Papers

Intervals with the same probability content can have different endpoint placements and widths. This matters for tolerance inference, where a reported interval must also satisfy a repeated-sampling content-confidence statement. We develop mean-tilted intervals (MTIs), a fixed-content family indexed by retained-mean balance. The zero-tilt member is the mean-preserving interval (MPI) induced by the residual-product criterion of Pouplin et al.; nonzero tilts move through admissible contiguous windows, including distribution-specific central and shortest intervals. For tolerance inference, we introduce TCSP, a tolerance-calibrated shortest-path action. TCSP chooses the retained order-statistic count by distribution-free scan calibration and reports the shortest closed window at that count. This keeps the certified interval action separate from generalized-posterior endpoint summaries. We also study a calibrated MTI-ECM comparator that profiles fitted content and tilt over a prespecified grid and applies an independent Dirichlet-process content-probability check. In iid simulations at tolerance confidence 0.95, we compare TCSP, MTI-ECM, Young-Mathew interpolation, and Wilks intervals across feasible content-sample-size cells and eight continuous distributions. The study emphasizes skewed distributions, where placement matters most, and excludes cells where the sample range cannot support the requested two-sided distribution-free statement.

Much like humans benefit from guidance while learning, reinforcement learning algorithms may benefit from additional supervision beyond rewards. Leveraging additional information during training to learn better representations and behaviors has been the focus of asymmetric reinforcement learning. This learning paradigm has proven effective under partial observability when additional state information is available, but also under full observability when more refined state information is available. Focusing on model-based reinforcement learning, we study the effect of asymmetric learning on observation representations and on privileged information representations. First, we identify a limitation in the privileged information representations learned by an asymmetric model-based algorithm known as the Informed Dreamer. Then, we propose a novel asymmetric representation learning objective using latent guidance, resulting in a new algorithm called the Reinformed Dreamer. Experiments across several benchmarks show a more consistent improvement over Dreamer than previous asymmetric approaches.

Understanding differences in microbial community structure is critical for uncovering risk factors and mechanisms underlying diseases such as colorectal cancer and preterm birth. Microbiome data present unique statistical challenges because they are compositional in nature, violating assumptions of many classical inference procedures. We propose an empirical bootstrap framework that enables robust hypothesis testing for equality of microbial community means across groups, including two-sample, paired, and multi-sample settings. The method accounts for the simplex structure of microbiome data and provides valid inference even in high-dimensional regimes. Through applications to two large-scale studies, fecal microbiota in colorectal adenoma and cancer patients, and vaginal microbiota in pregnancy with preterm birth outcomes-we demonstrate that our approach identifies clinically meaningful differences that conventional methods fail to detect, such as age-related differences in adenoma prevalence and race-associated disparities in vaginal microbiome composition. These results highlight the potential of resampling-based inference for advancing microbiome research, improving reproducibility, and uncovering clinically relevant microbial signatures.

Sharpness-Aware Minimization (SAM) aims to improve generalization by encouraging insensitivity to small, worst-case parameter perturbations. However, the notion of a "small" perturbation is inherently geometry-dependent: while existing SAM variants have explored a wide range of choices, a clear perspective on which geometries are most effective in practice remains elusive. Recent work on matrix-aware optimization, particularly the Muon optimizer, suggests that respecting the matrix structure of hidden-layer weights can lead to strong empirical performance. Motivated by this, we study matrix-aware geometry in both stages of SAM: we introduce a layerwise spectral inner perturbation for matrix-valued hidden-layer parameters and combine it with either AdamW/SGDW or Muon in the outer update. Across ImageNet-1K experiments on ViT-Small/16 and ResNet-50, we find that the combination of a spectral inner step with a Muon outer step performs consistently strongly, achieving the best validation accuracy on both models among the evaluated methods.

