Papers

We propose a scaling law that takes into account model size and training data while explicitly splitting the latter into training steps and batch size (called three-term law). Fitting the proposed law on a large set of training runs, we find that it correctly recovers the scaling of the optimal batch size. Moreover, because it makes use of training runs with suboptimal batch size, our proposed law can be robustly fit with a significantly smaller amount of training runs. We further show that the three-term law can be used to derive scaling laws for suboptimal batch sizes, and that it matches previous empirical findings related to the critical batch size.

To address sensitivity analysis and optimization for a discrete-time stochastic epidemic model, we derive unbiased gradient estimators that accommodate uncertainties represented as distributions over the parameters of interest, such as those arising from Bayesian calibration. Specifically, we estimate the sensitivity of total infections over a finite time horizon with respect to the proportion immunized ($v$) and the contact rate ($\beta$). Comparing the proposed estimators with deterministic limit approximations based on large populations reveals differences due to the finite population and time horizon. The estimators exhibit lower variance than finite-difference estimators for the derivative with respect to $\beta$, but higher variance for the derivative with respect to $v$. Simulation experiments indicate parameter uncertainty reduces sensitivity to the parameters of interest. In particular, indirect effects of vaccination, such as herd immunity, are less pronounced compared to when parameters are known. For optimization problems balancing intervention and infection costs, incorporating parametric uncertainty leads to more conservative policies.

Anomaly detection in geospatial data is a crucial tool in geographic information science (GIS), with applications ranging from national security to public-health surveillance to the study of societal disparities. This work focuses on spatial scan statistics and addresses a key mismatch: spatial counts are typically aggregated into predefined regions (census tracts, zip codes, counties), whereas the most efficient scan algorithms operate on spatial point data. The standard remedy – collapsing each region to its centroid, as in widely used tools such as SaTScan – is convenient but, as we show, discards the region's spatial extent and causes a significant loss in statistical power. To resolve this, we propose a simple yet scalable fix: replace each spatial region with 20-50 points sampled uniformly from its geometry, and divide the region's measured and baseline counts evenly among them. This approach improves statistical power while maintaining computational tractability. A convergence analysis explains why so few samples per region suffice. We recommend this sampling-based conversion as the default way to apply point-based spatial scan statistics to region-aggregated data for anomaly detection.

Conditional inference trees (CIT) and conditional inference forests (CIF) reduce split-selection bias by testing features before choosing split thresholds, but repeated permutation tests and threshold searches can make these methods computationally expensive. We study CIT and CIF as top-$k$ feature-ranking methods for downstream prediction using real-data benchmarks, runtime ablations, and synthetic feature-recovery experiments. At a fixed node, if the features and permutation budget do not depend on the node responses, Bonferroni-corrected $+1$ Monte Carlo permutation $p$-values control nodewise rejection under the complete permutation null. CIF ranks 4th among 17 classification methods on 22 datasets and 3rd among 18 regression methods on 8 datasets. With Bonferroni correction held fixed, the CIF runtime ablations indicate that adaptive stopping and the number of thresholds searched have the largest measured effect on runtime: turning off adaptive stopping and using exact threshold search increase fitting time by 4.0–8.4$\times$ and 1.9–10.8$\times$, respectively, while downstream score changes are at most 0.011. Sparse high-$p$ simulations indicate that forest feature sampling can leave informative features out of many split decisions. Overall, the results support CIF as a top-$k$ feature-ranking method in the evaluated downstream prediction benchmarks.

We propose specification tests for discriminating among non-nested parametric survival models with a cure fraction, focusing on models that differ only in their baseline distributions. The proposed approach augments the null log-likelihood with information from competing models and applies a score test to assess whether the additional information is redundant. Because the test relies only on restricted maximum likelihood estimates, it avoids fitting augmented models. For two competing models, the score statistic reduces to a quadratic form in the sample mean of the individual log-likelihood differences. We show that its signed square root coincides with Vuong's test statistic, although our framework differs in three important respects: it tests the specific null hypothesis that a given model is the true data-generating process, it uses an unsigned statistic that extends naturally to $M \ge 2$ competing models, and it estimates the Kullback-Leibler bias by parametric bootstrap. The resulting MJ statistic combines the individual J tests to assess the global null hypothesis that at least one candidate model is correctly specified, while also providing a model-selection criterion.

