Papers

We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.

We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt{np}$ with probability at least $1-n^{-r}$ whenever $np\ge c\log n$. We extend this bound to inhomogeneous Bernoulli sampling with deterministic entrywise weights. This removes the logarithmic factor in the work of Zhou and Zhu (2021). The proof follows the Kahn–Szemerédi light–heavy decomposition with a refined estimate on the heavy tuple part. We also obtain a log-free second eigenvalue bound for the random hypergraph model of Friedman and Wigderson (1995).

We construct a symmetric operad $\mathfrak{C}$ on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass $\mathfrak{C}^{ac}$—which coincides with the $L^1$ class of copula densities—forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula $g(v)=\phi\big(\Psi_1(v^{(1)}),\dots,\Psi_n(v^{(n)})\big)\prod_{k=1}^{n}\psi_{k}(v^{(k)})$. Furthermore, we show that copulas with finite copula entropy—identified with the $L\log L$ class of copula densities—are closed under substitution and hence constitute a suboperad $\mathfrak{C}^{L\log L}$. On this suboperad, copula entropy is strictly additive: $H(\gamma(\Phi;\Psi_{1},\ldots,\Psi_{n}))=H(\Phi)+\sum_{k=1}^{n}H(\Psi_{k})$.

Classical confidence intervals, when repeatedly obtained on accumulating data at different sample sizes, produce contradictory inferences with high probability. We propose a simple and efficient strategy for computing, instead, mixture confidence sequences for regression coefficients in generalized linear models under this sequential framework. Simulations demonstrate the computational convenience of our approach and the importance of drawing inferential conclusions based on these anytime-valid tools when observations become available in batches over time. The usefulness of the proposed procedure is also shown in the analysis of streaming data from the American National Automotive Sampling System.

One hospital runs bloods and imaging at the same time. Another runs them one after the other, in either order, equally often. Knowing which actually happened, and how it is recorded in data, is critical for all operational managers. In process mining, the standard approach is to construct an event log, and attempt to discover concurrent and sequential processes in a data-driven way. We show this standard approach, built on the stochastic language of an event log, reports only the assumptions of its discovery algorithm, because every such log is explained equally well by a model with no concurrency at all. Further, before any data is acquired, we characterise when data can and cannot distinguish concurrent behaviour. Where it cannot, the distinction is recoverable from evidence the stochastic language discards, such as the times at which activities start and end, or object-centric records that fix an order within an execution. The remedy is therefore a choice of what is recorded, rather than a larger sample. This impacts decision making, as planning resource for truly concurrent services is very different from sequential services.

We examine semiparametric solutions to contamination bias for nonbinary treatments. Deepening the discussion by Goldsmith-Pinkham et al. (2024), we detail how spline functions approximate conditional expectation and propensity score functions under weak functional-form assumptions. Reanalyzing 18 regressions across 11 studies, we compare standard linear regressions against parametric and semiparametric versions of three contamination-robust estimators. We document large point-estimate discrepancies between parametric and semiparametric approaches and find that adopting flexible semiparametric solutions may not increase statistical uncertainty substantially. We recommend that researchers verify the robustness of their conclusions to the use of semiparametric tools that address contamination bias.

The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many real-world signals exhibiting heavy-tailed or non-Gaussian behavior. In this paper, we extend Whittle's result to multidimensional, real-valued stationary compound Gaussian processes (CGPs) with arbitrary mean, under a unified framework encompassing fully known, parameterized, and completely unknown density generators. Building upon the Slepian-Bangs formula for $n$ consecutive observations and extending it to the semiparametric setting, we leverage the asymptotic properties of block Toeplitz matrices to obtain a closed-form spectral-domain expression for the asymptotic FIM rate. The resulting expression, common to all density generator families, generalizes the standard zero-mean Gaussian formula by incorporating a distribution-dependent contribution associated with the nonzero mean, and an additional covariance term that is invariant to the choice of non-Gaussian distribution. This extension enables efficient and theoretically grounded performance analysis for heavy-tailed signal processing applications.

