Papers

We study online estimation of spiked covariance eigenvalues from observations distributed across $L$ nodes with heterogeneous and persistent effective sample sizes. In the proportional high-dimensional regime, local Rayleigh statistics are deterministically distorted by node-specific aspect ratios $c_{\ell,t}=p/N^{\mathrm{eff}}_{\ell,t}$, and direct aggregation of uncorrected statistics converges to the wrong limit. We propose a correct-then-aggregate framework in which each node removes its deterministic bias via an inverse Rayleigh transfer map, and the server fuses corrected estimates using adaptive soft-max weights based on predictable fluctuation metrics, transmitting only $O(k)$ scalars per active node per round. We establish consistency and asymptotic normality of the global estimator, enabling valid online inference, and derive non-asymptotic bounds quantifying how accuracy improves with the number of nodes and their effective sample sizes. The adaptive weights achieve variance reduction comparable to oracle inverse-variance weighting, confirming the data-driven construction is nearly efficient. Simulation studies validate these properties. An application to cross-venue monitoring of a dominant market factor shows the method tracks systemic risk in real time while substantially reducing communication cost relative to a centralized pooled approach.

Despite the wide variety of existing Buy-`Til-You-Die (BTYD) models, nearly all rely upon the convenient assumption of transactions following a Poisson process. As modern customer bases grow larger and more diverse, a major gap in the marketing literature is BTYD models that can account for heterogeneity in timing patterns across millions of customers. This paper addresses that gap, introducing a family of models that assume transactions follow a Weibull renewal process and developing a highly scalable scheme for parameter estimation based on an amortized variational inference procedure. The proposed model fits to a proprietary dataset of 5 million online retail customers in 8 minutes which would take the current state-of-the-art an estimated 3-4 days. We show both theoretically and empirically that this dramatic improvement in computational performance comes with no appreciable change to either model interpretation or predictive performance. Beyond scalability, gradient-based variational inference also makes it easy to extend the model to covariates, which we illustrate on a public dataset of 4 million political donors during the 2020 US General election cycle. More generally, this paper demonstrates how to blend recent advances in approximate Bayesian inference and the tools of modern machine learning to dramatically improve the efficiency and expressivity of probabilistic models for customer base analysis.

The strong asymptotic freeness established in [28,53] allows for the study of norms of Gaussian matrix polynomials via corresponding free operator counterparts. For spectral edges of semicircular free counterparts to symmetric Kronecker-Gaussian matrices, Lehner in [39] determined closed-form analytical characterizations. As an alternative to classical spectral methods, in [62], we created a Random Duality Theory (RDT) based framework for studying these problems and reproved Lehner's formula for definite matrix coefficients. In this work, we develop the RDT machinery further and achieve strong progress in several key directions: (i) Symmetric Square Case: We consider indefinite matrix coefficients and provide spectral edges lower bounds that match Lehner's formula. (ii) Asymmetric Non-Square Variants: We establish RDT asymmetric analogues to Lehner formulas and prove that they lower-bound the spectral edges. (iii) Interlaced Decoupling Property: We uncover that the obtained analogues exhibit a remarkable interlaced decoupling property. Additionally, we consider semi-definite programming (SDP) formulations that allow for the practical solution of the obtained analytical characterizations. Theoretical predictions obtained through solving SDPs are compared to numerical simulations, showing a striking agreement even for problem dimensions on the order of a few hundreds.

Pairwise randomization can yield substantial efficiency gains in experiments. Yet methodological guidance cautions against pairwise randomization, especially in settings with attrition, partly because common practices for estimation (i.e., pair fixed effects) imply discarding data from incomplete pairs thus exacerbating data loss from attrition. This practice of dropping incomplete pairs reduces statistical power of tests as well as precision of estimates, in paired experiments, compared to designs with less finely stratified treatment assignment. We argue that this concern is misplaced if attrition is independent of treatment status and potential outcomes, and that these issues follow from an inefficient use of the data that remains post-attrition. First, we show how, by using a specific permutation test, it is possible to use all observed units for inference (complete pairs and incomplete pairs where one unit attrits) while still exploiting the pairwise randomization design structure. The test procedure we suggest provides exact size control under the sharp null. Second, we study an optimally weighted estimator that efficiently combines within-pair and across-pair comparisons. Finally, we show that combining these two insights yields a test procedure that dominates the two commonly used inference methods (a paired $t$-test and the two-sample $t$-test) in power, for any level of attrition. Usefully for applied researchers, we show that the efficient procedure can be implemented via a weighted fixed effects regression, straightforward in standard software. In sum, our results provide researchers with practical tools for conducting experiments with pairwise randomization without sacrificing observations or statistical power when facing independent attrition.

