Papers

Irregular errors such as heteroscedasticity and nonnormality remain major challenges in linear modeling. These issues often lead to biased inference and unreliable measures of uncertainty. Classical remedies, such as robust standard errors and weighted least squares, only partially address the problem and may fail when heteroscedasticity interacts with skewness or nonlinear mean structures. To address this, we propose a two-stage cumulative distribution function-based (CDF-based) beta regression framework that models the full conditional distribution of the response. The approach first transforms the outcome using a smoothed empirical CDF and then fits a flexible beta regression, allowing heteroscedasticity and nonnormality to be handled naturally through the mean-precision structure of the beta distribution. Predictions are mapped back to the original scale via the empirical quantile function, which preserves interpretability. A comprehensive Monte Carlo study shows that the proposed method consistently achieves good distributional accuracy and well-calibrated prediction intervals compared with OLS, WLS, and GLS. Application to the concrete compressive strength dataset demonstrates its stability and practical advantages.

Grokking, the abrupt transition from memorization to generalisation after extended training, suggests the presence of competing solution basins with distinct statistical properties. We study this phenomenon through the lens of Singular Learning Theory (SLT), a Bayesian framework that characterizes the geometry of the loss landscape. The key measure is the local learning coefficient (LLC) which quantifies the local degeneracy of the loss surface. SLT links lower-LLC basins to higher posterior mass concentration and lower expected generalisation error. Leveraging SLT, we develop a basin-selection perspective on grokking in quadratic networks: LLC ranks competing near-zero-loss basins by statistical preference, while the training-time transition between them is governed by optimisation dynamics. In this view, grokking corresponds to a transition from a higher-LLC (memorising) basin to a lower-LLC (generalising) basin that dominates the posterior. To support this, we derive analytic formulas for the LLC in shallow quadratic networks under both lazy and feature learning regimes. Empirically, we demonstrate that LLC trajectories estimated from training data track the onset of generalisation and provide an informative probe of the optimisation path.

Representation learning from complex data typically involves models with a large number of parameters, which in turn require large amounts of data samples. In neural network models, model complexity grows with the number of inputs to each neuron, with a trade-off between model expressivity and learning time. A precise characterization of this trade-off would help explain the connectivity and learning times observed in artificial and biological networks. We present a theoretical analysis of how learning time depends on input dimensionality for a Hebbian learning model performing independent component analysis. Based on the geometry of high-dimensional spaces, we show that the learning dynamics reduce to a unidimensional problem, with learning times dependent only on initial conditions. For higher input dimensions, initial parameters have smaller learning gradients and larger learning times. We find that learning times have supralinear scaling, becoming quickly prohibitive for high input dimensions. These results reveal a fundamental limitation for learning in high dimensions and help elucidate how the optimal design of neural networks depends on data complexity. Our approach outlines a new framework for analyzing learning dynamics and model complexity in neural network models.

Group relative policy optimization (GRPO), a core methodological component of DeepSeekMath and DeepSeek-R1, has emerged as a cornerstone for scaling reasoning capabilities of large language models. Despite its widespread adoption and the proliferation of follow-up works, the theoretical properties of GRPO remain less studied. This paper provides a unified framework to understand GRPO through the lens of classical U-statistics. We demonstrate that the GRPO policy gradient is inherently a U-statistic, allowing us to characterize its mean squared error (MSE), derive the finite-sample error bound and asymptotic distribution of the suboptimality gap for its learned policy. Our findings reveal that GRPO is asymptotically equivalent to an oracle policy gradient algorithm – one with access to a value function that quantifies the goodness of its learning policy at each training iteration – and achieves asymptotically optimal performance within a broad class of policy gradient algorithms. Furthermore, we establish a universal scaling law that offers principled guidance for selecting the optimal group size. Empirical experiments further validate our theoretical findings, demonstrating that the optimal group size is universal, and verify the oracle property of GRPO.