Graphon estimation requires structural assumptions to address its intrinsic non-identifiability. A standard approach is degree-based identifiability, where the degree function is assumed to be strictly monotonic. This assumption is rather restrictive and fails for graphons with constant or non-injective degree function, even when distinct latent positions have different connectivity profiles. In this paper, we introduce rooted motif signatures as higher-order node-level representations for graphons. They extend the degree function by recording, at each latent position, the densities of rooted motifs such as triangles, cycles, paths, and other local subgraph patterns. We study the extent to which these signatures can distinguish latent positions beyond degree information. For generic finite-rank graphons, we prove that suitable rooted motif signatures determine the connectivity profiles of latent positions. We also explain why such a property cannot hold for arbitrary graphons without additional assumptions, since different latent positions may have identical rooted motif signatures. On the statistical side, we define empirical rooted motif signatures from a single observed graph and prove uniform concentration bounds for these estimators. Simulation experiments illustrate that rooted motif signatures can reveal latent structure in settings where degree-based representations are uninformative, including graphons with constant or non-injective degree functions and stochastic block models with equal block degrees.

Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.

Sample-based generative models are increasingly used for probabilistic forecasting in high-stakes decision settings, yet their training objectives are blind to the decision maker's cost structure. These models are commonly trained with strictly proper scoring rules, such as the energy score, which allocate their training signal in proportion to data density, with no awareness of where forecast errors are most costly for downstream decisions. We therefore propose decision-aware training for sample-based generative models, augmenting the energy score objective with a differentiable decision loss that directly penalises the cost incurred by acting on the model's forecast. This combined loss is theoretically grounded, as the decision loss is itself a proper scoring rule. We compute the decision loss via a differentiable optimisation layer. Its gradient concentrates in cost-sensitive regions of the output space, making the method's effects interpretable and predictable from the cost structure. We validate the method on one synthetic and two real-world tasks. In the synthetic task, the method corrects the mode weights of a learned bimodal distribution; in a wind power dispatch task, it concentrates improvements in the rare but costly tail region, and in a frost protection task, it improves how well the decision costs are anticipated. Our method yields generative models that retain full probabilistic forecasts while being better aligned with the decision maker's specific cost structure.

We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.

The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.

Most existing multitask learning approaches are limited by their reliance on task-specific loss functions tailored to the scale and type of each outcome. When outcomes differ across tasks, these losses are generally not directly comparable, which makes it difficult to formulate a unified objective and may limit information sharing across tasks. We propose a multitask transformation framework in which task-specific responses may differ through unknown monotone transformations. Motivated by high-dimensional biological applications in which the predictor dimension may diverge with the sample size while only a common subset of predictors is informative, we consider shared sparsity across tasks. Under this framework, we estimate the target functions and identify important predictors by optimizing a smoothed rank-based criterion with a group-Lasso penalty, implemented through a multitask deep neural network with a shared first layer. We establish the nonasymptotic excess-risk bounds, and variable-selection consistency for the proposed estimator. Simulation studies show that the proposed method achieves competitive prediction and variable-selection performance compared with competing approaches. Analyses of gene-expression studies with continuous, binary, and mixed outcomes further illustrate that the proposed method improves prediction and identifies biologically meaningful shared predictors.

Reproducible and generalizable assessment of treatment decisions requires principled handling of subsequent treatment switching that may inform expected outcomes and shift across cohorts and over time. To effectively account for informative treatment switching, we propose an instrumental variable approach that characterizes the poorly documented expected outcomes at switching as unmeasured confounding. After establishing the baseline treatment as a viable instrumental variable, we constructed an estimating equation based on the association between the centered instrumental variable and a martingale style residual process that identifies the treatment effect under structural cumulative survival model. Our proposed method is doubly robust, i.e., valid whenever either of baseline propensity model or no-switching outcome model is consistently estimated. A co-training of treatment effect parameter and survival outcome regression model eliminated the requirement of observing a no-switching subset under semi-parametric additive hazards models. We further developed an baseline-survival-corrected cross-fitting approach to incorporate general machine learning models for estimating nuisance models. Numerical results demonstrated the validity of our method in various settings when a basket of benchmark solutions produced biased or contradictory results. We applied our method to comparison of high-efficacy vs standard efficacy disease modifying treatments as the second line therapy of multiple sclerosis.