Matrix-valued observations arise in multiplex networks, neuroimaging, and other domains where population-level patterns are often low-rank and subjects may express several latent patterns simultaneously. Existing tensor PCA methods provide continuous subject scores but their loading matrices can be difficult to interpret as population prototypes, while low-rank clustering yields interpretable prototypes with hard labels. We introduce a low-rank mixed membership model for matrix-valued data in which the expected value of each subject's matrix is a convex combination of latent low-rank basis matrices. The model yields both interpretable population-level extreme profiles and continuous subject-level memberships. Our multimodal extension shares memberships across modalities with modality-specific basis matrices and can restore identifiability when one modality is insufficient. We establish identifiability under a pure-subject condition, propose a constrained least-squares estimator and scalable algorithm with spectral initialization and low-rank refinement, and derive nonasymptotic error bounds. The estimator achieves a minimax-optimal reconstruction rate up to a logarithmic factor, with separate basis and membership convergence rates under a geometric condition. Simulations corroborate the theoretical rates and show strong performance. In an analysis of Human Connectome Project functional connectivity data, the proposed method identifies interpretable brain connectivity profiles whose estimated memberships are strongly associated with cognitive phenotypes.

Composite likelihood (CL) methods provide a computationally efficient alternative to full likelihood inference for complex multivariate models by replacing the joint likelihood with a product of lower-dimensional marginal or conditional components. Like the MLE, however, the maximum CL estimator (MCLE) is highly sensitive to data contamination. On the other hand, robust divergence-based procedures such as the minimum density power divergence (DPD) estimator require the full joint density and so scale poorly to complex multivariate models. We introduce the composite DPD (CDPD), a genuine statistical divergence built entirely from the low-dimensional component densities defining a CL, combining the computational scalability of CL with the robustness of the DPD. The resulting minimum CDPD estimator (MCDPDE) robustifies the MCLE without requiring integration over the full multivariate sample space. We establish consistency, asymptotic normality, and the influence function of the MCDPDE under regularity conditions on the component models alone, without requiring correct specification of the full joint distribution. We show that it is qualitatively robust for every positive value of its tuning parameter, unlike the MCLE recovered as the limit. Because its components can be chosen at the pairwise or cell level, the framework guards simultaneously against casewise and cellwise contamination. Operating directly on component densities rather than elliptical distance structures, it extends robust inference beyond the elliptical models to which most existing cellwise-robust procedures are confined. We develop computational algorithms implemented in the accompanying R package mvdpd. Simulation studies and real-data applications show that the MCDPDE achieves substantial robustness gains over the MCLE while retaining competitive efficiency under the assumed model.

Inference after variable selection in Cox regression is difficult because simple Wald-type intervals after selection can have poor finite-sample conditional coverage. We study a model-based bootstrap for inference after Cox-Lasso variable selection. The Cox-Lasso is fitted once to the original data to select a set of variables, after which an unpenalized Cox model is fitted using only those variables. Bootstrap samples are generated from a semiparametric plug-in Cox model specified by the coefficient estimate from this unpenalized Cox refit, the Breslow baseline cumulative hazard estimator, and a plug-in censoring distribution. In every bootstrap sample, the selected variable set is kept fixed and only the unpenalized Cox model is refitted. Under oracle-type sparse-model assumptions and standard Cox model regularity conditions, we prove first-order bootstrap validity for this procedure. In the simulation scenarios considered, percentile and studentized bootstrap intervals showed improved conditional coverage relative to the bootstrap-Wald interval in several small- and moderate-sample settings. Their performance was broadly competitive with debiased intervals, although the comparison depended on signal strength, tuning, and selection stability. A SEER breast cancer example illustrates that the procedure can be implemented in a realistic survival analysis and provides interpretable uncertainty quantification for effects reported after variable selection.