Accurate optimization of a supervised spectral objective need not produce an accurate population subspace or a better predictive representation. We investigate these distinctions for Online Kernel Supervised Principal Component Analysis (OKSPCA), which combines a centered cross-moment in finite random-feature coordinates with an Adam-style orthonormal basis update for an established objective. Fixed-map consistency, concentration and perturbation results describe the estimator and its exact subspace; same-target comparisons then assess the practical iterate separately. Across six predictive benchmarks, performance depends on the declared pipeline: replacing the tracker with the exact empirical target leaves the two regression deficits largely unchanged. Direct classification-rank models capture nearly all terminal objective energy on average, but a saved intermediate state exhibits substantial geometric deviation; a controlled sample-size study further separates empirical accuracy from population recovery. In distinct numerical-service workloads, exact on-request computation is faster in the tested classification settings, whereas Adam saves time relative to the tested full thin-SVD service for some dense wider-regression requests, alongside persistent geometric error. These diagnostics limit explanations based solely on terminal optimization accuracy and distinguish numerical cost from quality, rank coverage and freshness; they establish neither practical-tracker convergence nor predictive or deployment benefits from basis availability.

Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $\alpha\gg\beta$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(\alpha+\beta^2/\alpha^2)$, which we prove to be tight when $\beta\le\alpha^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $\beta^2/\alpha^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.

This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family \[ G_c(z) = \frac12 + z\,\frac{\Gamma(c-1/2)}{2\sqrt2\,\Gamma(c)}\,{}_1F_1\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \] contains the standard normal distribution at the boundary $c=3/2$. Away from the boundary, the density has algebraic tail behaviour $g_c(z)\sim(c-3/2)|z|^{-3}$, so the parameter $c$ indexes a directed heavy-tailed deformation of the Gaussian law. The distribution also admits an equivalent beta-precision normal scale-mixture representation, in which the Gaussian boundary corresponds to degenerate unit precision. I establish the admissibility of the hypergeometric family, derive minimum-distance estimators, and obtain both regular interior asymptotics and nonstandard boundary asymptotics under the normal null. The standardised fit-improvement statistic converges to the mixture distribution $\tfrac12\delta_0+\tfrac12\chi_1^2$. I extend the theory to plug-in location-scale procedures, including a robust median/IQR version. Simulations document accurate null size and directed power against heavy-tailed alternatives. An application to daily S&P 500 returns, both unconditionally and after GARCH(1,1) filtering, illustrates the empirical implications for tail fitting and risk quantiles, and documents that GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test.

Offline policy evaluation (OPE) is crucial in high-stakes reinforcement learning applications, where new policies must be assessed reliably before deployment. In such settings, point estimates alone are insufficient; principled uncertainty quantification, such as confidence intervals and variance estimates, is essential for safe and risk-aware decision-making. A comprehensive way to unify these tasks is to estimate the sampling distribution of the evaluation error. Existing approaches, however, often suffer from limited robustness, scalability, or finite-sample validity. In this paper, we propose a model-based bootstrap framework for uncertainty quantification of OPE in finite-horizon, time-inhomogeneous Markov decision processes (MDPs). Unlike classical bootstrap methods that rely on resampling complete episodes, the proposed method regenerates trajectories from an estimated MDP and can therefore accommodate a much broader range of offline data formats, including complete trajectories, transition-level observations, and trajectory fragments. This flexibility further improves finite-sample statistical efficiency. We establish bootstrap distributional consistency, asymptotically valid confidence intervals, and consistent variance estimation for the target policy value. Extensive simulations show that the proposed method accurately captures the sampling distribution of the OPE estimator, yielding tighter confidence intervals and more accurate variance estimates in most settings.

