Papers

We study empirical Bayes (EB) predictive density estimation in linear mixed models (LMMs) with large number of units, which induce a high dimensional random effects space. Focusing on Kullback Leibler (KL) risk minimization, we develop a calibration framework to optimally tune predictive densities derived from on a broad class of flexible priors. Our proposed method addresses two key challenges in predictive inference: (a) severe data scarcity leading to highly imbalanced designs, in which replicates are available for only a small subset of units; and (b) distributional shifts in future covariates. To estimate predictive KL risk in LMMs, we use a data-fission approach that leverages exchangeability in the covariate distribution. We establish convergence rates for our proposed risk estimators and show how their efficiency deteriorates as data scarcity increases. Our results imply the decision-theoretic optimality of the proposed EB predictive density estimator. The theoretical development relies on a novel probabilistic analysis of the interaction between data fission, sample reuse, and the predictive heat-equation representation of George et al. (2006), which expresses predictive KL risk through expected log-marginals. Extensive simulation studies demonstrate strong predictive performance and robustness of the proposed approach across diverse regimes with varying degrees of data scarcity and covariate shift.

Determining the number of change-points is a first-step and fundamental task in change-point detection problems, as it lays the groundwork for subsequent change-point position estimation. While the existing literature offers various methods for consistently estimating the number of change-points, these methods typically yield a single point estimate without any assurance that it recovers the true number of changes in a specific dataset. Moreover, achieving consistency often hinges on very stringent conditions that can be challenging to verify in practice. To address these issues, we introduce a unified test-inverse procedure to construct a confidence set for the number of change-points. The proposed confidence set provides a set of possible values within which the true number of change-points is guaranteed to lie with a specified level of confidence. We further proved that the confidence set is sufficiently narrow to be powerful and informative by deriving the order of its cardinality. Remarkably, this confidence set can be established under more relaxed conditions than those required by most point estimation techniques. We also advocate multiple-splitting procedures to enhance stability and extend the proposed method to heavy-tailed and dependent settings. As a byproduct, we may also leverage this constructed confidence set to assess the effectiveness of point-estimation algorithms. Through extensive simulation studies, we demonstrate the superior performance of our confidence set approach. Additionally, we apply this method to analyze a bladder tumor microarray dataset. Supplementary Material, including proofs of all theoretical results, computer code, the R package, and extended simulation studies, are available online.

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = \lambda_{k+1} - \lambda_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/\pi) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.

This study develops two robust, quantile-sliced moment systems, mean and median absolute deviation (MAD and MedAD moments), to serve as foundational tools in parametric modeling, statistical inference, and describing distributional location, scale, skewness, and tail behavior in settings where classical moments and L-moments fail. MAD moments use block-wise absolute deviations around the median and exist whenever the mean is finite, while MedAD moments replace expectations with medians, ensuring existence for all distributions, including heavy-tailed cases with undefined mean or variance. The systems exhibit strong consistency, slice-based robustness, and bounded influence. The results indicate that MAD and L moment ratios are efficient for light to moderate tails, whereas MedAD ratios remain uniquely stable when higher moments do not exist. Applications to Cauchy parameter estimation highlight the practical value of MedAD estimators as simple, fully robust alternatives to likelihood-based approaches. Together, these systems offer a unified, median-anchored framework for reliable distributional inference under heavy tails and contamination.

We study finite-sample statistical performance guarantees for distributionally robust optimization (DRO) with optimal transport (OT) and OT-regularized divergence model neighborhoods. Specifically, we derive concentration inequalities for supervised learning via DRO-based adversarial training, as commonly employed to enhance the adversarial robustness of machine learning models. Our results apply to a wide range of OT cost functions, beyond the $p$-Wasserstein case studied by previous authors. In particular, our results are the first to: 1) cover soft-constraint norm-ball OT cost functions; soft-constraint costs have been shown empirically to enhance robustness when used in adversarial training, 2) apply to the combination of adversarial sample generation and adversarial reweighting that is induced by using OT-regularized $f$-divergence model neighborhoods; the added reweighting mechanism has also been shown empirically to further improve performance. In addition, even in the $p$-Wasserstein case, our bounds exhibit better behavior as a function of the DRO neighborhood size than previous results when applied to the adversarial setting.