We study Bayesian optimization in a time-varying environment where the unknown reward function evolves according to a Gaussian process drift model. Existing GP-UCB analyses in this setting typically require the exploration parameter to grow with the horizon to maintain uniform confidence bounds. Using per-round local confidence events, we show that GP-UCB can instead be run with a constant exploration parameter and obtain an expected-regret bound whose coefficient depends on the drift rate. We also derive a sharper time-varying maximum-information-gain bound. For the squared exponential kernel, it yields $\tilde\gamma_T/T=\widetilde{\mathcal O}(\epsilon^{1/2})$ and expected average regret $\widetilde{\mathcal O}(\epsilon^{1/4})$ in the persistent-drift regime. The same constant-exploration analysis also yields realized-regret guarantees. Simulations support the predicted logarithmic dependence of the bound-suggested exploration parameter on $1/\epsilon$.

Background: Joint models for longitudinal and time-to-event data are widely used in clinical research. However, the choice of functional form linking the biomarker trajectory to event risk is often treated as a technical detail, despite its importance for model assumptions and interpretation. Default specifications may fail to capture clinically relevant features of biomarker trajectories. Methods: We provide a structured overview of functional forms linking longitudinal and survival processes in joint models. We compare association structures including instantaneous effects (current value, slope, and acceleration), cumulative and change-based formulations, shared random effects, and variability-based associations. Using longitudinal white blood cell measurements and overall survival data from the MIRAGE glioblastoma trial, we illustrate how different functional forms capture distinct features of biomarker trajectories and define different biomarker-risk relationships. Results: Instantaneous forms capture the biomarker's current level or short-term dynamics, whereas cumulative and change-based forms reflect longer-term exposure or trends. Variability-based structures quantify instability in the biomarker trajectory as an alternative prognostic signal. Association parameters depend on the functional form, biomarker scale, and time scale, and effect sizes are therefore not directly comparable. In the MIRAGE application, alternative functional forms produced different effect interpretations and, in some cases, different conclusions regarding the biomarker-risk relationship. Conclusions: The choice of functional form is a key modelling decision in joint models and determines the interpretation of the biomarker-risk association. Aligning the functional form with the scientific question is essential for valid interpretation and transparent reporting.

Tabular foundation models (TFMs) achieve strong performance through in-context learning, but context-dependent inference imposes substantial latency and memory costs, hindering large-scale deployment. We propose GEAR (Generative Expansion and Real Anchoring), a modular two-stage framework that distills TFMs into lightweight MLP or tree-based predictors that can be deployed on commodity CPUs. Stage 1 uses synthetic covariates solely as teacher-query locations and trains the student on soft TFM targets, expanding coverage beyond observed rows. Stage 2 re-anchors the student to the target distribution using real labels and out-of-fold teacher predictions, whitch avoids self-labeling leakage. We further derive a risk certificate characterizing the trade-off between generated-query volume and generator fidelity. Experiments on TALENT and TabArena demonstrate the broad applicability of GEAR. Two-stage MLPs outperform supervised MLPs by 1.81–2.00 AUC points on binary tasks and 1.19–1.35 points on multiclass tasks, with additional gains over real-data-only distillation of 1.76–2.19 and 2.09–2.40 points, respectively. On binary tasks, the gains also transfer to LightGBM and XGBoost, and all three student families outperform CatBoost, the strongest non-TFM baseline, in mean AUC. Ablations show gains beyond longer training or alternative warm starts, greater stability from staged than mixed optimization, and generator-dependent diminishing returns as query volume increases. Finally, GEAR reduces median inference time by 57–2866 times and peak prediction memory by 1.9–3.3 times, while retaining higher AUC than matched supervised baselines.