The NodeIdentifyR Algorithm (NIRA) has been increasingly applied in psychological network research as a simulated-manipulation approach for comparing the network-level impact associated with different nodes. However, NIRA is restricted to binary variables, and the theoretical interpretation of the manipulated node intercept is primarily grounded in the symptom-activation context of psychopathology. To address these limitations, we extend NIRA to Gaussian graphical model and propose the Gaussian Graphical Model NodeIdentifyR Algorithm (GGMNIRA). GGMNIRA systematically manipulates the conditional mean of each node and uses Kullback–Leibler (KL) divergence to quantify the resulting change in the joint network distribution. Because the conditional mean has a meaningful interpretation for variables across different areas of psychology, GGMNIRA can be applied to a broader range of psychological research contexts. We also developed complementary procedures for evaluating the uncertainty of GGMNIRA results, including a correlation-stability coefficient and a nonparametric bootstrap difference test for KL divergence. In addition, GGMNIRA was extended to bridge Gaussian graphical model, allowing simulated manipulation to be conducted at both the node and construct levels in multi-construct networks. All methods are implemented in the R package "GGMNIRA".

The study on shared micro-mobility is based on trip modeling and user data. User segmentation in shared micromobility systems is traditionally studied by aggregating trip-level observations into user-specific summary measures before applying clustering techniques. Such aggregation can obscure trip-level variability and lead to ecological fallacies if results are interpreted as applying to individual records. We propose a Bayesian finite mixture model for multivariate categorical count data that clusters users directly from repeated trip-level observations while preserving the full categorical structure of individual travel behavior. This approach focuses on identifying heterogeneous mobility users from high-dimensional categorical trip behavior while accounting for uncertainty in cluster assignments. Users are the fundamental unit of analysis for exploring latent cluster patterns. The model represents each user with a product-multinomial likelihood with latent cluster membership. The methodology is illustrated using a one-year trip record of shared bikes and e-bikes from the Municipality of Venice, Italy, comprising over 220,000 trips made by more than 11,000 recurrent users. The analysis identifies eight distinct latent mobility profiles corresponding to localized, commuter-oriented, tourist-oriented, central, and inter-zonal travel behaviors. The proposed framework provides a flexible and computationally scalable approach for clustering repeated categorical observations and is readily applicable to other large-scale behavioral and transportation datasets.

Understanding the fundamental limits of recovering planted subgraphs in random graphs is a central challenge in high-dimensional statistics and theoretical computer science. While existing work has largely focused on special subgraph families such as cliques, bicliques, or dense blocks, the exact recovery of a general planted subgraph in Erdős–Rényi random graphs remains poorly understood. In this paper, we study the exact recovery of an arbitrary planted subgraph $\Gamma = \Gamma_n$ embedded in a dense Erdős–Rényi random graph $\mathcal{G}(n,q_n)$, where edges within $\Gamma$ are present independently with probability $p_n > q_n$. Our main results identify sharp conditions under which exact recovery is possible with high probability, and we establish matching lower bounds showing the necessity of these conditions. The resulting statistical threshold is characterized by a new graph-theoretic quantity, which we term the minimal maximum subgraph density. This quantity is defined as the maximum subgraph density of the smallest induced balanced subgraph of $\Gamma$. We then turn to the problem of recovery under polynomial-time constraints. We propose a computationally efficient recovery algorithm that applies to arbitrary planted subgraphs and analyze its performance in terms of certain spectral properties of the adjacency matrix. In addition, we derive computational lower bounds for recovery using the low-degree polynomial framework, establishing regimes where recovery is statistically possible but computationally hard. Finally, we consider several extensions of our setting, including recovery in semi-random models and weaker notions of recovery.

Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.

We propose for a quantile regression an estimation method for transferring knowledge using two $L_1$ penalties based on an estimator obtained from a source database. The proposed transfer learning estimator satisfies the properties of consistency and sparsity. Its convergence rate and asymptotic behavior are studied in several scenarios. This knowledge transfer results in a shorter computation time than that of the standard adaptive LASSO estimator. Another advantage of our method is that it can be applied to models with non-Gaussian errors. In addition, in order to implement the computing of the adaptive transfer LASSO quantile estimator, we propose an algorithm. The simulations confirm the theoretical results and demonstrate that the adaptive learning estimator, calculated using the proposed algorithm, is more competitive than the LASSO estimators. Finally, we illustrate the practical utility of the proposed transfer learning estimator and algorithm using a real-data application involving the physicochemical properties of protein tertiary structures.

There is increasing evidence of a directional relationship from academic performance to self-efficacy. We develop a Bayesian model for investigating this relationship when academic performance is measured on an ordinal scale and self-efficacy on a continuous scale. The model allows latent academic achievement to enter the self-efficacy regression as a predictor, while Bayesian variable selection identifies factors associated with either response. The resulting conditional formulation yields an interpretable regression characterisation of how latent academic achievement relates to self-efficacy. Furthermore, it enables a tailored partially collapsed Gibbs sampler that analytically integrates out the regression coefficients when updating the variable inclusion indicators. Simulation studies demonstrate that the proposed conditional formulation and tailored sampler improve sampling efficiency and variable-selection performance relative to a recent, more general joint Gaussian copula regression formulation. We apply the methodology to data from the longitudinal study of Australian children, a landmark national cohort study covering children's education, social and emotional wellbeing, health and family circumstances. The model and analysis shed light on how latent academic achievement relates to self-efficacy in Australian children, and reveal that the two outcomes differ markedly in the range of covariates associated with each outcome.

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.

We present a simple, yet general approach to study the scaling properties as the dimensionality of Metropolised MCMC sampling algorithms increases. The study relies on the symmetries of the Hamiltonian formalism and ultimately on the symmetry of the Metropolis-Hastings formula. Our findings contain, as particular cases, many known results for the Random Walk Metropolis, MALA and other algorithms. In addition, they provide, in an easy way, new optimal scaling results for a variety of proposal mechanisms, including implicit proposals and proposals generated with the help of differential equation integrators. The analysis applies to targets that are products of a given, not necessarily univariate distribution, and also to cases where the different terms in the product are scaled differently. We show how to construct gradient-based MALA-like proposals where the variance of the proposal as the dimension $d$ increases may be taken as $O(1/d^\mu)$, with $\mu>0$ arbitrarily small, to be compared with the values $\mu = 1$ for Random Walk Metropolis and $\mu=1/3$ for MALA.

Scientific reasoning is an increasingly important capability of large language models, yet improving the robustness and efficiency of training such reasoning remains a key open challenge. We study this problem in instruction-based molecular optimization, where answer-only supervised fine-tuning (SFT) collapses multi-step reasoning and reinforcement learning with verifiable rewards (RLVR) suffers from sparse feedback. Reference-guided Policy Optimization mitigates both by anchoring policy updates to dataset-provided references, but its effectiveness is tightly coupled to reference quality: weak or misaligned references impose a performance ceiling. To overcome this ceiling, we propose active reasoning, a paradigm in which the policy actively decides, on a per-instance basis, when to imitate a reference and when to reinforce its own discoveries, while continuously upgrading what it imitates. We instantiate this paradigm as Active Group Relative Policy Optimization (Active-GRPO), realized through two coupled mechanisms: active imitate-reinforce and active referencing. The former performs imitation learning when the reference still outperforms the policy's own candidates, and shifts to self-improvement via reinforcement learning once the policy has generated molecules that surpass the reference. The latter continuously upgrades the reference itself by replacing it with the best policy-generated candidate discovered so far, progressively raising the imitation target and ensuring that reference guidance remains informative-rather than restrictive-throughout training. Across TOMG-Bench MOLOPT, Active-GRPO improves average SRxSim from 0.0959 for GRPO and 0.1665 for RePO to 0.1773 under matched three-seed evaluation, with statistically significant gains on LogP, MR, and QED.