Risk forecasts in financial regulation and internal management are calculated through historical data. The unknown structural changes of financial data pose a substantial challenge in selecting an appropriate look-back window for risk modeling and forecasting. We develop a data-driven online learning method, called the bootstrap-based adaptive window selection (BAWS), that adaptively determines the window size in a sequential manner. A central component of BAWS is to compare the realized scores against a data-dependent threshold based on the bootstrap method. We provide an asymptotic justification for the bootstrap threshold, covering non-smooth scores such as the VaR check loss and the joint VaR–ES score, with an extension to stationary weakly dependent data via the moving block bootstrap. A single-break analysis further shows that BAWS rejects overlong windows crossing sufficiently large breaks. The proposed method is applicable to the forecasting of risk measures that are elicitable individually or jointly, such as the Value-at-Risk (VaR) and the pair of VaR and the corresponding Expected Shortfall. Through simulation studies and an empirical analysis, we demonstrate that BAWS often improves upon the standard rolling window approach and the recently developed method of stability-based adaptive window selection, especially when there are structural changes in the data-generating process.

A fundamental challenge in causal inference with observational data is correct specification of a causal model. When there is model uncertainty, analysts may seek to use estimates from multiple candidate models that rely on distinct, and possibly partially overlapping, sets of identifying assumptions to infer the causal effect, a process known as triangulation. Principled methods for triangulation, however, remain underdeveloped. Here, we develop a framework for causal effect triangulation that combines model testability methods from causal discovery with statistical inference methods from semiparametric theory, while avoiding explicit model selection and post-selection inference problems. We propose a triangulation functional that combines identified functionals from each model with data-driven measures of model validity. We provide a bound on the distance of the functional from the true causal effect along with conditions under which this distance can be taken to zero. Finally, we derive valid statistical inference for this functional. Our framework formalizes robustness under causal pluralism without requiring agreement across models or commitment to a single specification. We demonstrate its performance through simulations and an empirical application.

How nations shape the scientific frontier matters for technological competition, but standard metrics, including publication counts, citations, and disruption indices, look backward and fail to distinguish between fundamentally different leadership strategies. We develop and validate two forward-looking model-based measures and apply them to tens of millions of articles since 1990. The first embeds research pathways within an evolving hypergraph of concepts and scientists to identify leadership in emerging areas–work that anticipates where the scientific crowd is heading. The second embeds evolving samples of ideas and disciplines drawn upon in past research to identify leadership in surprising new directions as unexpected combinations become routine and science reorganizes around them. China became the global leader in emerging areas roughly a decade ago, well before it led in volume, reflecting a capacity to detect and amplify nascent consensus at scale. The United States and Europe show the opposite profile: declining emergence shares but persistent leadership in prescient work, especially research bridging disciplinary boundaries. These patterns replicate across databases, attribution methods, and strategic domains, including AI, biotechnology, energy, and semiconductors. Nations lead science by reading the landscape or by reshaping it, and the institutional requirements for each strategy lie in tension. The distribution of these strategies promises to shape the global structure of technological leadership for decades.

We propose a three-stage framework named as Recovery-Informed Strategy Enhancement (RISE) to forecast the recovery of Chinese outbound tourism following the coronavirus disease 2019 pandemic. The framework decomposes the forecasts into three parts: the initial forecasts, the terminal forecasts and the recovery curve forecasts that connect the two points. We integrate multiple sources of information and employ forecast combination techniques in all stages, enhancing both the accuracy and robustness of recovery forecasts. Compared with conventional forecasting approaches, our framework provides a structured and transparent pipeline to integrate model-based forecasts with expert-informed judgment under structural breaks and high uncertainty. Our findings demonstrate the effectiveness of this framework, offering an adaptable tool for recovery trajectory forecasting in post-crisis contexts.

We develop a two-stage, cut-posterior Bayesian framework for quantifying issue-specific legislative alignment in multi-party systems. The approach integrates a Latent Space Item Response Model (LSIRM), embedding legislators and bills in a shared Euclidean space, with Bayesian beta regression using text-derived topic proportions as bill-level covariates. The resulting legislator- and issue-specific coefficients allow polarization and cohesion to be compared across policy domains. The beta-regression layer must not feed back into the estimated latent geometry, so that the two components target a cut posterior. We estimate it with a two-stage Multiple-Imputation procedure that propagates uncertainty in the latent positions into every downstream quantity. In the 17th Korean National Assembly, fiscal domains such as Taxation and Grants and Local Government Budget show sharp polarization with tight within-party clustering, whereas Armed Services, Patriots, and Veterans exhibits weak party structuring and greater intra-party variability. The Democratic Labor Party forms a distinct cluster on several issues even where the two major parties are not strongly polarized, showing that legislative conflict escapes a single left–right ordering. The framework supports analysis of issue-structured voting in legislatures where one-dimensional ideal point models are unreliable.