Graph neural networks (GNNs) are widely used, but how parameter sparsity affects the expressivity of relational (RGNNs) and temporal (TGNNs) variants is poorly understood. The Strong Expressive Lottery Ticket Hypothesis (SELTH) posits the existence of sparse GNNs that preserve Weisfeiler-Leman (WL) expressivity on static graphs. We generalize this existence result to a probabilistic statement for multi-relational and temporal domains via the relational WL (RWL). We prove that sufficiently parameterized RGNNs contain sparse subnetworks that maintain 1-RWL expressivity and derive a lower bound on the probability that a random pruning yields such a subnetwork. We show that common TGNNs and cross-graph message passing schemes admit RGNN reformulations such that they inherit these guarantees and, moreover, that the expressivity of a sparse RGNN is connected to its optimization behavior under common update regimes. Experiments instantiate the bound, compare it to empirical probabilities on synthetic data, and study how pre-training expressivity relates to optimization and prediction quality metrics on temporal and molecular benchmarks.

In some fields currently dominated by empirical approaches, such as state of health (SoH) prediction for lithium-ion batteries, phenomenological models motivated by quasi-physical thinking contain parameters to be estimated from experimental data. Often the structure of such models yields fully or partially confounded parameters, which are difficult or even impossible to estimate reliably. To preserve the desired model formulation and simultaneously improve the numerical conditioning for the problem we introduce a ridge regression scheme. An automated method is provided, based on information theoretic measures of model performance, which optimises the ridge regression hyper-parameter at each iteration. The formulae presented require fixed point iteration to solve for the hyper-parameter. Given a suitable starting value, analysis demonstrates convergence is very rapid. The optimal hyper-parameter selection mechanism is incorporated within an efficient regularised iterative generalised least squares mechanism, capable of fitting both heteroscedastic and serially correlated data as required. Simulation confirms the efficacy of the overall method.

Extreme value theory provides a fundamental framework for modeling rare and extreme events; however, threshold selection remains a persistent challenge, particularly in the presence of inliers such as instantaneous or early failures. Such observations commonly arise in applications including reliability studies and environmental data, where clusters of observations near the origin or at the origin can substantially distort classical threshold selection procedures and tail inference. In this paper, we propose a robust modeling framework that accounts for inliers, extremes, and the tail proportion. Parameter estimation is carried out using maximum likelihood. The proposed methodology is compared with classical numerical and graphical diagnostic tools, including the mean excess plot, parameter stability plot, Hill plot, and Pickands plot, as well as existing extreme value mixture models. The theoretical properties of the proposed model are established, and its performance is evaluated through extensive Monte Carlo simulations and real-data applications. The results demonstrate that the proposed methodology provides more accurate threshold estimation and more reliable extreme-value inference in the presence of inliers compared with existing classical approaches. Overall, the proposed methodology provides more accurate threshold estimation and tail inference in the presence of inliers, addressing key limitations of existing methods.

Online changepoint detection is the problem of detecting distributional changes in a data stream in real-time. A large body of methodology exists for the offline (fixed-size) setting, but applying these methods online quickly becomes infeasible since the per-observation computational cost and memory consumption typically grow at least linearly with the sample size. A recently proposed grid-based methodology (Moen, 2026) overcomes this by evaluating an offline test statistic over a sparse geometric grid of split points, with grid points spaced increasingly far apart further in the past. For a wide class of test statistics, this approach keeps update time and memory consumption growing logarithmic in the length of the data stream, while admitting finite-sample guarantees on the detection delay. Building on this methodology, we present gridcp, an open-source Python package that turns offline changepoint tests into efficient online detectors through a single, uniform interface. Users can choose from nine Numba-accelerated built-in tests, spanning changes in the mean, variance, covariance, and regression coefficients, as well as nonparametric tests and generalized likelihood-ratio tests for exponential-family models. Users can also supply their own test, which the package handles identically. For any test, gridcp provides Monte Carlo routines that calibrate the detection threshold to a target false alarm probability or average run length, including data-driven variants when no parametric null model is available. Through simulations and three real-data case studies, we show that calibration is accurate, that runtime scales favorably with both stream length and dimension, and that the complete pipeline, from calibration to deployment, runs efficiently on long real-world streams with only a short detection delay.