Deep generative models (DGMs) are widely used for complex high-dimensional data and increasingly applied to spatial and spatio-temporal modeling. Their generated samples implicitly represent the learned data distribution and associated uncertainty. However, for real-world data, assessing whether DGMs have learned the underlying process is difficult because the ground truth is unknown and evaluation often relies on observations alone. We evaluate representative DGMs, flow matching (FM), DDPM, score-SDE, and VAE, on a known non-stationary Gaussian random field. This paper provides comprehensive metrics to assess recovery of the ground-truth mean and covariance structures, with oracle samples and a stationary control as references. All four models recover the mean surface, while their covariance recovery differs across model families: DDPM and score-SDE recover the covariance structure reasonably well, FM exhibits mildly attenuated non-stationarity and slight variance under-dispersion, and VAE has difficulty recovering the covariance structure. An experiment on ERA5 temperature anomalies further demonstrates how the framework can support the validation and development of DGMs for complex real-world spatio-temporal data.

We illustrate in several examples that even neural networks of infinite width (specifically, Barron functions) may encounter substantial obstacles when used as a model class for problems in the calculus of variations. An instance of practical relevance concerns the bending, stretching and folding of a thin elastic shell with anchored or clamped boundary conditions where elastic energy could be reduced by folding along a circular line, but the neural networks can only describe straight folds along entire lines. Conversely, we show that there is no gap between the energy that Barron functions and Lipschitz functions can achieve for a large class of integral first-order functionals.

Large-scale observational studies increasingly rely on AI pipelines to extract structured variables from unstructured clinical records. A common workflow separates the data vendor, who validates extraction accuracy with a gold-standard sample, from the downstream researcher, who receives only the extracted dataset and summary accuracy statistics. We develop a bias-correction framework for the Cox proportional hazards model when covariates are subject to AI extraction error. Within a unified multivariate calibration framework, we show that the naive Cox estimator's bias decomposes into a leading-order calibration term and a second-order residual that vanishes as extraction accuracy improves. The leading-order term yields a corrected estimator that operates as a post-hoc matrix multiplication on the output of any standard Cox software. We further derive bias-adjusted confidence intervals that incorporate calibration uncertainty and a sensitivity diagnostic for assessing whether the neglected residual could materially affect inference. Synthetic data experiments with cross-dependent extraction errors and controlled nonlinear calibration violations confirm that the correction substantially reduces bias and achieves near-nominal coverage even under mild violations of the linear calibration assumption. The framework yields a concrete reporting specification: a short list of summary statistics that data vendors should provide alongside any AI-extracted covariate dataset used in survival analysis.

Wind-power variability is a major challenge for the reliable integration of utility-scale wind energy into modern power systems. Although wind-speed statistics are often described by simple parametric distributions, translating these statistics into turbine-level power fluctuations is nontrivial because the relationship between wind speed and power is highly nonlinear and changes across different turbine operating regimes. Here, we develop a two-regime statistical framework for wind-power distributions. In the aerodynamic operating regime, between the cut-in and rated speeds, the turbine power follows an approximate cubic dependence on wind speed. Starting from a physically motivated Rician model for the wind-speed magnitude, we derive an analytical expression for the corresponding wind-power distribution using a nonlinear change of variables. In the control-dominated near-rated regime, where active blade-pitch and generator control regulate the turbine output, the aerodynamic transformation is no longer applicable. Instead, we characterize the power deficit relative to the rated power and show empirically that its continuous tail is well described by a bounded stretched-exponential distribution for both individual turbines and wind-farm ensembles. These results provide a physically interpretable statistical description of wind-power fluctuations across the full operational range of utility-scale wind turbines.

Meta-analyzing natural indirect effect estimates from multiple studies is increas- ingly used to synthesize evidence on causal pathways of interest. However, stan- dard mediation meta-analysis approaches are typically based on structural equation modeling, which fails to account for mediator-outcome confounding, is not read- ily extended to address missing mediator and outcome data, and is often unclear about the target population to which the summary indirect effect pertains. In this work, we propose a novel method that addresses these limitations. Our ap- proach transports study-specific natural indirect effect estimates to a well-defined target population prior to evidence synthesis. The proposed methods enable the integration of studies that do not explicitly investigate mediation but collect data on the mediator to improve extensiveness. Using semiparametric theory, we con- struct flexible, data-adaptive estimators for the target parameter. Novel random- effects models and non-parametric analogues based on ANOVA sums of squares are also developed to decompose between-study heterogeneity into distinct sources that may affect the causal interpretability of the obtained findings. Finite-sample per- formance of the proposed methods is evaluated through simulated and real-world data.