Semi-supervised learning has attracted significant attention due to the proliferation of applications featuring limited labeled data but abundant unlabeled data. In this paper, we examine the statistical inference problem in an assumption-lean framework which involves a high-dimensional regression parameter, defined by minimizing the least squares, within the context of semi-supervised learning. We investigate when and how unlabeled data can enhance the estimation efficiency of a regression parameter functional. First, we demonstrate that a straightforward debiased estimator can only be more efficient than its supervised counterpart if the unknown conditional mean function can be consistently estimated at an appropriate rate. Otherwise, incorporating unlabeled data can actually be counterproductive. To address this vulnerability, we propose a novel estimator guaranteed to be at least as efficient as the supervised baseline, even when the conditional mean function is misspecified. This ensures the dependable use of unlabeled data for statistical inference. Finally, we extend our approach to the general M-estimation framework, and demonstrate the effectiveness of our methodology through comprehensive simulation studies and a real data application.

We review recently proposed Bayesian approaches for clustering high-dimensional data. After identifying the main limitations of available approaches, we introduce an alternative framework based on vertical consensus inference (VCI) to mitigate the curse of dimensionality in high-dimensional Bayesian clustering. VCI builds on the idea of consensus Monte Carlo by dividing the data into multiple shards (smaller subsets of variables), performing posterior inference on each shard, and then combining the shard-level posteriors to obtain a consensus posterior. The key distinction is that VCI splits the data vertically, producing vertical shards that retain the same number of observations but have lower dimensionality. We use an entropic regularized Wasserstein barycenter to define a consensus posterior. The shard-specific barycenter weights are constructed to favor shards that provide meaningful partitions, distinct from a trivial single cluster or all singleton clusters, favoring balanced cluster sizes and precise shard-specific posterior random partitions. We show that VCI can be interpreted as a variational approximation to the posterior under a hierarchical model with a generalized Bayes prior. For relatively low-dimensional problems, experiments suggest that VCI closely approximates inference based on clustering the entire multivariate data. For high-dimensional data and in the presence of many noninformative dimensions, VCI introduces a new framework for model-based and principled inference on random partitions. Although our focus here is on random partitions, VCI can be applied to any dimension-independent parameters and serves as a bridge to emerging areas in statistics such as consensus Monte Carlo, optimal transport, variational inference, and generalized Bayes.

The empirical Bayes $g$-modeling approach based on the nonparametric maximum likelihood estimator (NPMLE) has been central to large-scale estimation and inference in the normal means problem. However, theoretical guarantees for uncertainty quantification remain scarce. A key obstacle is that the NPMLE is necessarily discrete, which yields discrete posterior credible sets and a slow logarithmic deconvolution rate. We address both limitations by introducing a hierarchical Gaussian smoothing layer that restricts the mixing distribution to a Gaussian location mixture. Our smooth NPMLE inherits the favorable properties of the classical NPMLE: it is computable via convex optimization and achieves nearly parametric denoising performance. Moreover, it achieves a polynomial deconvolution rate that is asymptotically minimax over the corresponding class. Our procedure also leads to estimated smooth posteriors that converge to the true posteriors at a polynomial rate. Further, we characterize marginal coverage sets that are optimal in expected length, construct plug-in estimators of these sets, and establish theoretical guarantees for the estimated sets in terms of both coverage probability and expected length. We also extend the theory to settings with model misspecification and heteroscedastic Gaussian observations, and study identifiability of the proposed hierarchical model.