We extend the unified kernel framework for transport equations and Koopman eigenfunctions, developed in previous work by the authors for deterministic systems, to stochastic differential equations (SDEs). In the deterministic setting, three analytically grounded constructions-Lions-type variational principles, Green's function convolution, and resolvent operators along characteristic flows–were shown to yield identical reproducing kernels. For stochastic systems, the Koopman generator includes a second-order diffusion term, transforming the first-order hyperbolic transport equation into a second-order elliptic-parabolic PDE. This fundamental change necessitates replacing the method of characteristics with probabilistic representations based on the Feynman–Kac formula. Our main contributions include: (i) extension of all three kernel constructions to stochastic systems via Feynman–Kac path-integral representations; (ii) proof of kernel equivalence under uniform ellipticity assumptions; (iii) a collocation-based computational framework incorporating second-order differential operators; (iv) error bounds separating RKHS approximation error from Monte Carlo sampling error; (v) analysis of how diffusion affects numerical conditioning; and (vi) connections to generator EDMD, diffusion maps, and kernel analog forecasting. Numerical experiments on Ornstein–Uhlenbeck processes, nonlinear SDEs with varying diffusion strength, and multi-dimensional systems validate the theoretical developments and demonstrate that moderate diffusion can improve numerical stability through elliptic regularization.

Precursor peaks in the wavenumber $k_{\mathrm{peak}}(t)$ maximizing the curl-of-vorticity spectrum have been observed to precede the dissipation peak in decaying turbulence. Because small perturbations in the initial condition can shift peak times, the associated lead time should be characterized statistically. We perform a pseudospectral DNS ensemble of $N_s=1000$ perturbed Taylor–Green vortex realizations at $N=256^3$ and $\nu=10^{-3}$. For each run we extract $k_{\mathrm{peak}}(t)$, several definitions of the precursor time $t_k$, the dissipation-peak time $t_\varepsilon$, and run-wise extrema including $K_{\max}=\max_t k_{\mathrm{peak}}(t)$ and $M_{\max}=\max_t\max_k \mathcal{C}(k,t)$, where $\mathcal{C}(k,t)$ is the isotropic curl-of-vorticity spectrum. The distribution of $\Delta t_{\varepsilon,k}=t_\varepsilon-t_k$ shows that the precursor typically leads, while rare lagging realizations occur and are strongly conditioned on $K_{\max}$. Using peaks-over-threshold extreme-value theory, we fit generalized Pareto models to the right tails of $X=-\Delta t_{\varepsilon,k}$ and $M_{\max}$; the negative shape estimates are consistent with effective bounded tails under the present finite-resolution sampling protocol and provide protocol-dependent endpoint estimates. Finally, $M_{\max}$ correlates strongly with $\varepsilon_{\max}$ and ensemble cross-correlations reveal a reproducible phase offset, consistent with an empirical association between high-curvature activity and dissipation bursts.

Generative Flow Networks (GFlowNets) were developed to learn policies for efficiently sampling combinatorial candidates by interpreting their generative processes as trajectories in directed acyclic graphs. In the value-based training workflow, the objective is to enforce the balance over partial episodes between the flows of the learned policy and the estimated flows of the desired policy, implicitly encouraging policy divergence minimization. The policy-based strategy alternates between estimating the policy divergence and updating the policy, but reliable estimation of the divergence under directed acyclic graphs remains a major challenge. This work bridges the two perspectives by showing that flow balance also yields a principled policy evaluator that measures the divergence, and an evaluation balance objective over partial episodes is proposed for learning the evaluator. As demonstrated on both synthetic and real-world tasks, evaluation balance not only strengthens the reliability of policy-based training but also broadens its flexibility by seamlessly supporting parameterized backward policies and enabling the integration of offline data-collection techniques.

