Papers

Existing panel beta regression estimators do not accommodate higher-order autoregressive dependence within a joint likelihood framework. We develop such an estimator: a joint conditional maximum-likelihood beta-AR(k) model extended from a single time series to panel data, and extend it to panel inference through two new covariance estimators - a generalization of Beck-Katz panel-corrected standard errors (PCSE) and a Driscoll-Kraay-style alternative (DK) - to account for the cross-panel dependence the extension introduces. The estimator is evaluated by Monte Carlo simulation spanning AR(1)-AR(3) error structures, three panel sizes, and five series lengths, under both independent panels and genuine cross-panel dependence, against an uncorrected baseline. PCSE generally exhibited superior Type I error control and coverage relative to DK, at a power cost that was modest and partly illusory given DK's own miscalibration. Under genuine cross-panel dependence, the uncorrected estimator's Type I error was severe and did not diminish with additional data, while both corrections resolved it, with PCSE again preferred. An illustrative multi-site disease management example is provided, along with a Stata implementation (xtbetark).

We study the multiplayer multi-armed bandit problem with information asymmetry under Bernoulli rewards, for three information structures: asymmetry in actions, in rewards, and in both. Replacing the Hoeffding-style confidence intervals of prior work with Kullback–Leibler (KL) divergence-based bounds gives strictly tighter regret guarantees in each case. We propose mKL-UCB, mKL-UCB-Intervals and mKL-DSEE, and show that the improvement factor is at least two by Pinsker's inequality and far larger when reward means are near zero or one. For asymmetry in rewards we prove that two arms' KL intervals separate after a deterministic number of samples, and that $M$ independent players accelerate elimination further.

Collections of diverse observations often share one acquisition, processing, geometric, or channel corruption, while only an unpaired clean reference set is available. For a prescribed low-dimensional correction shared across observations, the observed and clean reference sets support inference only through the response of fixed moments. We formulate this problem as two-sample moment calibration and report a rank-aware information state combining local rank, scaled moment sensitivity, source-separated covariance, and a moment compatibility residual. Full rank gives local moment identifiability, whereas kernel directions remain unresolved to first order. A unified linearization separates observed-set and reference-set uncertainty. Under covariance weighting, the weakest scaled singular value determines worst-direction asymptotic amplification. For an orientation-preserving planar-similarity correction shared across observations, ensemble centroids and a nonzero third-order complex moment yield closed-form global population identification of translation, rotation, and isotropic scale under matched-population and no-clipping assumptions. Controlled validation tests the predicted $N^{-1}$ and $\sigma_{\min}^{-2}$ laws, Gaussian efficiency, and interval coverage. Bounded applications report color corrected-output quality, channel magnitude-response calibration, and a separate paired geometric de-beautification result. The framework therefore reports missing or weak information instead of treating every fitted correction as identified.

Cramér's V, the effect size reported beside almost every chi-square test, is almost never accompanied by a confidence interval, because the interval is a hard nuisance-parameter problem: infinitely many tables share one effect-size value. The two intervals an analyst can reach for are unsatisfactory. The projection of a joint confidence region onto the effect size is guaranteed for every table but wide, over-covering at essentially 1.000 on larger tables; the noncentral inversion the software prints, and the bootstrap, are narrow but do not cover (median coverage 0.31 on this study's grid). This paper supplies an interval that is both valid and informative. Conditioning on the observed margins makes the exact conditional distribution of the Pearson statistic under a non-null association computable with no asymptotics and no Monte Carlo; inverting it yields a near-nominal confidence interval for Cramér's V that is a third to a half the width of the projection, the advantage growing with table size. The one price, stated plainly, is a change of estimand to the effect size at the observed margins; the mid-p default undercovers small effects, where a conservative variant or the projection remains the fallback.

Cramer's V and the other chi-square-family effect sizes are positive under exact independence, so a reported value can be smaller than what the same statistic produces on a table with no association at all. The upward bias is classical (Tschuprow 1925; Bartlett 1937; corrected by Bergsma 2013), and reporting the boundary value, the critical effect size, beside the observed one has been recommended in general form (Perugini et al. 2025); we claim neither. Our contribution is two things a general recommendation does not supply. First, an exact per-table reporting gate: the noise floor V_0.95, the value the effect size reaches by chance 5% of the time under independence, computed from the exact margin-conditional law rather than the asymptotic critical value. The asymptotic floor miscalibrates where reporting is riskiest: on heterogeneous tables its false-alarm rate reaches 0.10 against a 0.05 target, while the exact floor holds nominal. Second, a measurement: across 4,129 real two-way tables from 291 public datasets, 21.8% of labeled effects (cluster-robust 95% CI 17 to 27%) sit on tables with no significant association. We recommend reporting V_0.95, in exact form on sparse or heterogeneous tables, beside every association measure and at the design stage.