Test-time adaptation (TTA) enables neural forecasters to adapt to distribution shifts in streaming time series, but existing methods apply the same adaptation intensity regardless of the nature of the shift. We propose Regime-Guided Test-Time Adaptation (RG-TTA), a meta-controller that continuously modulates adaptation intensity based on distributional similarity to previously-seen regimes. Using an ensemble of Kolmogorov-Smirnov, Wasserstein-1, feature-distance, and variance-ratio metrics, RG-TTA computes a similarity score for each incoming batch and uses it to (i) smoothly scale the learning rate – more aggressive for novel distributions, conservative for familiar ones – and (ii) control gradient effort via loss-driven early stopping rather than fixed budgets, allowing the system to allocate exactly the effort each batch requires. As a supplementary mechanism, RG-TTA gates checkpoint reuse from a regime memory, loading stored specialist models only when they demonstrably outperform the current model (loss improvement >= 30%). RG-TTA is model-agnostic and strategy-composable: it wraps any forecaster exposing train/predict/save/load interfaces and enhances any gradient-based TTA method. We demonstrate three compositions – RG-TTA, RG-EWC, and RG-DynaTTA – and evaluate 6 update policies (3 baselines + 3 regime-guided variants) across 4 compact architectures (GRU, iTransformer, PatchTST, DLinear), 14 datasets (6 real-world multivariate benchmarks + 8 synthetic regime scenarios), and 4 forecast horizons (96, 192, 336, 720) under a streaming evaluation protocol with 3 random seeds (672 experiments total). Regime-guided policies achieve the lowest MSE in 156 of 224 seed-averaged experiments (69.6%), with RG-EWC winning 30.4% and RG-TTA winning 29.0%. Overall, RG-TTA reduces MSE by 5.7% vs TTA while running 5.5% faster; RG-EWC reduces MSE by 14.1% vs standalone EWC.

Counterfactual explanations emerge as a powerful approach in explainable AI, providing what-if scenarios that reveal how minimal changes to an input time series can alter the model's prediction. This work presents a survey of recent algorithms for counterfactual explanations for time series classification. We review state-of-the-art methods, spanning instance-based nearest-neighbor techniques, pattern-driven algorithms, gradient-based optimization, and generative models. For each, we discuss the underlying methodology, the models and classifiers they target, and the datasets on which they are evaluated. We highlight unique challenges in generating counterfactuals for temporal data, such as maintaining temporal coherence, plausibility, and actionable interpretability, which distinguish the temporal from tabular or image domains. We analyze the strengths and limitations of existing approaches and compare their effectiveness along key dimensions (validity, proximity, sparsity, plausibility, etc.). In addition, we implemented an open-source implementation library, Counterfactual Explanations for Time Series (CFTS), as a reference framework that includes many algorithms and evaluation metrics. We discuss this library's contributions in standardizing evaluation and enabling practical adoption of explainable time series techniques. Finally, based on the literature and identified gaps, we propose future research directions, including improved user-centered design, integration of domain knowledge, and counterfactuals for time series forecasting.

Randomized controlled trials (RCTs) yield internally valid causal effect estimates, but generalizing these results to target populations with different characteristics requires an untestable selection ignorability assumption: conditional on observed covariates, trial participation must be independent of potential outcomes. This assumption fails when unobserved effect modifiers are distributed differently between trial and target populations. We develop a sensitivity analysis framework for trial generalization grounded in omitted variable bias (OVB). Our key theoretical contribution is an exact decomposition showing that external-validity bias equals moderation strength $\times$ moderator imbalance: (i) how strongly an unobserved variable shifts the treatment effect, times (ii) how differently that variable is distributed across populations after covariate adjustment. We introduce scale-free sensitivity parameters based on partial $R^2$ values, enabling closed-form bounds and benchmarking against observed covariates – practitioners can assess whether conclusions would change if an unobserved moderator were "as strong as" a particular observed variable. Simulations demonstrate that our bounds achieve nominal coverage and remain conservative under model misspecification, while comparisons with alternative sensitivity frameworks highlight the interpretive advantages of the OVB decomposition.

We present AutoStan, a framework in which a command-line interface (CLI) coding agent autonomously builds and iteratively improves Bayesian models written in Stan. The agent operates in a loop, writing a Stan model file, executing MCMC sampling, then deciding whether to keep or revert each change based on two complementary feedback signals: the negative log predictive density (NLPD) on held-out data and the sampler's own diagnostics (divergences, R-hat, effective sample size). We evaluate AutoStan on five datasets with diverse modeling structures. On a synthetic regression dataset with outliers, the agent progresses from naive linear regression to a model with Student-t robustness, nonlinear heteroscedastic structure, and an explicit contamination mixture, matching or outperforming TabPFN, a state-of-the-art black-box method, while remaining fully interpretable. Across four additional experiments, the same mechanism discovers hierarchical partial pooling, varying-slope models with correlated random effects, and a Poisson attack/defense model for soccer. No search algorithm, critic module, or domain-specific instructions are needed. This is, to our knowledge, the first demonstration that a CLI coding agent can autonomously write and iteratively improve Stan code for diverse Bayesian modeling problems.

