamics. Our synthesis positions RGPs as both a principled computational tool and a candidate model for the brain's predictive machinery, generating testable predictions for laminar-specific dynamics and spectral asymmetries between feedforward and feedback processing.