Two-study meta-analyses are common in evidence synthesis but pose major statistical challenges. With only two studies, the between-study variance cannot be reliably estimated, rendering standard random-effects methods unstable. Here, we investigate meta-analyses based on Edgington's p-value combination method as an alternative approach, applying it to 1226 two-study meta-analyses from the German Institute for Quality and Efficiency in Health Care (IQWiG). Like fixed-effect meta-analysis, Edgington's method is calibrated under homogeneity. However, it adapts confidence interval width to observed between-study discrepancy without requiring explicit heterogeneity estimation. In all of the examined meta-analyses, this leads to confidence intervals that contain both study-specific estimates but remain informative. Edgington's method agrees with fixed-effect meta-analysis on statistical significance (at two-sided $\alpha$ = 0.05) in 91% of all meta-analyses, but can give wider intervals when study results are discrepant and narrower intervals when results are highly consistent. Weighted extensions of Edgington's method shift point estimates toward the more precise study while preserving much of this adaptive behavior. We conclude that Edgington's method offers a principled and practically useful complement to existing approaches for two-study meta-analysis, occupying a middle ground between standard fixed-effect and random-effects approaches.

The Expected Value of Sample Information (EVSI) is a powerful instrument to determine the value of additional evidence to inform an economic model. However, EVSI has been applied only to idealized data collection mechanisms, thereby reducing its potential applications in realistic studies. In this paper, we define a methodology to calculate EVSI when the additional evidence we aim to collect exhibits missing data; a very common challenge in real-world studies. First, we define how to simulate individual-level data and how to induce missingness inside the simulated data. We will reproduce Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR) missing data mechanisms. Then, we will apply the multiple imputation method to adjust for the bias related to the missing data. Finally, we use the imputed data to compute the EVSI using nonparametric regression methods. We apply the novel methodology to two different health economic models and compare the EVSI computed on data without missingness with the EVSI with different types of missing data. We show that the EVSI decreases when the additional evidence suffers from missingness, and therefore, we define a method to efficiently compute the sample size we need to collect to recover the idealized EVSI (without missingness). We find out that the number of additional samples needed to correct for missing data exceeds that coming from standard approaches. With this methodology, we compute EVSI when the additional data are affected by non-trivial forms of missingness, modeling both the MAR and MNAR mechanisms, and extending EVSI calculation to more realistic scenarios. The fact that the necessary sample size to recover the idealized EVSI exceeds that of standard methods suggests that this methodology potentially represents a novel technique for sample size calculation in realistic studies.

Many statistical problems concern the analysis of species distributions or, more generally, of discrete labeled quantities. Assessing species diversity constitutes a key step toward understanding population structure, and the Gini-Simpson index is among the most widely adopted diversity measures. In this manuscript, we examine several well-established nonparametric prior models for species frequencies and compare them with a newly proposed distribution within this framework. Specifically, we demonstrate that the conventional symmetric Dirichlet distribution leads to certain undesirable properties in terms of expectation and dispersion. These limitations can be mitigated by adopting an alternative symmetric Dirichlet specification, in which the parameter depends on the number of species. This modified formulation is characterized by analytical tractability and interpretability of its main summaries. Furthermore, when the number of distinct species is infinite, the classical Ferguson Dirichlet process exhibits unsatisfactory behavior compared to the more general Poisson-Dirichlet model. Notably, within this latter model, the posterior mean of the diversity index can be expressed as a convex combination of the optimal classical unbiased estimator and the prior expectation. Theoretical results are further supported by asymptotic analyses and systematically compared with their classical counterparts.