Empirical likelihood is an attractive inferential framework that respects natural parameter boundaries, but existing approaches typically require smoothness of the functional and miscalibrate substantially when these assumptions are violated. For the optimal-value functional central to policy evaluation, smoothness holds only when the optimum is unique – a condition that fails exactly when rigorous inference is most needed where more complex policies have modest gains. In this work, we develop a bootstrap empirical likelihood method for partially nonsmooth functionals. Our analytic workhorse is a geometric reduction of the profile likelihood to the distance between the score mean and a level set whose shape (a tangent cone given by nonsmoothness patterns) determines the asymptotic distribution. Unlike the classical proof technology based on Taylor expansions on the dual optima, our geometric approach leverages properties of a deterministic convex program and can directly apply to nonsmooth functionals. Since the ordinary bootstrap is not valid in the presence of nonsmoothness, we derive a corrected multiplier bootstrap approach that adapts to the unknown level-set geometry.

This paper develops an interpretive framework for divergence P-values and S-values within a descriptive frequentist perspective. Statistical analysis is framed as operating within idealized worlds defined by a set of assumptions and a target hypothesis, where probabilities describe the behavior of data under the model but do not assign truth values to hypotheses. Within this view, P-values are interpreted as graded indices of compatibility between the observed result and the predictions generated by the assumed model; accordingly, small P-values should not be read as indicating logical impossibility or strict inconsistency of the model itself. Building on this distinction, the paper argues that practical inference requires moving beyond the internal logic of the model toward judgments of overall acceptability, which depend not only on data-model compatibility but also on multiple contextual considerations such as subject-matter knowledge, plausibility of assumptions, data quality, usefulness, and loss - all interpreted through the competence, intentions, perceptions, and moral values of the specific analyst. S-values are therefore interpreted not as evidence against the epistemic status of the model, but as a specific form of refutational information that contributes to the broader body of information used by the analyst to judge whether a model remains acceptable for an intended practical purpose. The paper also examines the linguistic and conceptual risks associated with the language of incompatibility, distinguishes probability from rarity, and clarifies different notions of surprise - including a possible definition of Shannon-type surprise, to be distinguished from Bayesian belief revision. Overall, the article proposes a more cautious and explicit interpretation of frequentist measures, centered on model-based description, analyst responsibility, and decision acceptability.

We study the assessment of semiparametric and other highly-parametrised models from the perspective of foundational principles of parametric statistical inference. In doing so, we highlight the possibility of avoiding the usual semiparametric considerations, which typically require estimation of nuisance components through kernel smoothing or basis expansion, with the associated difficulties of tuning-parameter choice that blur the distinction between estimation and model assessment. A key aspect is the inducement of replication under the postulated model. This can be cast in terms of some non-standard inferential separations, in the vein of Fisherian ancillarity/co-ancillarity and sufficiency/co-sufficiency separations, allowing the replacement of out-of-sample prediction error as a criterion for semiparametric model assessment by a type of within-sample prediction error. Framed in this light are new methodological contributions in multiple example settings, including model assessment for the proportional hazards model, for a time-dependent Poisson process with semiparametric intensity function, and for matched-pair and two-group examples. Also subsumed within the framework is a post-reduction inference approach to the construction of confidence sets of sparse regression models. Numerical work confirms recovery of nominal error rates under the postulated model and high sensitivity to departures in the direction of semiparametric alternatives. We conclude by emphasising open challenges and unifying perspectives.