Researchers often choose a proxy dataset from many releases, transformations, or seeds. Search can make an invalid release appear adequate, while one adequate release does not establish that its generator is reliable. ProxyGuard controls both errors using prespecified bounded risks and a sealed target set. Named-release mode corrects for multiplicity and certifies specific releases. Direct shared-target mode evaluates independent mechanism draws on a common target, lower-bounds their favorable-score rate, and subtracts a bound on favorable scores contributed by invalid releases. Conditional on the target, release scores are independent, yielding a finite-sample mechanism-reliability guarantee without independent target batches or assumptions on release-level $p$-value dependence. We show that the mean-only penalty is sharp and derive a smooth-score certificate with additive target concentration. In a registered three-requirement study, direct mode raises power from 5.6% to 64.2% at reliability 0.95, while named mode remains stronger under high-signal evidence. Prospective audits span full-pipeline Rice–TVAE, which retrains on every draw, and a non-tabular text mechanism.

This paper develops asymptotic theory and feasible inference for unbounded-kernel order-k U-statistics under clustered sampling and weakly dependent time-series. The analysis first builds the complete order-2 pipeline, moving from clustered data to exact $m$-dependence and then to near-epoch dependence. The same logic is subsequently extended to general order k greater or equal to 2. Under clustered sampling, the theory allows arbitrary within-cluster dependence and growing, unbalanced cluster sizes. Under weak dependence, an i.i.d.-based approximating sequence carries the exact-m theory to near-epoch-dependent processes. The common combinatorial device partitions the sample into columns, separating sampling-generic tuples, where the first-order Hoeffding projection is analysed, from collision terms and higher-order Hoeffding projection remainders, which are controlled explicitly. Cluster-robust and HAC estimators of the covariance of the first-order projection, needed for feasible inference, are shown to be consistent. Empirical applications and data-calibrated simulations for inequality, L-moment, and rank-dependence statistics illustrate the finite-sample performance of the proposed procedures.

Building on a reparameterization for multivariate linear regression that yields a jointly convex penalized likelihood in the reparameterized regression coefficient matrix and the precision matrix, we show that the resulting scaled Gaussian loss is standard self-concordant. This places the joint estimation problem within composite self-concordant optimization and leads to two algorithms: a proximal gradient method and a damped proximal Newton method. In simulations, we evaluate algorithmic robustness, iterations to convergence, and elapsed time. In a protein expression application, compared with the classical-parameterization formulation, the proposed convex formulation attains similar mean squared prediction error and can be substantially faster when the fitted precision matrix is dense.

The emergence of molecular-targeted agents and immune-oncology therapies has fundamentally transformed oncology drug development, necessitating evolution beyond traditional dose-finding approaches designed for cytotoxic agents. While conventional agents exhibit predictable monotonic dose-response relationships, novel anticancer agents often demonstrate plateau-effect patterns where higher doses may compromise therapeutic benefit, requiring identification of optimal biological doses that balance efficacy and tolerability. The FDA's Project Optimus initiative emphasizes comprehensive dose optimization through parallel randomized cohorts and patient backfilling to better understand pharmacological profiles across multiple dose levels. Contemporary drug development increasingly prioritizes combination therapy alongside monotherapy evaluation, yet existing designs typically assume equivalent roles for both agents, diverging from clinical practice where novel agents combine with established treatments having limited dose options. This paper proposes a seamless dose-optimization design that adaptively evaluates both monotherapy and combination therapy based on efficacy and toxicity outcomes through adaptive subtrials with patient backfilling capabilities. The model-assisted framework employs predetermined Bayesian optimal boundaries, eliminating real-time model fitting while accommodating evaluation of both monotherapy and combination therapy and enabling sequential enrollment with strategic backfilling. Simulation studies demonstrate robust performance across diverse dose-response patterns relevant to contemporary oncology. The design addresses critical gaps between methodological assumptions and clinical reality, offering a practical approach that integrates monotherapy and combination therapy evaluation with efficacy-toxicity-based backfilling.