Patient-specific biomechanical models of the left ventricle can relate cardiac magnetic resonance imaging to regional myocardial material properties, but existing emulator-based studies typically treat the myocardium as mechanically homogeneous, limiting representation of localised dysfunction. We propose a Bayesian surrogate-modelling framework for inferring regional Holzapfel-Ogden material parameters in a left ventricle partitioned into five physiological zones derived from the American Heart Association 17-segment model. Eight emulator strategies spanning single- versus multi-output, local versus global, and Gaussian-process- versus neural-network-based architectures were screened using parameter point-estimation accuracy; the three retained models were evaluated using empirical marginal credible-interval coverage and posterior contraction. We found that models with comparable point accuracy nevertheless differed markedly in uncertainty. A multi-output variational Gaussian process provided the most favourable balance across these criteria and was retained for the subsequent analyses. In synthetic local and global stiffening scenarios, maximum a posteriori estimates generally distinguished stiffened from baseline zones, but the nonlinear-stiffening parameters were more difficult to identify from end-diastolic observations than the stiffness-magnitude parameters. A healthy-volunteer analysis demonstrates feasibility with an incomplete observation vector and jointly inferred strain-noise scales. These results suggest that the proposed framework provides a computationally feasible, uncertainty-aware approach to regional left ventricle parameter inference.

We study smooth strongly convex–strongly concave minimax optimization in the deterministic unconstrained setting, without assuming a bilinear or separable structure. Although existing multi-loop methods attain near-optimal condition-number dependence, standard single-loop methods generally exhibit a substantial complexity gap. To close this gap, we propose the Single-Loop Predictor–Corrector Extragradient Method with Damped Momentum (PCE-DM), which combines an extragradient prediction–correction scheme with a novel auxiliary feedback recursion for the weaker-curvature variable. PCE-DM uses fixed parameters and two new full-gradient evaluations per iteration after one initialization query, while requiring no inner solves, accuracy schedules, or staged restarts. We develop a Lyapunov analysis that controls the predictor–corrector mismatch through corrected-gradient increments and establish last-iterate linear convergence. Specifically, PCE-DM computes an $\varepsilon$-accurate relative solution, measured by the squared Euclidean distance to the saddle point, within $\mathcal{O}\!\left(\sqrt{\kappa_x\kappa_y} \log(2\kappa_x\kappa_y/\varepsilon)\right)$ full-gradient queries. This result closes the condition-number complexity gap between standard single-loop methods and near-optimal multi-loop methods, matching the known lower-bound order up to logarithmic factors while retaining fixed, explicit single-loop updates. Numerical experiments on regularized linear regression and AUC maximization demonstrate the computational efficiency of PCE-DM.

Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves a fixed fraction of the way along the geodesic toward each new observation. The smoothing parameter is estimated by minimizing the average squared distance between observations and their forecasts. We further introduce an innovations mechanism under which each observation has conditional Fréchet mean equal to the current level, providing the metric-space analog of the innovations state-space model. In contrast to autoregressive models for object-valued time series, the framework involves a single scalar parameter, assumes no stationarity, and updates online in constant time per observation. Under this mechanism, we establish sample-path properties of the generative process via the quasilinearization available in Hadamard spaces, and prove almost-sure consistency of the smoothing-parameter estimator. Three real-data applications, spanning covariance-matrix, distributional, and functional time series, assess the forecasting performance of the method against structurally heavier alternatives.

Missing not at random (MNAR) data pose significant challenges for causal inference, particularly when both confounders and the outcome are partially observed. Without additional assumptions beyond those required for causal inference, causal estimands are generally not identifiable under MNAR mechanisms. This paper first develops identification results for the causal estimand, the average treatment effect, under several plausible MNAR mechanisms using completeness conditions, and proposes an estimation approach based on the Expectation-Maximization (EM) algorithm. We further extend this identification and estimation framework to mediation analysis, enabling the estimation of natural direct and indirect effects under MNAR mechanisms. Through extensive simulation studies, we compare the proposed method with two widely used approaches for handling missing data, complete-case analysis and multiple imputation. The results show that the proposed method yields substantially lower bias under the considered MNAR mechanisms. Finally, we apply the proposed approach to NHANES data to estimate the causal effect of education on depression, with health condition as the mediator.