Tuning parameters are parameters involved in an estimating procedure for the purpose of reducing the risk of some other estimator. Examples include the degree of penalization in penalized regression and likelihood problems, as well as the balance parameter in hybrid methods. Typically tuning parameters are set to the minimizers of some estimator of the risk, a step which introduces additional randomness and makes standard methodology inapplicable. We derive precise asymptotic theory for this situation. Our framework allows for smooth, but otherwise arbitrary, loss functions and for the risk to be estimated by cross-validation procedures. Results include consistency of the optimal estimator towards a well-defined quantity and asymptotic normality after proper scaling and centring. We give explicit forms and estimators for the limiting variance matrix and results sharply characterizing the distance from the training error to the cross-validated estimator of the risk.

Quantifying predictive uncertainty is essential for real world machine learning applications, especially in scenarios requiring reliable and interpretable predictions. Many common parametric approaches rely on neural networks to estimate distribution parameters by optimizing the negative log likelihood. However, these methods often encounter challenges like training instability and mode collapse, leading to poor estimates of the mean and variance of the target output distribution. In this work, we propose the Neural Energy Gaussian Mixture Model (NE-GMM), a novel framework that integrates Gaussian Mixture Model (GMM) with Energy Score (ES) to enhance predictive uncertainty quantification. NE-GMM leverages the flexibility of GMM to capture complex multimodal distributions and leverages the robustness of ES to ensure well calibrated predictions in diverse scenarios. We theoretically prove that the hybrid loss function satisfies the properties of a strictly proper scoring rule, ensuring alignment with the true data distribution, and establish generalization error bounds, demonstrating that the model's empirical performance closely aligns with its expected performance on unseen data. Extensive experiments on both synthetic and real world datasets demonstrate the superiority of NE-GMM in terms of both predictive accuracy and uncertainty quantification.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

We investigate the structural foundations of statistical efficiency under $\alpha$-local differential privacy, with a focus on maximizing Fisher information. Building on the role of continuous staircase mechanisms, we identify a fundamental symmetry regarding the extremal values $1$ and $e^{\alpha}$. We demonstrate that when the optimal measure satisfies this symmetry, the Fisher information admits a closed-form expression. More generally, we derive a decomposition of the Fisher information into symmetric and asymmetric components, scaling as $\alpha^{2}$ and $\alpha^{3}$, respectively, for $\alpha \to 0$. This reveals that, if in the high-privacy regime asymmetry is negligible, it is no longer the case as privacy constraints are relaxed. Motivated by this, we introduce a class of fully asymmetric privacy mechanisms constructed via pushforward mappings, proving that-unlike their symmetric counterparts-they recover the full Fisher information of the non-private model as $\alpha \to \infty$. We bridge the gap between theory and practice by providing a tractable implementation of these mechanisms, governed by a tuning parameter $c$. This parameter allows for a smooth interpolation between the symmetric regime and the fully asymmetric regime. Furthermore, we demonstrate the versatility of this framework by showing that it encompasses the binomial mechanism as a limiting case.

We propose a scalable, provably accurate method for localizing an unknown number of multiple axis-aligned anomalous patches in spatial data under a general class of spatial dependence. Motivated by the practical need to detect localized changes rather than completely segment large spatial grids, we first introduce both a naive and a significantly faster intelligent-sampling-based estimator for a single patch. We then extend this methodology to the highly challenging multiple-patch setting and propose a two-stage Spatial Patch Localization of Anomalies under DEpendence procedure (SPLADE). Under mild conditions on signal strength, separation from the boundary, inter-patch separation, and a uniform Gaussian approximation, we establish simultaneous consistency for the estimated number of patches and for each individual patch boundary. Extensive numerical results based on synthetic data scenarios demonstrate that the proposed method exhibits significant computational and accuracy gains over competing approaches, as well as robustness to moderate and severe spatial dependence. Finally, we demonstrate the real-world utility of the proposed method by applying it to frame-to-frame video surveillance data, where it accurately detects small, closely separated subjects, a task where existing methods are significantly slower and highly prone to spurious detections due to not accounting for spatial dependence. A second application on 3D fibrous media is deferred to the Appendix.