Each row of a transformer's attention matrix is a probability distribution over tokens, and in trained models most of that probability lands on a single sink token, usually the first. Standard tools for comparing attention rows (cosine similarity, Jensen–Shannon divergence, Shannon entropy) therefore hinge on a choice papers rarely report: keep the sink, or drop it and renormalize. This choice can reverse conclusions. On ten pretrained models from five families, 17–47% of verdicts about which of two heads is more similar flip with the convention, and the most prominent structure in a standard BERT head-clustering pipeline is an artifact of it. The reason is that one-number summaries mix two questions: how much attention the sink takes, and how the rest is divided among the content tokens. Treating rows as compositional data separates them exactly: the Aitchison distance splits orthogonally into a sink term and a content term, entropy splits by an exact identity, and the content distance is characterized by invariances the transformer itself possesses. The separation matters in practice: most measured entropy collapse during training is the sink growing, not attention sharpening (30% of the drop at 70M parameters, 95% at 1B, 79% at 1.4B), and pruning heads with the wrong channel can inflate perplexity more than a hundredfold. We map where each convention is safe, test a frozen out-of-sample predictor (one confirmation, one abstention, one failure), and release code regenerating every number.

Studies of association-football forecasting routinely report three-way accuracy in the low fifties and present it as competitive with the betting market. Accuracy against a uniform benchmark answers the wrong question; the question worth asking is whether a model carries information a margin-free closing price has not already absorbed. We formalise that test as the fitted weight in a logarithmic opinion pool and apply it to nineteen complete Serie A seasons (7,220 matches). The answer is negative and stable. A Dixon-Coles model with tuned exponential decay attains 53.4% accuracy and a Ranked Probability Score of 0.1972 against the market's 0.1905; the paired difference is +0.0067 (95% CI [0.0046, 0.0088]) and the market wins in all seven test seasons. The fitted pooling weight on the structural model is 0.000, and the log-loss profile is monotone increasing in that weight on validation and test alike, so this is a boundary solution, not an optimisation artefact. Refitting the same machinery to shots on target yields a variant earning weight 0.35 against the goals model – it carries information the goals model lacks – and 0.000 against the market. Two structural signals, each informative about the other, both priced. The structural model is better calibrated than the market on the home-win margin (slope 0.995 versus 1.103) while clearly less sharp: the market's advantage is discrimination rather than honesty, which accuracy alone cannot distinguish. Value lies not in a better forecast but in what is built on a calibrated one. We define match leverage, the change in a club's probability of achieving a season objective between winning and losing a fixture, and compute it for ACF Fiorentina: an away fixture against a relegation rival carried 2.25x the leverage of hosting the eventual champions. The paper also documents and corrects errors in an earlier study of our own.

We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_\alpha(x,y) = \bigl[\alpha\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_\alpha\leq\beta$, the particle iterates remain within $O(\sqrt{\beta/\alpha})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.

We propose an ARMA model that allows for multiple seasonal periods and time varying parameters in both regular and seasonal components, building upon previous work for pure AR processes and the conditional likelihood. The model is parameterized to ensure stability and invertibility at each time point. The parameter evolution is governed by dynamic shrinkage processes, enabling extended periods of essentially constant parameters, gradual changes, and abrupt shifts. The model includes a stochastic volatility component to account for potentially heterogeneous noise, also modeled by a dynamic shrinkage process. A Gibbs sampler is developed using the exact likelihood, with separate updating steps for the latent errors and the unobserved pre-sample history of the process. The time-varying AR and MA parameters are sampled jointly using a fast posterior sampler based on the extended Kalman filter. The model and the efficiency of the Gibbs sampler are evaluated using simulated and real data. A case study on monthly air passenger data in the US during 1990-2024 reveals significant changes in seasonality during the Covid-19 pandemic.