The framework of recanting twin path-specific effects has recently been propose to address the issue of intermediate confounding in causal mediation analysis, enabling the decomposition of the average treatment effect into identifiable fine-grained path-specific effects (PSEs). An open question, however, is the extent to which recanting-twin PSEs reflect the direction and relative magnitude of their corresponding natural PSEs. In this paper, we systematically characterize the behaviors of the recanting-twin PSEs in terms of magnitude, sign and ranking, benchmarking against natural PSEs. To achieve this, we first derive non-parametric, identifiable upper and lower bounds for the absolute difference between recanting-twin and natural PSEs in binary outcome settings. These bounds provide a practical way to assess whether discrepancies in magnitude between the two types of effects are substantial, thereby facilitating the use of recanting-twin PSEs as informative approximations of their natural counterparts. A simulation study is then conducted to evaluate frequency of disagreements in sign and ranking between recanting-twin and natural effects in non-linear, non-monotonic scenarios. Across four scenarios and 320 numerical settings, we find that disagreement ranges from 15% to 56% for ranking and from 5% to 43% for sign. Disagreement is strongly influenced by the non-observable correlation between counterfactual values of the intermediate confounder, suggesting that caution is needed when interpreting the magnitude and ranking of recanting-twin PSEs, and motivating future work to identify conditions under which the two types of PSEs yield consistent conclusions when non-monotonicity presents.

Robust generalised Bayesian methods protect parameter inference from contamination but do not generally infer its proportion or identify individual outliers. We propose Hölder–Bayes, which applies the Hölder divergence to a scaled model to construct a joint generalised posterior over model parameters and the contamination proportion, without specifying an outlier distribution. Each posterior draw flags a corresponding number of observations least compatible with the fitted model; averaging selections propagates joint uncertainty to outlier-detection scores. Under fixed, non-negligible contamination, we establish uniform boundedness of the posterior influence function, a finite-sample $n^{-1/2}$ bound on posterior-averaged excess risk, and a Bernstein–von Mises theorem. Tail overlap between the outlier distribution and the fitted model controls the population risk minimiser's bias and the associated change in limiting posterior covariance. Interpreting the posterior temperature as an affine volume scaling yields a closed-form, data-driven choice. Simulations and real-data experiments demonstrate robust estimation, contamination-proportion recovery, and posterior-guided data cleaning.

Machine learning demand forecasts optimize statistical accuracy yet leave excess operational volatility that inflates safety stock and amplifies the Bullwhip effect. We introduce Contextual Deconvolution (CD), a two-stage estimator that reframes demand sensing as a convex decomposition: a kernel-modulated banded operator separates transient promotion-driven shocks from a smooth structural baseline, and hierarchical partial pooling enables catalog-scale deployment without per-SKU training. The operator is data-derived, not imposed—it reduces to the identity wherever the promotional response is impulsive (most of M5, all of Favorita) and contributes only where genuine multi-day carryover exists, so the gains rest on the structural decomposition itself. Evaluating strictly out-of-sample on 30,490 M5 SKUs and 2,845 Favorita items, with calendar-aware baselines given CD's identical future calendar, we anchor the contribution on a full inventory-cost accounting: CD lowers safety stock, holding cost, and order variance but under-provisions event spikes, reducing total cost only when holding costs exceed $\sim$20% of stockout costs (95% CI $[17\%,25\%]$); otherwise it is an operational-stability and inventory-capital layer, not an expected-cost minimizer. Its accuracy contribution is reliability rather than central tendency: across eleven baselines, CD attains the lowest cross-sectional dispersion of per-SKU error and mis-forecasts by more than 200% on 0.8% of SKUs versus 9.9–20.6% for every baseline, ranking first on both in all four M5 draws. Because the Variance Ratio and std-based safety stock are minimized by any sufficiently smooth forecast, we treat them as diagnostics, not objectives. A supporting analysis shows the learned demand operators are non-normal, yet CD's compact parametric kernel matches their operational performance interpretably.