In many statistical settings, two types of data are available: coupled data, which preserve the joint structure among variables but are limited in size due to cost or privacy constraints, and marginal data, which are available at larger scales but lack joint structure. Since standard methods require coupled data, marginal information is often discarded. We propose a fully nonparametric procedure that integrates decoupled marginal data with a limited amount of coupled data to improve the downstream analysis. The approach can be understood as an extension of coupling via projection in optimal transport. Specifically, the estimator is a solution for the optimal transport projection over the space of probability measures, which genuinely provides a natural geometric interpretation. Not only is its stability established, but its sample complexity is also derived using recent advances in statistical optimal transport. In addition to this, we present its explicit formula based on “shadow," a notion introduced by Eckstein and Nutz. Furthermore, the estimator can be approximated in almost linear time and in parallel by entropic shadow, which demonstrates the theoretical and practical strengths of our methods. Lastly, we present experiments with real and synthetic data to justify the performance of our method.

The rivalry between two football superstars Cristiano Ronaldo and Lionel Messi has always been a subject of extensive discussion. This study aimed to compare the level of consistency between the two players in scoring goals through 6 ways: right-footed kicks, left-footed kicks, penalty kicks, direct free kicks, long-range kicks, and headers. The data analyzed was the duration of time (minutes) each player took to score a goal in every match they played. The data was obtained from a football website called Transfermarkt.com. Competing Failure Modes (CFM) was used to measure the reliability of the two players in scoring goals based on those various ways. The results of CFM exploratory analysis showed that Ronaldo and Messi had the same level of consistency in scoring goals for more than 17 years of their professional football career. Both have been among most talented players in the modern football era with individual and team achievements that are far above other footballers around the world.

In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.

Considering recent advances in remote sensing satellite systems and computer vision algorithms, many satellite sensing platforms and sensors have been used to monitor the condition and usage of transportation infrastructure systems. The level of details that can be detected increases significantly with the increase of ground sample distance (GSD), which is around 15 cm - 30 cm for high-resolution satellite images. In this study, we analyzed data acquired from high-resolution satellite imagery to provide insights, predictive signals, and trend for travel demand estimation. More specifically, we estimate the impact of COVID-19 in the metropolitan area of Houston using satellite imagery from Google Earth Engine datasets. We developed a car-counting model through Detectron2 and Faster R-CNN to monitor the presence of cars within different locations (i.e., university, shopping mall, community plaza, restaurant, supermarket) before and during the COVID-19. The results show that the number of cars detected at these selected locations reduced on average 30% in 2020 compared with the previous year 2019. The results also show that satellite imagery provides rich information for travel demand and economic activity estimation. Together with advanced computer vision and deep learning algorithms, it can generate reliable and accurate information for transportation agency decision makers.

Risk assessment of hurricane-driven storm surge relies on deterministic computer models that produce outputs over a large spatial domain. The surge models can often be run at a range of fidelity levels, with greater precision yielding more accurate simulations. Improved accuracy comes with a significant increase in computational expense, necessitating the development of an emulator which leverages information from the more plentiful low-fidelity outputs to provide fast and accurate predictions of high-fidelity simulations. To properly assess the risk of storm surge over a geographic region at aggregated spatial resolution, an emulator must account for spatial dependence between outputs yet remain computationally feasible for high-dimensional simulations. To address this challenge, we exploit the autoregressive cokriging framework to develop two cross-covariance structures to account for spatial dependence. One approach uses a separable covariance structure with a sparse Cholesky prior for the inverse of the cross-covariance matrix; the other involves a low-rank approximation via basis representations. We demonstrate their predictive performance in the storm surge application and a testbed example.

We develop an extreme value framework for CoVaR centered on $v(q \mid p ; C)$, the copula-adjusted probability level, or equivalently, the CoVaR on the uniform (0,1) scale. We characterize the possible tail regimes of $v(q \mid p ; C)$ through the limit behavior of the copula conditional distribution and show that these regimes are determined by the joint tail expansions of the copula. This leads to tractable conditions for identifying the tail regime and deriving the asymptotic behavior of $v(q | p ; C)$. Building on this characterization, we propose a minimum-distance estimation approach for CoVaR that accommodates multiple tail regimes. The methodology links CoVaR and $\Delta$CoVaR to the underlying joint tail behavior, thereby providing a clear interpretation of these measures in systemic risk analysis. An empirical analysis across U.S. sectors demonstrates the practical value of the approach for assessing systemic risk contributions and exposures with important implications for macroprudential surveillance and risk management.