We provide an estimator for the perturbed utility route choice (PURC) model that works with data at the level of individual trips. The estimator is a nested fixed-point algorithm that combines an upper bias-corrected linear regression problem with a lower individual-level perturbed utility maximization problem. We establish the statistical properties of the microPURC estimator and confirm these results with an experiment using simulated data. Finally, we demonstrate the estimator in practice using a large real-world dataset.

We introduce a simple and performant approach for rapidly and accurately performing polyharmonic spline (PHS) interpolation using a combination of the fast multipole method (FMM) from computational electrostatics and the Vecchia approximation from the Gaussian process and sparse approximate inverse literatures. Using basic properties about Hadamard products and low-rank matrices, we demonstrate that an FMM with two kernels, the logarithmic and distance kernels, results in fast PHS interpolation for all orders. Furthermore, we demonstrate the exceptional performance of sparse inverse approximation methods with the Matérn covariance model for preconditioning. Combined with careful management of disallowed subspaces, we describe a procedure for obtaining prediction weights using preconditioned conjugate gradient that converges in less than $15$ iterations, even for problem sizes with over one million points. As a result, thin-plate spline interpolation—a particularly popular method that does not require parameter tuning—that matches the fully dense $\mathcal{O}(n^3)$ computation in accuracy can be done at the cost of approximately $50-60$ FMMs. A high-performance software library for odd-order PHS interpolation in two dimensions is made available as a companion to this work.

The Cox model and the Cox regression coefficient are widely used in applied work, although the corresponding statistical theory is principally developed in the continuous case where there can be no tied failure times. In this work, we derive the efficiency bound of the Cox regression coefficient in the Cox model, without imposing continuity or discreteness assumptions on the failure distribution. Then, under a technical assumption that the population failure mass points belong to an unknown finite set, we show that the “exact” estimator in \citet{cox1972regression} is asymptotically efficient. Next, we propose an asymptotically equivalent estimator which solves the efficient score which has lower computational complexity than the exact Cox score. The development employs recently introduced martingale theory, and throughout examples of the general theory are given for both the absolutely continuous case and the discrete case.

The classical multinomial, negative multinomial, hypergeometric, and negative hypergeometric distributions are naturally organized by two features of the sampling scheme: sampling with or without replacement and stopping after a fixed number of draws or a fixed number of failures. We complete the graphical analogue of this scheme for decomposable graphs by adding graphical hypergeometric and graphical negative hypergeometric distributions to the previously introduced graphical multinomial and graphical negative multinomial models in Danielewska et al. (2025). The resulting four families provide a unified parametric framework for graphical modeling of multivariate count data, in which dependence and admissible configurations are encoded by a graph. They interpolate between products of univariate distributions for the empty graph and the corresponding classical multivariate distributions for the complete graph, while retaining explicit Markov factorizations and tractable sampling representations. We further develop a unified Bayesian hierarchy based on graphical Dirichlet-type distributions, obtaining explicit posterior and predictive laws. The framework is particularly natural for count data arising under exclusion or incompatibility constraints. We discuss several such applications and illustrate its practical potential using Rydberg-atom excitation data.

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

Systems often need to combine two numerical assessments of the same yes/no question. The appropriate formula depends on what the numbers represent and on how the sources are related. Averaging is correct when one of several alternative interpretations applies; multiplying odds is correct when probability reports are based on conditionally independent evidence and a common prior; and probabilities of alternative successful derivations require their dependence or shared evidence to be taken into account. We state the assumptions behind several common combination rules and derive the corresponding combined probabilities. Two groups of Monte Carlo experiments address different questions. First, controlled generating mechanisms verify that the derived rule recovers the correct probability in the situations for which its assumptions hold. Second, the same mechanisms measure the consequences of using a mismatched rule, using logarithmic score and threshold decisions with different costs. Distinct pooling rules can produce the same binary decision at threshold 1/2 while assigning substantially different probabilities, so binary accuracy alone can conceal important differences. We also give probabilistic interpretations of conflicting-evidence rules and show that, for overlapping derivations, retaining the identities of shared uncertain premises permits direct calculation of the probability that at least one derivation is available. Pairwise combination of proof probabilities loses information when there are three or more derivations.