Among chess opening positions that a strong engine judges essentially equal (Stockfish 18 evaluation within 10 centipawns of zero, depth-stable) and that humans actually reach on Lichess (October 2025; 1,661 positions, 16.1M occurrences), human results are not balanced. Positions carry outcome skews, each the gap between its games' actual results and what the players' ratings predict, whose directions are stable properties of the naturally-reached position: some positions favour White, others Black. These skews reproduce across three re-partitions – disjoint player-account sets (primary), time, and disjoint rating bands – and on an out-of-sample month eight months later. On the primary split, each position's skew is measured once in each account group, and the replication slope asks how well one measurement predicts the other after removing rating and opening-family effects: one means undiminished carry-over; zero, no linear relation. We find 0.69 (family-clustered 95% CI [0.65, 0.74]), rising to 0.94 on the most-popular, best-measured positions. The slope's value depends on the position mix. Existence is the invariant claim: it survives every tighter evaluation band, search depth, calibration, and popularity cutoff we test, and replicates within blitz and rapid separately. The typical skew is small (median $|\delta| \approx 0.018$, about two percentage points of White score), yet it reproduces, position by position, across disjoint accounts. At these positions the disfavoured side also thinks longer. Even where the evaluation is most confident, it is not a sufficient statistic for human outcomes. The result is observational, and the causal question is left to a pre-registered randomised companion study.

The collective risk model is a fundamental framework in insurance ratemaking for modeling aggregate losses by combining claim frequency and claim severity components. A key structural requirement for a reliable experience rating system is a monotone ordering property: policyholders with worse past experience should receive a stochastically larger prediction for future losses, and hence a higher premium. Such credibility-type monotonicity is well documented in the insurance literature for univariate outcomes under classical random-effect models. However, extending this principle to the collective risk model is nontrivial because the relevant history is inherently multivariate, involving both claim counts and individual claim amounts. Hence, despite its practical importance, a corresponding credibility-type ordering has not been systematically developed for predictive distributions of aggregate loss. In this paper, we provide an example showing that standard collective risk model specifications can violate monotonicity: adding an additional but sufficiently small claim may decrease the premium, thereby creating perverse incentives for strategic reporting and potentially undermining the integrity of experience rating. Motivated by this pathology in view of insurance, we formalize a credibility order tailored to collective risk models and derive tractable sufficient conditions under which the predictive distribution of aggregate loss is monotone in past experience, ruling out such pathological violations. Numerical studies and an empirical illustration using real insurance data accompany our theoretical results.

In this article, a comprehensive framework for goodness-of-fit testing for distributions on Stiefel manifolds is developed. The approach is based on integrals of the squared differences between empirical and theoretical characteristic functions, yielding test statistics that are consistent against all fixed alternatives. For the Fisher-Bingham family of distributions, explicit computable forms of the test statistic are derived. Simplified expressions for important special cases, including the matrix Fisher, matrix Bingham, and uniform distributions are provided. In the case of testing uniformity on hyperspheres, we obtain the complete asymptotic distribution of the test statistic, enabling computationally efficient asymptotic testing. For general Fisher-Bingham distributions, we establish theoretically justified Monte Carlo testing procedures for both simple and composite hypotheses. Simulation studies demonstrate accurate Type I error control and strong power across a wide range of alternatives. The practical relevance of the proposed methodology is illustrated by an application to data on the orbits of comets.

We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=\sigma_t \xi_t \in \mathbb{R}^N,$ where the coordinates of $\xi_t$ are i.i.d.\ and the scalar mixture variable $\sigma_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Marčenko–Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.

In the field of building energy efficiency, the measurement and verification (M&V) of energy savings following energy efficiency measures often relies on the use of a calibrated statistical model. In order to obtain reliable estimates, the estimation of uncertainties associated with this procedure is recognized as a crucial aspect of M&V. Several approaches have been proposed in the literature to compute uncertainties but a recent work has raised serious doubts on their accuracy. This is especially the case for energy data with fine time resolution (hourly and daily), for which the issues associated with autocorrelations become more important. This work focuses on this issue for daily data with three main contributions: (i) a detailed comprehensive review of several methods proposed in the literature for linear and non-linear regression models (including exact formulas for linear models, the ASHRAE 14 approximate formula and resampling approaches) ; (ii) the improvement of approaches based on Bayesian modeling for non-linear regression models with autocorrelated residuals ; (iii) a large scale numerical test of the different approaches based on a synthetic dataset containing thousands of stochastic thermal dynamic simulations. Our results permit to test the accuracy of different approaches in a controlled numerical setting. In contrast with previous work, we find in particular that our Bayesian approach leads on our dataset to consistent estimates of uncertainties. This suggests that the proposed Bayesian approach offers a convincing method to estimate uncertainties in non-linear regression models with autocorrelated residuals often encountered M&V.