Motivated by applications in statistics and machine learning, we consider a problem of unmixing convex combinations of nonparametric densities. Suppose we observe $n$ groups of samples, where the $i$th group consists of $N_i$ independent samples from a $d$-variate density $f_i(x)=\sum_{k=1}^K \pi_i(k)g_k(x)$. Here, each $g_k(x)$ is a nonparametric density, and each $\pi_i$ is a $K$-dimensional mixed membership vector. We aim to estimate $g_1(x), \ldots,g_K(x)$. This problem generalizes topic modeling from discrete to continuous variables and finds its applications in LLMs with word embeddings. In this paper, we propose an estimator for the above problem, which modifies the classical kernel density estimator by assigning group-specific weights that are computed by topic modeling on histogram vectors and de-biased by U-statistics. For any $\beta>0$, assuming that each $g_k(x)$ is in the Nikol'ski class with a smooth parameter $\beta$, we show that the sum of integrated squared errors of the constructed estimators has a convergence rate that depends on $n$, $K$, $d$, and the per-group sample size $N$. We also provide a matching lower bound, which suggests that our estimator is rate-optimal.

Classical optimization algorithms–hill climbing, simulated annealing, population-based methods–generate candidate solutions via random perturbations. We replace the random proposal generator with an LLM agent that reasons about evaluation diagnostics to propose informed candidates, and ask: does the classical optimization machinery still help when the proposer is no longer random? We evaluate on four tasks spanning discrete, mixed, and continuous search spaces (all replicated across 3 independent runs): rule-based classification on Breast Cancer (test accuracy 86.0% to 96.5%), mixed hyperparameter optimization for MobileNetV3-Small on STL-10 (84.5% to 85.8%, zero catastrophic failures vs. 60% for random search), LoRA fine-tuning of Qwen2.5-0.5B on SST-2 (89.5% to 92.7%, matching Optuna TPE with 2x efficiency), and XGBoost on Adult Census (AUC 0.9297 to 0.9317, tying CMA-ES with 3x fewer evaluations). Empirically, on these tasks: a cross-task ablation shows that simulated annealing, parallel investigators, and even a second LLM model (OpenAI Codex) provide no benefit over greedy hill climbing while requiring 2-3x more evaluations. In our setting, the LLM's learned prior appears strong enough that acceptance-rule sophistication has limited impact–round 1 alone delivers the majority of improvement, and variants converge to similar configurations across strategies. The practical implication is surprising simplicity: greedy hill climbing with early stopping is a strong default. Beyond accuracy, the framework produces human-interpretable artifacts–the discovered cancer classification rules independently recapitulate established cytopathology principles.

Statistical estimation often involves tradeoffs between expensive, high-quality measurements and a variety of lower-quality proxies. We introduce Multiple-Prediction-Powered Inference (MultiPPI): a general framework for constructing statistically efficient estimates by optimally allocating resources across these diverse data sources. This work provides theoretical guarantees about the minimax optimality, finite-sample performance, and asymptotic normality of the MultiPPI estimator. Through experiments across three diverse large language model (LLM) evaluation scenarios, we show that MultiPPI consistently achieves lower estimation error than existing baselines. This advantage stems from its budget-adaptive allocation strategy, which strategically combines subsets of models by learning their complex cost and correlation structures.

We propose a topological framework for detecting Hopf-type dynamical transitions directly from scalar time series. The method combines delay-coordinate reconstruction with persistent homology and uses the maximum persistence of one-dimensional homology classes as a scalar descriptor of cyclic structure. For the supercritical Hopf setting, we derive finite-resolution persistence bounds that relate detectability of the reconstructed periodic orbit to its geometry, sampling quality, and finite-data perturbations. A derivative-based estimator is then introduced to localize the critical parameter from the sampled topological functional. The approach is evaluated on the Hopf normal form, the Lorenz system, and a reduced Belousov–Zhabotinsky model. The numerical experiments show accurate finite-resolution localization of the corresponding transitions and illustrate the effects of embedding parameters, temporal sampling, smoothing, observational noise, and transient removal. These results support persistent homology as an interpretable data-driven tool for detecting geometric reorganizations in nonlinear time series.