Net survival is conventionally defined as “survival if cancer were the only possible cause of death”, an estimand corresponding to cancer-specific mortality alone. The Pohar Perme estimator targets this by removing general population other-cause mortality from observed total mortality, but achieves it only when cancer patients experience the same other-cause mortality as the general population. However, cancer patients often experience elevated other-cause mortality due to baseline health differences and treatment-induced effects. Using recent theoretical work decomposing total mortality into four components (cancer deaths, baseline health differences, treatment-induced other-cause deaths, and general population other-cause mortality), we show that the Pohar Perme estimator delivers the sum of cancer deaths, baseline differences, and treatment-induced deaths, falling short of its intended estimand whenever either source of excess is present. From Botta et al, we present empirical evidence showing relative risk of other-cause deaths ranging from 1.0 (colorectal cancer) to 4.0+ (head and neck cancers), and calculations demonstrating that net survival can substantially underestimate cancer-specific survival probability when relative risk exceeds 1.0. Critically, treatment-induced other-cause deaths represent irreducible causal pathways from cancer to death that cannot be eliminated through better stratification. We recommend interpreting net survival as “survival where general population other-cause mortality is removed” rather than as a causal counterfactual, and call for more precise language in cancer epidemiology.

Separate modelling of cause specific mortality rates and their projections can yield inconsistent forecasts when the sum of deaths by cause does not match the total observed in a population. We develop a hierarchical probabilistic framework for cause specific mortality counts in which both the total number of deaths and the occurrence of deaths across causes are treated as random. Conditional on the total number of deaths, cause specific counts follow a multinomial distribution, whereas the total count is modelled using a Poisson distribution, and the vector of cause of death probabilities is assigned a Dirichlet distribution. The variation in cause specific mortality rates by age and calendar year is captured in both the Poisson and Dirichlet models, allowing interpretable demographic patterns while preserving coherence by construction. This model construction naturally preserves the coherence between the sum of deaths by cause and the total mortality. The method is exhibited through the analysis of cause specific mortality rates in the United States and France, sourced from the Human Mortality Database from 1979 to 2023, separately by sex and across ages, with deaths grouped into major cause categories. The empirical analysis uses a rolling 15 year out o fsample evaluation and compares the proposed model with the standard Lee Carter model and its compositional extension. The results show that coherent projections can be obtained across countries and sexes, that competitive predictive accuracy is achieved, and that uncertainty is well calibrated for both total and cause specific mortality.

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.

The Nash-Sutcliffe efficiency ($\text{NSE}$) is a widely used, positively oriented relative measure for evaluating forecasts across multiple time series. However, it lacks a decision-theoretic foundation for this purpose. To address this, we examine its negatively oriented counterpart, which we refer to as Nash-Sutcliffe loss, defined as $L_{\text{NS}} = 1 - \text{NSE}$. We prove that $L_{\text{NS}}$ is strictly consistent for an elicitable and identifiable multi-dimensional functional, which we name the Nash-Sutcliffe functional. This functional is a data-weighted component-wise mean. The common practice of maximizing the average $\text{NSE}$ across multiple series is the sample analog of minimizing the expected $L_{\text{NS}}$. Consequently, this operation implicitly assumes that all series originate from a single non-stationary, stochastic process. We introduce Nash-Sutcliffe linear regression, a multi-dimensional model estimated by minimizing the average $L_{\text{NS}}$, which reduces to a data-weighted least squares formulation. By reorienting the sample average loss function, we extend the previously proposed evaluation and estimation framework to forecasting multiple stationary dependent time series with differing stochastic properties. This constitutes a more natural empirical implementation of the $\text{NSE}$ than the earlier formulation. Our results establish a decision-theoretic foundation for $\text{NSE}$-based model estimation and forecast evaluation in large datasets, while further clarifying the benefits of global over local machine learning models.