Meta-analyses of proportions often involve sparse event counts and zero-event studies. The beta-binomial model has been used as a flexible random-effects model for pooling overdispersed and rare-event proportions. However, the ordinary maximum likelihood estimator (MLE) may suffer from finite-sample bias when few studies are available or the mean event probability is close to the boundary. In this article, we propose a maximum penalized likelihood estimator based on the Jeffreys-prior penalty, which has shown favorable finite-sample bias and stability properties in sparse-data models. We also develop Wald-type and profile penalized likelihood confidence intervals (CIs). In a simulation study, the proposed estimator generally achieved higher convergence rates, lower bias, and lower root mean squared error than the ordinary MLE, particularly with fewer studies or lower event probabilities. Additionally, the profile penalized likelihood CIs maintained coverage close to the nominal level. The proposed method provides a useful alternative for meta-analyses of rare-event proportions.

We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and warm-starts each subproblem from the previous solution. A final refinement stage is then performed on the full-scale problem. Since most iterations are carried out on smaller subproblems, the overall computational cost is significantly reduced. We establish the convergence theory, prove that SPCG attains OLS prediction accuracy, and derive per-subproblem iteration bounds and complexity estimates. Numerical experiments show that SPCG reaches the target prediction accuracy with fewer iterations and less CPU time than full-data PCG and Iterative Double Sketching (IDS).

Item Response Theory (IRT) has recently been proposed as a framework for evaluating large language model (LLM) benchmarks by separating a model's latent ability from the properties of individual benchmark items. Existing neural IRT approaches, including PSN-IRT, estimate these quantities using point estimates, limiting uncertainty quantification and downstream statistical inference. We introduce Laplace-PSN-IRT, a post-hoc last-layer Laplace approximation that augments a trained PSN-IRT model with approximate Bayesian posterior inference, recovering calibrated uncertainty over model ability and item difficulty without retraining. The resulting posterior enables credible intervals, probabilistic comparisons between models, and propagation of parameter uncertainty into Fisher-information-based item selection. We show that most pairwise comparisons among 12 models on a standard LLM benchmark leaderboard are not statistically distinguishable despite differing point-estimate ranks. We further show that point-estimate Fisher information can become nearly zero for many benchmark items because it is evaluated at a single reference ability, whereas posterior-expected Fisher information remains substantially more stable across the ability range. Finally, posterior-expected Fisher information more accurately recovers full-benchmark ability rankings from small benchmark subsets in most experimental settings while matching point-estimate performance for the smallest subsets. We validate the calibration of the approximate posterior using held-out predictive coverage and find that modeling item difficulty as random while treating item discrimination as fixed produces well-calibrated uncertainty in this architecture.

Classical regression approaches, including ordinary least squares, rely on strong assumptions such as constant variance and normality of residuals, which are often violated in real-world data. Although log-transformation is commonly used to stabilise variance, it may introduce re-transformation bias and fail to address heteroscedasticity and asymmetric dependence structures adequately. To overcome these limitations, this study proposes a copula-based regression framework for modelling data in the presence of heteroscedastic error structures. The proposed copula-based regression framework separates marginal distributions of the response and explanatory variables from their dependence structure, allowing flexible modelling of different tail-dependent relationships. The proposed approach explicitly accounts for heteroscedasticity without requiring restrictive distributional assumptions. A comprehensive simulation study and two real-world applications were considered under heteroscedastic scenarios to compare the performance of the proposed method with existing methods. The simulation results demonstrated that the proposed copula-based model consistently outperformed conventional approaches, achieving an average mean absolute percentage error of 0.21, compared with 0.27 and 0.36 for the linear and log-linear models, respectively. In the first application, which exhibited clear heteroscedasticity, the copula-based model achieved the lowest MAPE, although the overall differences between the copula-based and GAMLSS models were not substantial. In Application 2, all models achieved low predictive performance, as there was a moderate linear relationship between the response and predictor variables. Overall, the findings indicate that no single model consistently dominates across all settings, while copula-based regression provides a flexible and competitive alternative for heteroscedastic data.