We show that logarithmic Voronoi boundaries for one-dimensional algebraic models in the probability simplex need not be algebraic. We give two explicit examples. For a union of two lines, the logarithmic Voronoi cell at a specific smooth rational point has a transcendental boundary point. For a cuspidal curve, the cell at the singular point contains a non-algebraic real-analytic boundary branch.

This paper proposes a multi-level Bayesian calibration approach that fuses information from heterogeneous sources and accounts for uncertainties in modeling and measurements for time-dependent multi-component systems. The developed methodology has two elements: quantifying the uncertainty at component and system levels, by fusing all available information, and corrected model prediction. A multi-level Bayesian calibration approach is developed to estimate component-level and system-level parameters using measurement data that are obtained at different time instances for different system components. Such heterogeneous data are consumed in a sequential manner, and an iterative strategy is developed to calibrate the parameters at the two levels. This calibration strategy is implemented for two scenarios: offline and online. The offline calibration uses data that is collected over all the time-steps, whereas online calibration is performed in real-time as new measurements are obtained at each time-step. Analysis models and observation data for the thermo-mechanical behavior of gas turbine engine rotor blades are used to analyze the effectiveness of the proposed approach.

Matching for causal inference can be computationally expensive at scale and can silently change the target population when overlap is limited. We propose Centroid-Referenced Mahalanobis Matching (CRM), which replaces global pairwise search with stratified sampling in two reference coordinates: each unit's Mahalanobis distance from the treated centroid and its Fisher coordinate along the treated-control mean shift. All covariates enter through the treated covariance geometry; CRM is therefore not principal-component preprocessing followed by nearest-neighbor matching. For $n$ units and $p$ pretreatment covariates, its implemented cost is $O(np^2+p^3+n\log n)$, simplifying to $O(np^2+n\log n)$ when $n \ge p$. We derive an error decomposition separating representation, support, discretization, and stochastic components. A pre-matching shortage fraction $\hat{\pi}$ estimates the population support restriction $\pi$, which enters a gap bound under bounded treatment-effect heterogeneity. Final retention is reported separately for capacity-driven exclusions. Under representation sufficiency, smoothness, and adequate cell capacity, CRM has a conservative two-dimensional histogram mean-squared-error bound $O(n_T^{-1/2})$; representation sufficiency is an additional assumption, not a consequence of ignorability given the original covariates. On Criteo, CRM retains at least 99.4% of treated units, has lower MaxSMD than corrected propensity-score matching in 31 of 36 large-scale configurations, and is roughly an order of magnitude faster. Moderate-size simulations favor some pairwise and weighting baselines on balance, locating CRM's contribution in scalability and explicit support diagnostics rather than universal finite-sample dominance.

We study finite-sample linear regression in the presence of varied and unknown label noise, focusing on the heteroskedastic and adaptive linear regression models. Heteroskedastic linear regression models settings where the labels are of varying quality. We receive $n$ pairs $(X_i,Y_i)$ with labels $Y_i=X_i^\top\beta+\varepsilon_i$, where $\varepsilon_i\sim N(0,\sigma_i^2)$ and the variances are unknown to the estimator. One natural measurement of the difficulty of this problem is the number of samples $m$ for which $\sigma_i^2\le1$ (larger $m$ is easier). We obtain a polynomial-time estimator with rate $\tilde{O}((nd^3/m^4)^{1/6})$ when $m\gg d^{3/4}n^{1/4}$, as well as nearly-matching lower bounds. For $d=O(1)$, our estimator achieves error $o(1)$ when $m\gg n^{1/4}$, whereas $L_1$ regression and other traditional approaches require $m\gg n^{1/2}$. In adaptive linear regression, the errors are drawn i.i.d. from an unknown distribution $p$, and our goal is to design a generic estimator that performs nearly as well as the best custom estimator that knows $p$. We introduce a (computationally inefficient) adaptive estimator that, so long as $p$ is a mixture of $k$ symmetric log-concave densities, achieves error comparable with the optimal estimator that knows $p$ and has $\tilde\Theta(n/k)$ samples. For $k=1$, we show that $L_q$ regression (with data-dependent $q$) gives a polynomial-time estimator. Finally, to study the computational limits of both problems, we introduce the planted linear regression problem, where $X_i\sim N(0,I_d)$, $m$ unknown samples are noiseless, and the rest have error $\varepsilon_i\sim N(0,1)$. We conjecture that recovering $\beta$ up to error $\ll\sqrt{d/n}$ (or exactly) may have an information-computation gap between $m=d+1$ and $m\sim d^{3/4}n^{1/4}$, as is suggested by our near-matching polynomial-time estimator and statistical query (SQ) lower bound.