High-dimensional feature selection is routinely required to balance statistical power with strict control of multiple-error metrics such as the k-Family-Wise Error Rate (k-FWER) and the False Discovery Proportion (FDP), yet some existing frameworks are confined to the narrower goal of controlling the expected False Discovery Rate (FDR) and can not exploit the group-structure of the covariates, such as Sorted L-One Penalized Estimation (SLOPE). We introduce the Group Stepdown SLOPE, a unified optimization procedure which is capable of embedding the Lehmann-Romano stepdown rules into SLOPE to achieve finite-sample guarantees under k-FWER and FDP thresholds. Specifically, we derive closed-form regularization sequences under orthogonal designs that provably bound k-FWER and FDP at user-specified levels, and extend these results to grouped settings via gk-SLOPE and gF-SLOPE, which control the analogous group-level errors gk-FWER and gFDP. For non-orthogonal general designs, we provide a calibrated data-driven sequence inspired by Gaussian approximation and Monte-Carlo correction, preserving convexity and scalability. Extensive simulations are conducted across sparse, correlated, and group-structured regimes. Empirical results corroborate our theoretical findings that the proposed methods achieve nominal error control, while yielding markedly higher power than competing stepdown procedures, thereby confirming the practical value of the theoretical advances.

We study non-rectangular robust Markov decision processes under the average-reward criterion, where the ambiguity set couples transition probabilities across states and the adversary commits to a stationary kernel for the entire horizon. We show that any history-dependent policy achieving sublinear expected regret uniformly over the ambiguity set is robust-optimal, and that the robust value admits a minimax representation as the infimum over the ambiguity set of the classical optimal gains, without requiring any form of rectangularity or robust dynamic programming principle. Under the weak communication assumption, we establish the existence of such policies by converting high-probability regret bounds from the average-reward reinforcement learning literature into the expected-regret criterion. We then introduce a transient-value framework to evaluate finite-time performance of robust optimal policies, proving that average-reward optimality alone can mask arbitrarily poor transients and deriving regret-based lower bounds on transient values. Finally, we construct an epoch-based policy that combines an optimal stationary policy for the worst-case model with an anytime-valid sequential test and an online learning fallback, achieving a constant-order transient value.

Latent-space Bayesian optimization (LSBO) extends Bayesian optimization to structured domains, such as molecular design, by searching in the continuous latent space of a generative model. However, most LSBO methods assume a fixed objective, whereas real design campaigns often face temporal drift (e.g., evolving preferences or shifting targets). Bringing time-varying BO into LSBO is nontrivial: drift can affect not only the surrogate, but also the latent search space geometry induced by the representation. We propose Time-Aware Latent-space Bayesian Optimization (TALBO), which incorporates time in both the surrogate and the learned generative representation via a GP-prior variational autoencoder, yielding a latent space aligned as objectives evolve. To evaluate timevarying LSBO systematically, we adapt widely used molecular design tasks to drifting multi-property objectives and introduce metrics tailored to changing targets. Across these benchmarks, TALBO consistently outperforms strong LSBO baselines and remains robust across drift speeds and design choices, while remaining competitive under actually time-invariant objectives.

Envelope models provide a sufficient dimension reduction framework for multivariate regression analysis. Bayesian inference for these models has been developed primarily using Markov chain Monte Carlo (MCMC) methods. Specifically, Gibbs sampling and Metropolis-Hastings algorithms suffer from slow mixing and high computational cost. Although automatic differentiation variational inference (ADVI) has been explored for Bayesian envelope models, the resulting gradient-based optimization is often numerically unstable due to severe ill-conditioning of the posterior distribution. To address this issue, we propose a novel reparameterization of the posterior distribution that alleviates the ill-conditioning inherent in conventional variational approaches. Building on this reparameterization, we develop an efficient variational inference procedure. Since the resulting likelihood remains nonconjugate, we approximate the corresponding variational factor using a Laplace approximation within a coordinate-ascent variational inference (CAVI) framework. We establish theoretical results showing that, at each one-step coordinate update, the Laplace approximation error relative to the exact variational inference coordinate update converges to zero. Simulation studies and a real-data analysis demonstrate that the proposed method substantially improves computational efficiency while maintaining estimation accuracy and model-selection performance relative to existing approaches.