This paper develops a design-based asymptotic theory for two-stage sampling over a continuous spatial domain. The target parameters are integral totals, or smooth functions of such totals, defined over a fixed bounded territory partitioned into an increasingly fine collection of primary sampling units. Within this fixed-area framework, we derive the order of the variance components of the Horvitz-Thompson estimator when secondary sampling units are points selected from continuous sub-regions. We establish design consistency of the estimator, as well as consistency of variance estimators under explicit regularity conditions on inclusion probabilities, sampling densities, and the study variable. The results are extended to plug-in estimators of smooth scaleinvariant functions of totals. For high-entropy first-stage designs, we further show that a Hájek-type variance estimator based only on first-order inclusion probabilities is consistent, and we prove asymptotic normality of both total and plug-in estimators. A simulation study inspired by forest inventory applications illustrates the finite-sample performance of the proposed estimators and variance estimators.

Economic networks are often estimated separately over two periods, and changes in their edge sets are interpreted as structural rewiring. Since both networks are estimated, observed turnover also reflects graph-selection error. We study the two-snapshot Hamming-turnover functional under a homogeneous edge-misclassification model. With known sensitivity and specificity and conditional independence of the estimated edge indicators across periods, latent turnover admits a closed-form unbiased adjustment based only on observed turnover and the two estimated graph sizes. We then examine the effects of sparsity, calibration error and dependence across periods. When the number of true links is proportional to $p$, a false-positive probability of order $p^{-1}$ generates expected spurious turnover of order $p$. An error of the same order in calibrating the false-positive probability can likewise leave order-$p$ bias after adjustment. We also derive the bias induced by cross-period dependence and give sufficient conditions for consistency relative to network size. Numerical results illustrate the finite-sample implications.

This paper proposes an Expectation–Maximization algorithm with Metropolis–Hastings sampling for parameter estimation in a Mixed Fractional Merton Jump Diffusion model. The model combines fractional Brownian motion to capture long-range dependence with a compound Poisson jump process to describe abrupt movements in financial returns. The latent jump process is inferred during the E-step using Markov Chain Monte Carlo sampling, while the M-step updates the model parameters by maximizing the expected complete-data likelihood. The consistency and asymptotic normality of the proposed estimator are established under suitable regularity conditions. The methodology is applied to the Helsinki Stock Index (OMXH25). The proposed estimation framework provides a reliable and computationally efficient approach for modeling financial time series exhibiting both long-memory dependence and jump behavior.

Wastewater-based epidemiology (WBE) is an increasingly important tool for infectious disease surveillance, but there has been limited direct comparison of modelling approaches for predicting pathogen concentrations across space and time. We compare the predictive performance of six modelling approaches using norovirus in England as a case study, considering 3,232 wastewater samples from 152 sewage treatment works collected between May 2021 and March 2022 as part of the UK Environmental Monitoring for Health Protection programme. The benchmark was a Bayesian spatio-temporal model using Integrated Nested Laplace Approximation (INLA) with the Stochastic Partial Differential Equation approach (SPDE), compared with Lasso regression, Generalised Additive Models (GAM), Bayesian GAM, Extreme Gradient Boosting (XGBoost), and Random Forest. Models were evaluated using 10-fold spatial-block cross-validation with metrics including mean squared error, bias, correlation, empirical coverage of 95% prediction intervals, interval score, and computational cost. Random Forest was the best overall performing model, achieving the best interval score and nearest to nominal empirical coverage (95.3%), while maintaining point prediction accuracy comparable to XGBoost. The INLA-SPDE model also performed well, with near-nominal empirical coverage, the third best interval score, and consistently low bias across all evaluated metrics. Comparing predicted spatio-temporal trends, both models yielded similar spatial patterns but notable differences in uncertainty estimation. Our findings highlight a trade-off between predictive accuracy, uncertainty quantification, and computational efficiency. Ensemble machine learning methods are well suited to rapid prediction, whereas Bayesian geostatistical models remain valuable when probabilistic decision support is a priority for public health surveillance

Modern tabular datasets are becoming increasingly large, both in the number of samples and covariates, posing significant challenges for Bayesian variable selection due to the resulting computational burden. While there is extensive literature on scaling Bayesian inference to large numbers of observations or high-dimensional covariate spaces, comparatively little work addresses scalable Bayesian variable selection when both dimensions are large simultaneously in a practical setting. This paper presents a novel Bayesian variable selection framework for efficiently analysing data with a large number of both rows and columns. The proposed framework operates via a divide-and-conquer approach, splitting data into batches along both directions and analysing each batch independently in parallel, after which the results are combined together in a two-phase consensus procedure. Experiments demonstrate the computational gains while retaining strong variable selection performance, successfully identifying relevant signals in noisy, high-dimensional settings despite the loss of information induced by data splitting. Practical guidelines are also provided, including recommendations for tuning parameter choices and effective data partitioning strategies, followed by a real application to an H3N2 influenza dataset.