In this paper, we first consider a Lasso estimator incorporating a weighted IPW factor,and obtain certain conditions for the Lasso variable selection to be consistent or inconsistent. Then we provide a modified version of the adaptive Lasso equipped with a weighted IPW factor and a KRR penalty function, establishing that the proposed adaptive Lasso possesses the Oracle property. Numerical simulations demonstrate that the proposed adaptive Lasso exhibits near-minimax optimality in performance

Ocean eddies are highly dynamic structures marked by frequent splitting and merging events. They exhibit complex spatio-temporal clustering that traditional Poisson models fail to capture. In this paper, a novel spatio-temporal Hawkes process is introduced to model self-exciting eddy fields on the basis of high-frequency ocean flow data. We formulate a triggering kernel that couples the self-excitation intensity with spatial deformation caused by the strain rate magnitude. To characterize the asymptotic behavior of the eddy field, we derive a Volterra integral equation that governs the expected eddy intensity over time and space. We develop an Expectation-Maximization (EM) algorithm that treats the unobserved parent-child relationships as latent branching structures for parameter estimation. We further extend this framework to accommodate a time-varying, non-homogeneous background intensity, modifying both the EM updates and the analytical Volterra solution accordingly. Finally, we propose a two-stage simulation framework utilizing a cluster representation algorithm. The simulated empirical paths of eddy formation are compared together with numerical solution of the Volterra equation for mean rate as validation of the Hawkes model.

We consider the quickest change detection from 1-bit quantized observations, where the post-change distribution is a parametric model with unknown parameters and the quantization thresholds are jointly chosen with the detection statistic. We propose an Adaptive-Quantization-Threshold CUSUM (AQuTeCUSUM) algorithm, which estimates the post-change parameter online and adaptively selects the quantization threshold to maximize the induced Kullback-Leibler divergence. Under suitable regularity conditions, we characterize the average run length and worst-case average detection delay of the AQuTe-CUSUM procedure, and show that it is asymptotically optimal in first-order as the average run length goes to infinity. Finally, we assess the performance of AQuTe-CUSUM for two distributions, namely the Gaussian and Poisson.

Adaptive therapy has emerged as a promising treatment strategy that exploits within-tumour competition to delay disease progression. Implementation, however, typically relies on indirect measurements of tumour burden and must account for potentially substantial patient heterogeneity. In this work, we capture patient-to-patient variability, parameter uncertainty, and imperfect biomarker measurements with a mathematical and statistical model that we calibrate to clinical prostate cancer data using a Bayesian inference framework. We use the resulting virtual cohort to demonstrate that, within the simple but now well-established Lotka-Volterra-based model, adaptive therapy robustly improves time-to-progression for the subset of patients that are predicted to eventually progress by the model. To account for other risk factors associated with larger tumour volumes, we introduce a new metric based on the risk of metastasis that demonstrates how adaptive therapy may be disadvantageous when sustained tumour burden is also considered. Given the ubiquity of uncertainty in oncology, we then describe several future modelling directions that also capture uncertainty in the temporal evolution of the underlying tumour or biomarker dynamics. Finally, we demonstrate how model misspecification and non-identifiability can lead to unreliable predictions, especially if uncertainty is inadequately captured.