Despite exceptional predictive performance of Deep sequence models (DSMs), the main concern of their deployment centers around the lack of uncertainty awareness. In contrast, probabilistic models quantify the uncertainty associated with unobserved variables with rules of probability. Notably, Bayesian methods leverage Bayes' rule to express our belief of unobserved variables in a principled way. Since exact Bayesian inference is computationally infeasible at scale, approximate inference is required in practice. Two major bottlenecks of Bayesian methods, especially when applied in deep neural networks, are prior specification and approximation quality. In Chapter 3 & 4, we investigate how the architectures of DSMs themselves can be informative for the design of priors or approximations in probabilistic models. We first develop an approximate Bayesian inference method tailored to the Transformer based on the similarity between attention and sparse Gaussian process. Next, we exploit the long-range memory preservation capability of HiPPOs (High-order Polynomial Projection Operators) to construct an interdomain inducing point for Gaussian process, which successfully memorizes the history in online learning. In addition to the progress of DSMs in predictive tasks, sequential generative models consisting of a sequence of latent variables are popularized in the domain of deep generative models. Inspired by the explicit self-supervised signals for these latent variables in diffusion models, in Chapter 5, we explore the possibility of improving other generative models with self-supervision for their sequential latent states, and investigate desired probabilistic structures over them. Overall, this thesis leverages inductive biases in DSMs to design probabilistic inference or structure, which bridges the gap between DSMs and probabilistic models, leading to mutually reinforced improvement.

LLMs increasingly excel on AI benchmarks, but doing so does not guarantee validity for downstream tasks. This study contrasts LLM alignment on benchmarks, downstream tasks, and, importantly the intended impact of those tasks. We evaluate the performance of leading LLMs (i.e., generative pre-trained base models) on difficult-to-verify tasks of the teaching and learning of schoolchildren. Across all LLMs, inter-model behaviors on disparate tasks correlate higher than they do with expert human behaviors on target tasks. These biases shared across LLMs are poorly aligned with downstream measures of teaching quality and often negatively aligned with the intended impact of student learning outcomes. Further, we find multi-model ensembles, both unanimous model voting and expert-weighting by benchmark performance, further exacerbate misalignment with learning. We measure that selection of LLM and/or prompting strategy only reliably accounts for $15\%$ of all measured misalignment error and that variation in misalignment error is shared across LLMs, suggesting that common pretraining accounts for much of the misalignment in these tasks. We demonstrate methods for robustly measuring alignment of complex tasks and provide unique insights into practical applications of LLMs in high-noise contexts.

Batteries are ubiquitous today, with applications ranging from smartphones, watches, and laptops to electric cars, drones, and electric aircraft. Lithium-ion batteries are widely used in these applications due to their high energy density, rechargeability, and low lifecycle cost. Understanding the lifetime of lithium-ion batteries is essential for their effective utilization across many domains. In this study, data-driven modeling approaches are explored to predict the lifetime of lithium-ion batteries using various measurable battery parameters. A battery dataset from NASA's electric aircraft experiments was used, which included 17 predictor variables and remaining flight time as the response variable representing battery lifetime. The dataset contained more than 4,000,000 rows. However, the original dataset provided limited directly useful information about battery utilization over time; therefore, feature engineering was performed to generate more informative variables. Additionally, dimensionality reduction using principal component analysis (PCA) was applied to reduce computational cost and model complexity by selecting a smaller number of principal components as predictors for model development. Random forest and neural network models were explored for battery lifetime prediction using the engineered features. Multiple neural network configurations were evaluated, including single- and double-hidden-layer architectures with varying numbers of nodes. Mean squared error (MSE) on the test dataset was used as the performance metric for model comparison. The results indicate that data-driven modeling approaches are effective for battery lifetime prediction, with neural network models outperforming other models based on the MSE metric. Furthermore, neural networks demonstrate robustness in handling high-dimensional battery data.

Comparing multivariate yield quality distributions across spatially referenced agricultural fields is complicated by two pervasive features: non-normality and spatial autocorrelation. Classical procedures such as ANOVA, MANOVA, and standard rank tests assume independence and therefore exhibit severe Type I error inflation when spatial dependence is present. We propose a nonparametric spatial Cramer-von Mises-type test based on kernel-smoothed empirical copula processes constructed from pooled componentwise ranks. Spatial kernel weights account explicitly for local dependence, while the rank transformation removes sensitivity to marginal distributional form. Under fixed-domain infill asymptotics and polynomial alpha-mixing conditions, we establish weak convergence of the smoothed empirical copula process to a mean-zero Gaussian limit and show that the resulting quadratic test statistic converges to a weighted sum of chi-squared random variables restricted to the K-1-dimensional contrast subspace. Practical inference is obtained through a Satterthwaite approximation calibrated using the exact discrete spatial covariance operator under a Gaussian copula model. Monte Carlo experiments with bivariate log-normal spatial data demonstrate that the proposed test maintains nominal size across varying strengths of spatial dependence, in contrast to classical parametric and non-spatial rank-based methods, which become severely anti-conservative. The procedure provides a theoretically justified and computationally tractable framework for comparing multivariate spatial yield distributions in precision agriculture and related applied settings.