The issue of model misspecification is critical, yet it is often regarded as unavoidable in applied statistical modeling. Model misspecification can be mitigated by incorporating informative features and strengthening model formulations, such as through the integration of domain knowledge or structural constraints. In this paper, we propose a regression framework constrained by differential equations, which leverages first-order differential equations and adapts local polynomial regression techniques. Specifically, we focus on the local exponential growth model, characterized by an exponential-type differential equation. For this model, we examine the asymptotic biases and variances of kernel estimators constructed using Taylor polynomials of varying degrees. To evaluate model robustness, we conduct simulation studies comparing different estimators under two misspecification scenarios varying the levels of misspecification.

Standard offline reinforcement learning (RL) algorithms typically assume that the actions in the dataset are observed without error. However, in many real-world applications, the true actions are unobserved and only noisy proxies are available, causing existing RL methods to yield biased and potentially misleading conclusions. We study off-policy evaluation in infinite-horizon discounted Markov decision processes with hidden actions. By leveraging the next-state variable as a natural proxy for the unobserved action, we establish identification of the policy value and propose an influence-function-based estimator called LURE (Learning from the Unseen: Robust Estimator). LURE is multiply robust, remaining consistent under several combinations of correctly specified nuisance components, and is asymptotically normal, enabling valid statistical inference. To our knowledge, this is the first work to address offline RL with hidden actions. We demonstrate LURE's effectiveness through simulations and a sepsis management application using the MIMIC-III database.

Despite the empirical advantages of deep networks over shallow ones, theoretical depth separations largely concern approximation power, while algorithmic results are mostly limited to comparisons between two- and three-layer networks. In this work, we prove the first algorithmic separation between constant-depth and logarithmic-depth networks. Specifically, we identify a class of Boolean functions with hierarchically structured Fourier spectra that logarithmic-depth networks can learn efficiently using layerwise coordinate descent by reconstructing the spectra hierarchically and adaptively. We also exhibit a subclass for which every constant-depth, polynomial-width network with sufficiently regular activations and controlled spectral norms must incur constant $L^2$ approximation error under the uniform distribution over the hypercube.

Lloyd's $K$-means algorithm, also known as naïve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.

This paper proposes a GARCH-type volatility model in which level-and-slope updates of a latent power-law kernel generate state-dependent decay of past shocks within a two-dimensional Markov state. We derive a joint Foster–Lyapunov condition and establish positive Harris recurrence and uniqueness of the invariant distribution. Simulations show substantial low-frequency persistence in log-squared innovations, especially near the diagnostic stability boundary. Empirically, the model captures a substantial portion of observed volatility persistence and delivers competitive out-of-sample forecast accuracy using only a two-dimensional Markov state.

Externally controlled single-arm trials provide an option when limited patient populations or ethical constraints make concurrent randomized controls impractical, including in rare diseases and investigator-initiated trials with limited recruitment. Standardization improves comparability by aligning external controls with the enrolled population, but does not by itself preserve planned power. Differences between anticipated and enrolled covariate distributions can change the precision of the standardized control estimate and leave a fixed-size trial underpowered. We propose an outcome-blinded, covariate-adaptive sample size re-estimation procedure that translates these changes in precision into updated recruitment targets. Historical-control data and accumulating active-arm baseline covariates are used to update external-control standardization and its estimated precision. Adaptation requires no active-arm outcomes and retains the prespecified clinically meaningful effect to be detected. We state sufficient conditions for type I error calibration and power after adaptive stopping, assess operating characteristics through simulations, and illustrate implementation using Alzheimer's Disease Cooperative Study data. In the primary simulations, adaptive recruitment restored power lost under standardized fixed designs, achieving at least the target power across the distribution-shift settings and outcome models examined. Type I error rates were close to nominal, although mild inflation remained with a smaller historical-control sample. The application illustrated how accumulating baseline information guided recruitment revisions. By adapting sample size to the precision of the population-standardized control estimate, the proposed procedure addresses a source of power loss that standardization alone does not resolve.