Omics datasets, particularly single-cell RNA sequencing data, are high-dimensional, sparse, noisy, and dominated by zero values, making faithful low-dimensional representation challenging. Existing dimensionality-reduction methods may distort local neighbourhoods, global organization, or the cohesion of meaningful populations, with similar limitations arising in genealogical data. We introduce Contrastive Manifold Approximation and Projection (CosMAP), a graph-based unsupervised dimensionality-reduction method for producing faithful and interpretable embeddings. CosMAP extends the graph-based framework of UMAP by combining cosine-similarity neighbourhoods with temperature-normalized contrastive affinities, which are optimized in the embedding space using an attractive–repulsive objective. It further employs a two-phase refinement strategy: an intermediate higher-dimensional representation is first learned and then used to reconstruct the neighbourhood graph and initialize the final low-dimensional embedding. We evaluate CosMAP on MNIST and USPS handwritten-digit datasets, mouse retina and cortex single-cell RNA-sequencing datasets, and a large genealogical kinship dataset derived from BALSAC-CARTaGENE. Compared with state-of-the-art dimensionality-reduction methods, CosMAP produces more coherent visual representations, improves neighbourhood preservation, and provides clearer global organization of digit classes, biological cell populations, and regional genealogical patterns. These results indicate that CosMAP offers a robust framework for exploratory analysis of complex, sparse, high-dimensional data. The implementation is publicly available at https://github.com/FenosoaRandrianjatovo/CosMAP-dr.

Beta diversity quantifies variation in species composition across ecological communities and is fundamental for understanding biodiversity patterns across space and environmental gradients. Inference is complicated by incomplete sampling, differences among communities and imperfect detection. We develop a partially exchangeable Bayesian feature allocation model for species occurrence data that accommodates unobserved species and incorporates site-specific covariates to relate variation in species composition to environmental conditions. The model yields explicit conditional distributions for species richness and sharing, together with predictions of the species newly recorded at individual sites or newly shared between sites under additional sampling. We also introduce a novel probabilistic beta diversity index with posterior uncertainty quantification. An occupancy–detection representation preserves the tractability of the occurrence model and allows sensitivity analysis under specified detectability assumptions. This tractability makes the approach scalable, enabling inference in large biodiversity studies. Simulation studies assess predictive performance, and an application to global airborne fungal data illustrates how accounting for unobserved species and borrowing information across sites affect inference on compositional heterogeneity.

There exist tests calibrated under a null narrower than the one implied by their test statistic – the detection-null set. The part of the detection-null set not in the null is the test's blind spot. A framework for assessing detection coherence – whether a test's blind spot is empty – is introduced, built on new concepts of calibration statistic, detector, detection functional, detection-null set, and blind spot. It is demonstrated that the Wilcoxon–Mann–Whitney, Kruskal–Wallis, Friedman, and logrank tests are detection incoherent as unrestricted tests. Detection incoherent tests may become coherent under restrictions that make the blind spot empty, though domain restriction is not a reliable route to coherence in practice, since the boundary of the coherence-restoring domain is unknown and unverifiable. The Kolmogorov–Smirnov and Zaremba tests, and the Maximum Mean Discrepancy test with a characteristic kernel, are shown to be universally detection coherent. Due to the blind spot, a detection incoherent test misses discoveries when non-rejecting and yields spurious discoveries when rejecting, in both cases regardless of sample size. Detection incoherent tests should be abandoned.

Conditional Independence (CI) tests are the statistical engine of constraint-based causal discovery: in algorithms such as PC (Peter-Clark) and FCI (Fast Causal Inference), skeleton pruning and key orientations follow directly from CI decisions. This survey reviews CI testing with emphasis on assumptions, robustness, and scalability in high-dimensional and mixed-type settings common in biomedical domains. The survey organizes widely used CI methods into six families: partial-correlation, contingency-table, regression, nearest-neighbor, kernel, and machine-learning-based. Special emphasis is provided on the robustness layers that address the limitations of these families. For each family, the survey examines when CI decisions reflect the data-generating distribution and when they fail. By this, we link test-level properties, including power decay with conditioning set size and asymmetric type I/II error consequences, to graph-level errors in skeleton recovery and v-structure orientation. The survey also compares adoption across major R and Python libraries and summarizes open challenges, including mixed-type CI testing without discretization, small-sample error control, and strategies for improving scalability of CI-testing.