We present an HSIC-based approach for global sensitivity analysis of broad classes of models with correlated and possibly function-valued inputs and outputs. To this end, we define the first-order and total HSIC sensitivity indices, which are bounded, interpretable, and moment-independent analogues to the first-order and total-effect Sobol' indices, respectively. These desirable qualities hinge upon the key property of monotonicity under marginalization for the HSIC. We rigorously establish this monotonicity property by using a suitable class of augmented kernels. Furthermore, we provide an efficient algorithm for computing an empirical estimator of the HSIC that significantly reduces computational complexity and storage requirements. The effectiveness and interpretability of the first-order and total HSIC sensitivity indices are demonstrated through computational experiments on models that feature nonlinear relationships, correlated inputs, and functional outputs.

In complex real-world settings, optimization is challenged by the presence of diverse models of differing fidelity. In many optimization problems, a single model is treated as the most accurate representation of the underlying system, while other models are evaluated primarily by their agreement with this presumed most accurate model. Yet in real-world applications, model accuracy is rarely known a priori and assuming a single most accurate model can be misleading. This paper addresses this gap by proposing a flexible set-based optimization methodology called Set-Based Optimization with Multiple Models (S-BOMM) that works with multiple models without the assumption of a most accurate high-fidelity model. Unlike traditional optimization approaches that focus on finding an optimal solution according to the high-fidelity model, our methodology utilizes consistency between models to identify good solutions across multiple models. A probabilistic analysis of the consistency method is provided that bounds the likelihood of the methodology producing correct or incorrect results. Empirical results demonstrate the effectiveness of S-BOMM on test problems. By focusing on the consistency across models rather than relying on a single best solution, this set-based approach offers a practical alternative to optimization problems where multiple models must be considered without assuming a single most accurate high-fidelity model.

The study of minimax convergence rates for classification procedures adapted to SDE paths is rarely addressed in the literature. Only one paper established optimal convergence rates for a binary classifier for SDE paths constructed from the white noise model. In this paper, we consider a more complex diffusion model with space-dependent drift and diffusion coefficients where the drift depends on the class and the diffusion coefficient is common to all classes. We establish, under the low-noise condition, a faster convergence rate over a Holder space. This result will require the establishment of an exponential inequality, which is essential to obtain the expected rate. We then study the lower bound on the excess risk of the empirical classifier.

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.

Nonparametric curve estimation by kernel methods has attracted widespread interest in theoretical and applied statistics. One area of conflict between theory and application relates to the evaluation of the performance of the estimators. Recently, Marron and Tsybakov (1995) proposed visual error criteria for addressing this issue of controversy in density estimation. Their core idea consists in using integrated alternatives to the Hausdorff distance for measuring the closeness of two sets based onthe Euclidean distance. In this paper, we transfer these ideas to hazard rate estimation from censored data. We are able to derive similar results that help to understand when the application of the new criteria will lead to answers that differ from those given by the conventional approaches.

For an arbitrary diffusion process $X$ with time-homogeneous drift and variance parameters $\mu(x)$ and $\sigma^2(x)$, let $V_\varepsilon$ be $1/\varepsilon$ times the total time $X(t)$ spends in the strip $[a+bt-(1/2)\varepsilon,a+bt+(1/2)\varepsilon]$.The limit $V$ as $\varepsilon\rightarrow0$ is the full halfline version of the local time of $X(t)-a-bt$ at zero, and can be thought of as the time $X$ spends along the straight line $x=a+bt$. We prove that $V$ is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of $\mu(\cdot)$, $\sigma(\cdot)$, $a$, $b$, and the starting point $X(0)$. The special case ofa Brownian motion is studied in more detail, leading in particular to a full process $V(b)$ with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such `total relative time' variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).