Structural equation modelling (SEM) is widely used in survey-based business and information systems research to assess latent constructs and theory-driven structural relationships. However, SEM path significance is obtained within a particular model specification and may not show whether findings remain stable under alternative estimation frameworks. This study develops and demonstrates a staged robustness analysis framework that connects SEM, ordinary least squares (OLS) regression, and Double Machine Learning (DML). SEM is first used to refine the measurement structure and estimate the robustness-baseline SEM model, in which the full theory-specified structural path system is retained for downstream robustness analysis before final structural path evaluation. OLS regression is then applied to SEM-derived construct scores as a transparent regression benchmark. Finally, DML-style residualisation is used to examine whether each tested focal relationship remains stable after flexible machine-learning-based adjustment for observed controls. Learner-sensitivity checks compare Random Forest, Gradient Boosting, and Support Vector Machine learners, and selected reverse-direction diagnostics are used to examine directional sensitivity. The framework is demonstrated using a FinTech Digital Customer Intimacy survey model. The findings identify which relationships are stable across SEM, OLS, and DML-style checks, and which require more cautious interpretation. A reproducible Google Colab workbook and generated result files are publicly available, providing a reusable template that researchers and students can adapt to other survey-based latent-construct studies. The paper contributes a practical robustness workflow and interpretation guide for survey-based researchers seeking to complement SEM with conventional and machine-learning-based robustness checks.

Knowing which training examples drive outputs is fundamental to auditing, correcting, and understanding language models, yet for modern LLMs this remains expensive, approximate, and largely post-hoc. Standard language models generate tokens through a dense network pathway, causing training data's influence to be distributed across parameters rather than organized along explicit, traceable components. We introduce a prototype language model architecture, Prototypes for Interpretable Sequence Modeling (PRISM), that forms each prediction via a sparse, non-negative mixture of learned prototypes, trained with clustering objectives that anchor each prototype to coherent neighborhoods of training examples. Across architectures from 130M to 1.6B parameters trained on up to 50B tokens, prototype language models either surpass or remain within 2.5 percentage points on average downstream accuracy of matched dense baselines. We show that sparse prototype structure localizes curvature in the loss landscape, yielding a more tractable Hessian and enabling training data attribution that is ~500x faster than post hoc baselines when consuming equivalent memory. Calibrating linear prototype controllers can improve downstream accuracy by roughly 3 points while tracing those corrections back to training neighborhoods, and targeted prototype suppression can remove model behaviors without finetuning or measurable loss in generation quality.

Transformer-based large models have demonstrated remarkable generalization abilities across different tasks by leveraging a context-aware attention module for in-context learning. With richer context, transformers adapt more effectively to the current use case without any parameter updates. However, the quadratic computational and memory complexity with respect to context length significantly slows data processing in softmax transformers. Linear transformers were proposed to address this issue by reducing the complexity to linear dependence on context length, but the design and understanding of the feature mapping in linear attention, from a theoretical viewpoint, remain unclear. In this paper, we investigate the approximation and generalization abilities of linear transformers under a two-staged sampling process from domain generalization. We show that linear transformers perform in-context learning as learning a mapping from context distributions to response functions. A dimension-independent convergence rate is obtained for our generalization analysis, which also exhibits the tradeoff between the regularities of data distributions and latent features. Guided by our theoretical framework, we propose a new perspective on activation and loss design for linearizing pretrained softmax large language models.

We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.