Graphical multiple testing procedures are a flexible and transparent way to control the family-wise error rate when a confirmatory trial pursues several label claims, but they leave open the question of which graph to use. In practice sponsors often fall back on fixed-sequence or Holm procedures that may poorly reflect what the trial is actually trying to achieve. We propose to instead choose the graph that maximises an explicit gain function, which states what each possible set of rejected hypotheses is worth to the sponsor: typically nothing until a regulatory hurdle is cleared, then an incremental amount for each further claim. Conventional power criteria are recovered as special cases, and because the search is confined to graphical procedures, family-wise error rate control holds whichever graph is selected. We set out a structured procedure for eliciting the gain function from the clinical, commercial, and regulatory members of a trial team. The proposed framework also handles uncertainty in the treatment effects and correlations assumed at the design stage, by averaging performance over their plausible values rather than fixing a single assumption. For group sequential designs, we describe how it can further reward early claims. Five examples, drawn from real and hypothetical pharmaceutical trials, show that the best graph depends on how trial success is defined, how uncertain the design assumptions are, and, in a group sequential setting, when claims can be established.

In this paper, three operations on Gibbs probability measures are studied. The first operation, often referred to as renormalization, takes one Gibbs probability measure and generates a new Gibbs measure by normalizing a power of its density. This normalization has a twofold effect: it changes the regularization factor and concentrates the support within a subset of the original support. Interestingly, these effects can be independently controlled by different parameters. The second operation consists of a normalized log-linear combination of the densities of Gibbs probability measures. The third operation takes two Gibbs probability measures and changes the reference measure of the latter with the former. Hence, the former is said to be "nested" within the latter, yielding a new Gibbs probability measure. The resulting measures from the second and third operations are also Gibbs probability measures and are shown, respectively, to solve optimization problems involving the expectations of linear combinations of the objective functions of the given measures, subject to a relative entropy regularization. These optimization problems differ exclusively in the coefficients of the linear combinations. This leads to the conclusion that there exists a set of parameters for which nesting one Gibbs probability measure into another has the same effect as log-linearly combining them. These operations have relevant applications in statistical learning. As an example, a one-shot federated learning system in which clients send their locally trained Gibbs algorithms to the server for log-linear combination is shown to achieve the same performance as a Gibbs algorithm trained upon the aggregation of all local training datasets.

The Brunner-Munzel (BM) test is a nonparametric test for two independent samples that evaluates whether observations from one group tend to be greater than observations from another group, or vice versa. The BM test has a broader scope of application than the Mann-Whitney $U$ test because it does not assume equal variances between the two groups. However, the meaning of the BM test statistic is difficult to understand intuitively, which may be one of the factors hindering the widespread use of the BM test. To alleviate this problem, in this paper, I introduce an alternative interpretation of the BM test statistic from the viewpoint of local linear approximation. It is shown that the variance estimator for the sample stochastic superiority used in the BM test can be derived using local linear approximation, in which the influence of each observation on the sample stochastic superiority is assumed to be additive. This simple interpretation will help practitioners decide to use the BM test without hesitation.

Exploration in centralized serial-dictatorship matching bandits must use complete matchings, so learning one player-arm pair can impose regret on others. We study this externality under a known common priority order and Gaussian rewards with unit variance. We show that the matching-level Graves-Lai constraints reduce to finitely many pairwise exploration quotas and, at top-choice-separated instances, yield a polynomial-size marginal linear program. At these instances, the exact attainable set of expected logarithmic regret coefficients is $G(\theta)\mathcal{X}(\theta)$, where $\mathcal{X}$ is the feasible matching-allocation set and $G$ maps allocations to player regret. The usual upper-closed Graves-Lai region can be strictly larger despite having the same Pareto-minimal boundary. We further show that identical exploration quotas can induce very different regret through their scheduling. Finally, we construct estimate-solve-track policies, uniformly good on the full row-strict class, that attain every fixed positively weighted optimum without assuming optimizer uniqueness. Every Pareto-minimal point is pointwise attainable, possibly through an instance-calibrated target.