Assessing the goodness-of-fit of a logistic regression model is a critical prerequisite before the model is used for inference. However, goodness-of-fit (GOF) tests such as the chi-square and deviance tests often give invalid results when the data are "sparse" – a common issue with continuous predictors like age or weight, where the asymptotic distributional assumptions are not satisfied. This thesis studies classical GOF tests for binary logistic regression under both grouped and sparse data, comparing about 30 statistical tests and machine-learning calibration algorithms. These span the classical chi-square and Hosmer-Lemeshow variants, standardized Pearson statistics, covariate-space partitioning, smoothing-based methods, and contemporary calibration machine-learning and bootstrap procedures. At a fixed size, the GiViTI calibration test (2016), McCullagh (1989), Osius-Rojek (1992), le Cessie (1995) and Stute-Zhu (2002) proved empirically powerful, balancing correct identification of bad models (high empirical power) against not raising false alarms on good models (correct empirical Type I error). Relying on formal methods alone is insufficient: visual diagnostics such as calibration plots are a vital exploratory step for detecting model deficiencies that formal tests often overlook. An application to real data (the Low Birth Weight dataset) shows that many of these tests fail to give valid conclusions when exposed to the complexities of actual datasets. The main conclusion is that model assessment requires a combination of several powerful statistical tests alongside careful visual inspection of model calibration.

We study the geometry of simplicial cochains on Vietoris-Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

The simplicial depth (SD) is a commonly used indicator of the centrality of points $x\in\mathbb{R}^d$ with respect to distributions $P$ on $\mathbb{R}^d$. Asymptotic theory for the sample SD is based on its representation as a $U$-statistic, which can be either non-degenerate or degenerate. For $d=2$, we prove under mild conditions that this $U$-statistic is degenerate with rate $n$ if and only if $x$ is a center of symmetry of $P$. Otherwise, the asymptotic distribution of SD is non-degenerate with rate $\sqrt{n}$. Because the location of the center of symmetry of $P$ is usually unknown, these two modes of behavior complicate the estimation of the sample distribution of SD at $x$. We propose a two-step adaptive subsampling procedure for estimating that distribution. First, an estimator $\widehat \gamma$ of a parameter $\gamma\in\{1/2, 1\}$ characterizing the correct rate of convergence $n^\gamma$ of SD is constructed based on subsampling. Our estimator uses a bias correction suitable for $U$-statistics. Second, $\widehat \gamma$ is employed for approximating the distribution of the sample SD. We prove the consistency of this subsampling approach and illustrate its usefulness (i) in the construction of confidence intervals for SD, and (ii) in an SD-based supervised classification task

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024, Probab. Theory Relat. Fields). It combines Laplace-type approximations for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation under the logistic parametrisation, Poisson intensity estimation under the exponential link, and mildly ill-posed linear inverse problems observed in Gaussian white noise. The resulting minimax rates yield optimal recovery of density score functions, derivatives of Poisson intensities and, in a concrete elliptic inverse problem, the unknown source function.

Generative artificial intelligence (GenAI) is a large language model (LLM) that has the ability to generate media based on user-provided prompts. Given the demonstrated capabilities of models such as ChatGPT in information synthesis and programming, there is growing interest in their potential role within the research process. However, little work has evaluated recent GenAI models for research tasks in the domain of statistical research. This case study examines GenAI as a tool for developing a literature review and translating methodology from academic papers into code, for the topic of dynamic treatment regime (DTR) estimation via the dynamic weighted ordinary least squares (dWOLS) approach. Specifically, we utilize ChatGPT-5 and ScholarAI (Sept-Nov 2025 release) in the processes of identifying relevant sources for the literature review, creating summaries of papers, identifying gaps in research, and R code generation to implement methodology. Our findings show that current GenAI models lack the depth and contextual understanding required to accomplish these tasks without careful prompting and supervision of a knowledgeable researcher. Nonetheless, GenAI has potential to increase efficiency of tasks which take advantage of its search and summarization abilities, as well as basic code debugging and algorithm formation. We demonstrate that under a knowledgeable guide, GenAI can function as a research tool, but not as a substitute for methodological expertise.

Research on the use of medications during pregnancy has two primary goals: to detect signals that medications may be harmful to a pregnant individual or fetus, and to support better treatment of pregnant people who require pharmacotherapy. Target trial emulation has been proposed as an approach to estimate the effects of interventions in real world data, with recent extensions to the pregnancy setting. This approach focuses on aligning eligibility and treatment initiation with start of follow up, which is particularly desirable given methodological challenges specific to pregnancy, such as right and left censoring and truncation, competing events, differences in gestational length, and varying etiologically susceptible periods. While previous work on target trial emulation in pregnancy has focused on initiation versus non-initiation of point treatments such as vaccines or antibiotics, this paper focuses on research questions regarding changes to pregestational treatment regimes, and aims to highlight opportunities and approaches to designing studies that align with relevant time points during early pregnancy at which treatment decisions occur in clinical practice. Using the example of treatment for type 2 diabetes mellitus, we review methods for identifying pregnancy episodes in routinely collected healthcare data, introduce possible time zero candidates, and discuss analytic approaches that minimize potential bias due to selection and immortal person time.