Predicting a response based on covariates is a fundamental problem in statistics and machine learning. However, profound difficulties arise when the underlying low-dimensional structural parameters are unidentifiable, as typified in dimension reduction contexts. Specifically,estimating these non-identifiable parameters inherently introduces severe nonconvexity. In distributed settings, this difficulty is further compounded by the challenges of data heterogeneity and communication cost. To overcome these intertwined barriers, we propose a novel distributed semiparametric framework. We formulate an adaptive homogeneity pursuit utilizing a trace-similarity penalty to effectively address data heterogeneity. To resolve the ensuing severe nonconvexity and communication bottlenecks, we introduce an invex relaxation technique coupled with a multi-step local update algorithm, ensuring stable convergence to global optimality with significantly reduced communication overhead. Theoretically, we establish a non-asymptotic model-free prediction error bound and prove that our estimator achieves a two-phase minimax optimal convergence rate and an sharper model-free prediction error bound. Furthermore, we provide theoretical guarantees for algorithmic convergence and communication efficiency. Extensive simulations and a real-world multi-center medical application validate the superiority of our method.

Combined unilateral and bilateral binary outcomes frequently arise in studies involving paired organs. The risk difference is a clinically interpretable measure for comparing treatment effects between groups. Existing confidence interval methods are primarily based on asymptotic normality and may fail to adequately reflect finite-sample distributional features, particularly skewness. To address this issue, we propose a distribution-based confidence interval derived from the probability distribution of the risk difference estimator and a modified MOVER procedure that accounts for intra-subject correlation. Their performances are compared with those of commonly used asymptotic methods through extensive simulation studies. Across a broad range of parameter settings, all methods exhibited satisfactory performance as sample size increased. The proposed distribution-based interval achieved coverage probabilities close to the nominal level with interval widths comparable to those of existing procedures. In small sample settings, it was able to capture skewness in the sampling distribution that was not reflected by methods relying on asymptotic normality. Analyses of two real-world datasets demonstrated the practical applicability of the competing methods and yielded consistent inferential conclusions. The proposed approach provides an alternative framework for interval estimation of the risk difference in studies involving combined unilateral and bilateral binary outcomes.

Hybrid controlled trials (HCTs) augment randomized controlled trials (RCTs) with external controls (ECs) to improve statistical efficiency when RCTs face limited sample sizes, slow accrual, or ethical constraints. However, valid use of ECs requires careful adjustment for covariate shift and outcome drift, as inappropriate borrowing may introduce bias and compromise inference. This tutorial provides a practical workflow for estimation and inference in HCTs. We first present a statistical analysis roadmap covering estimands, identification assumptions, eligibility alignment, matching, full and selective borrowing strategies, and both asymptotic inference and randomization tests. We then demonstrate step-by-step implementation using the SelectiveIntegrative and intFRT packages. The workflow is illustrated using a synthetic lung cancer dataset included in the intFRT package that mimics the CALGB 9633 trial and ECs from the National Cancer Database. The tutorial aims to help applied statisticians conduct transparent, interpretable, and reproducible HCT analyses that improve efficiency while maintaining valid inference.

Precipitation forecasts are judged by accuracy, but the decisions they support – when to restrict water, when to warn of drought – turn on noticing when a local regime is becoming abnormal, which forecast scores alone do not reveal. We ask whether one recurrent model can do both with little or no loss in forecast skill. We add a backward-coherence (reverse-martingale) penalty that keeps the network's hidden state smooth when read backward in time; the size of the resulting reconstruction defect becomes an online warning signal, monitored by a sequential change-point detector. The design is deliberately conservative. On real daily station data from four contrasting climates – monsoonal Taiwan, semi-arid Texas, temperate Germany, and Mediterranean Anatolia (Turkey) – the model matches a standard network's forecast skill everywhere, and makes the hidden state markedly steadier in every region. The novelty is the added information: on these real droughts the signal can alarm well ahead of the operational SPI-3 index, giving lead that neither the forecast nor the index provides. This benefit is not uniform across the four regions – large in one, partial in two others, and near-absent in the fourth. We offer the hydroclimatic character of drought onset, whether it precedes or merely coincides with the rainfall deficit, as a plausible explanation to be tested in future work, supported by a controlled synthetic study with known onset times. The contribution is thus a new and conservative way to read precipitation records: no loss in forecast skill, a steadier model, and an early-warning signal beyond the standard index.