We address two open questions in streaming PCA via Oja's algorithm: sharp operator-norm convergence for general rank under sub-Gaussian data, and distributional inference for the resulting subspace estimator. Existing convergence analyses, even in the rank-one case, either assume bounded data or leave non-vanishing remainder terms that prevent adaptation to a polynomially vanishing tail spectrum, while existing distributional results are confined to the rank-one case. Our convergence theory removes these remainder terms and yields a sharp rate. In the dense-tail spiked covariance regime, this rate matches the minimax rate up to logarithmic factors. More generally, we prove a matching lower bound, up to logarithmic factors, across both dense-tail and sparse-tail regimes under a mild nondegeneracy condition. The analysis yields a linearization of Oja's iterates, which in turn enables a high-dimensional Gaussian approximation for the general-rank subspace estimation error with an explicit limiting covariance. We also establish a row-wise Gaussian approximation over convex sets for the aligned difference, recovering prior rank-one results as special cases. For practical inference, we develop an online multiplier bootstrap algorithm and prove its consistency. Beyond streaming PCA, our techniques contribute to Gaussian approximation and bootstrap inference for nonconvex stochastic approximation.

Bayesian targets may converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this probability–sensitivity mismatch and its consequences for Metropolized Hamiltonian proposals. A vanishing-amplitude wiggly-energy model first gives the basic analytic obstruction: the potential perturbation tends to zero while its classical derivative is of order $r_\varepsilon/\varepsilon$. We then show that the same scaling arises naturally in a periodic elliptic inverse problem, where homogenization makes the forward map and Gaussian likelihood converge while differentiation with respect to a microscopic scale parameter retains an $O(1)$ oscillatory contribution. This provides a PDE origin for the single-scale wiggly mechanism. The main construction concerns a more demanding nested Weierstrass hierarchy, interpreted as an analytically tractable prototype for repeated corrector contributions across geometrically separated scales. There all previously resolved scales persist, adjacent classical-force increments grow geometrically like $(ab)^N$, and the limiting rough component may fail to possess a classical derivative. In this self-similar setting the matched Jackson quotient is structurally adapted to the refinement through dilation covariance and exact finite closure. Uniform negative-log-likelihood approximation yields explicit total-variation, Hellinger, and bounded quantity-of-interest bounds. Measurable kick–drift–kick maps remain exact after Metropolis correction, local field convergence propagates to fixed-length proposals and kernels, and the first classical HMC half-kick can have no fixed-step refinement limit. Numerical experiments on scale-structured inverse problems test the resulting resolution-stability mechanism across one- and two-dimensional inverse problems.

When experienced mayors govern better-performing municipalities, it is tempting to credit the leader. Yet municipalities with stronger administrative capacity may also be more likely to attract and elect experienced mayors, generating selection on pre-existing municipal capacity. We examine this identification problem using a balanced panel of 1,642 Peruvian district municipalities from 2015 to 2022. Within-municipality estimators and panel double machine learning recover a positive association between prior public-management experience and investment-budget execution, while formal education is substantially weaker. Negative-outcome controls reveal an important distinction. Across municipalities, mayoral human capital predicts GDP and HDI measured before the mayor took office, providing evidence of selection in cross-sectional comparisons. By contrast, the first-difference negative control aligned with the within-municipality design shows that changes in mayoral experience do not predict pre-determined district GDP, while they remain associated with budget execution. Sensitivity analysis and partial-identification bounds nevertheless do not support a clean causal interpretation of that within association. The results therefore separate two sources of variation often conflated in studies of political leadership and show how negative-outcome controls can sharpen the identification of leader effects.

Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $\eta(\xi)=D | \xi |^{\alpha}$ ($0<\alpha\le2$), so coherences decay at $\Gamma_{ab}=D | q_a-q_b |^{\alpha}$. For integer charge differences, the process descends to the wrapped variable $\Theta_u=L_u\bmod 2\pi$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<\alpha\le2$, while $\alpha>2$ yields an analytic obstruction. This boundary is exact, verified numerically here. A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise ($\alpha=d/p$) and Bochner subordination of Brownian phase by an $\alpha/2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($\alpha=0.5,1,1.5,2$) confirm this law-independence. In continuous time, $\alpha$-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.