This paper establishes sharp dimension-free Frobenius-norm concentration inequalities for sample moment tensors. Our bounds are optimal over the sub-Gaussian class, while for Gaussian data we obtain matching two-sided estimates. We also identify a parity effect: the intermediate Gaussian and sub-Gaussian scales coincide at odd tensor orders but differ at even orders. Our second main result, which supplies the weak-moment estimate behind these bounds, proves a dimension-free moment bound for Hilbert-valued polynomials under Gaussian convex domination. The proof of this bound combines a recently developed structured martingale coupling theorem with new estimates for moment tensors of uniformly log-concave distributions.

Inverse reinforcement learning (IRL) aims to recover a reward function under which the resulting policy reproduces the behavior observed in expert demonstrations. A natural approach is to formulate IRL as a bilevel optimization problem, in which the inner level corresponds to policy optimization under the learned reward and the outer level measures the discrepancy between the induced policy and expert data. However, this formulation is computationally challenging in practice because the outer update requires a hypergradient involving an inverse-Hessian-vector product for the inner objective. We address this challenge by showing that, at the inner optimum, the Hessian of the inner objective is proportional to the Fisher information matrix of the policy, yielding a structured Fisher-based hypergradient closely related to Natural Hypergradient Descent. To address the resulting scalability bottleneck associated with large Fisher matrices, we approximate the required inverse-Fisher-vector product using a streaming spectral sketch, avoiding explicit construction of the Fisher matrix. We evaluate our approach against a first-order stochastic bilevel baseline across discrete- and continuous-control environments. The results demonstrate competitive policy performance and strong reward-ranking quality, while Fisher sketching reduces curvature-storage complexity and can improve computational efficiency relative to an explicit Fisher solver.

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.

Reliable estimation of cure fractions depends critically on adequate follow-up. Classical procedures for assessing follow-up sufficiency are primarily inferential, whereas a separate body of literature estimates time to cure using excess-hazard, conditional cure-probability, or residual-survival definitions. These two approaches remain largely disconnected: assessing the adequacy of observed follow-up does not generally quantify the additional follow-up required under an explicit tolerance, whereas estimating time to cure does not itself provide a formal inferential assessment of the available follow-up. We develop a unified parametric framework that combines the Parametric Follow-up Sufficiency Test, which compares the terminal Kaplan-Meier estimate with the cure fraction estimated from a parametric mixture cure model, with two complementary tolerance-based formulations: the Plateau Distance Criterion on the population survival scale and the Residual Survival Criterion on the susceptible survival scale. We derive closed-form expressions for both criteria under the Weibull mixture cure model and establish that they yield identical minimum follow-up times under an appropriate transformation of their tolerance parameters. We evaluate the finite-sample performance of the framework through Monte Carlo simulations across sample sizes, cure fractions, and administrative censoring scenarios. Applications to prostate cancer and triple-negative breast cancer data illustrate how the framework assesses available follow-up, estimates additional observation time under prespecified tolerances, and identifies settings in which follow-up is already adequate.

We prove multiple central limit theorems for serial tail dependence estimators in heavy-tailed long memory linear time series. The main theoretical tools are two novel multivariate reduction principles for partial sums of heavy-tailed long memory linear time series, subordinated over sliding windows and above a threshold growing with sample size. This requires addressing several substantial difficulties, including handling a nonlinear, sample-size dependent, and multivariate subordination mechanism, the dependence between several overlapping linear processes, and the lack of higher-order moments of the marginal distribution. Despite these obstacles, our assumptions are mild and, in particular, the innovation process is allowed to have infinite variance. A key feature of our theory is that our second reduction principle holds uniformly in the threshold, allowing central limit theory for empirical extremograms with sample quantiles as thresholds. This question has received little attention in the literature on serial extremal dependence estimation even though the version of empirical extremograms with random thresholds is ubiquitous in practice. We compare our results in several respects with those that may be obtained under short-range dependence, thereby discovering markedly different convergence rates and limit laws in our long memory setting.