Recent studies have shown that smooth functions can be well approximated by ReLU neural networks with path norm constraint on the weights. We extend these results from uniform approximation to approximation in Sobolev norm. Specifically, we analyze how well Sobolev functions in $W^{n,p}$ can be approximated by neural networks with width $W$, depth $L$ and path norm bounded by $K$, when the approximation error is measured in the $W^{1,p}$-norm. For shallow networks with depth $L=1$, we derive the approximation error bound $\mathcal{O}(\max\{W^{-(n-1)/d}, K^{-(n-1)/(s-n)}\})$, when the smoothness index satisfies $n<s=(d+3)/2$ and the input is $d$-dimensional. For deep networks, we remove the restriction on the smoothness by showing that the approximation bound $\mathcal{O}(K^{-(n-1)/(d+d/p+1)})$ holds if the width $W$ and depth $L$ are sufficiently large.

Phase-type (PH) distributions are versatile semi-parametric models for lifetime duration and can be used in survival and reliability analysis. In this paper we put forward methods and software for using PH distributions in a competing-risks model. The resulting multi-absorbing phase-type (MAPH) distribution records both the time until absorption and its cause. After formulating basic properties of this family, we develop an EM algorithm for parameter inference from exact event-time/cause observations and independently right-censored observations. For a censored subject, the E-step conditions on survival up to the censoring time and imputes both the latent transient path and its eventual cause of absorption; closed-form cause-resolved sufficient-statistic expectations are obtained from matrix exponentials. We illustrate applications and numerical properties and describe two accompanying Julia packages, in which both the exact-event and the censored E-step are implemented. Applied to intensive-care length-of-stay data in full, censored records included, the method attains a higher likelihood than the phase-type competing-risks fit previously published for those data.

We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of $|\rho-\gamma|$ and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of $\rho$ and $\gamma$ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.

We study the mean function of longitudinal functional data, where each subject contributes a small number of complete profiles over a general domain, observed at random visit times. The mean is projected onto an orthonormal basis in the time direction, and each coefficient function is estimated by a weighted average of the profiles. We consider deterministic and random weights, including closed-form, data-driven weights that dispense with the design density entirely. For each weighting scheme we derive non-asymptotic bounds on the integrated quadratic risk and an explicit optimal truncation level. All schemes share the same convergence rate, which we show to be minimax optimal over the corresponding coefficient-decay class, only the leading constants differ. We also provide a goodness-of-fit test for the mean function, and derive non-asymptotic bounds for the Gaussian approximation of its distribution under both the null and alternative hypotheses. Our estimation and testing procedures are easy to implement, fast, and perform well in applications.

Reinforcement learning for LLM agents involves two distinct optimization di- mensions: how environment feedback is exploited within a trajectory, and how complete trajectories are aggregated across a batch. We formulate these dimen- sions as Intra-Trajectory Feedback Attribution and Inter-Trajectory Objec- tive Aggregation, and introduce BATON (Bayesian Attribution and Trajectory Objective Normalization), a dual-axis policy optimization framework. BATON instantiates the first axis with Bayesian Feedback Attribution, which constructs a feedback-conditioned posterior over sampled actions, and the second with Trajec- tory Mass Normalization (TMN), which assigns equal optimization mass to com- plete trajectories. Experiments with GRPO and GiGPO on ALFWorld, WebShop, and SearchQA show that both axes provide independent gains and that their combi- nation consistently achieves the strongest overall performance across model scales.

DNA data storage offers extraordinary information density and long-term durability, but its reliability is limited by sequence-dependent errors introduced during synthesis and accumulated during storage. It remains unclear how sequence composition is associated with storage failure through specific molecular damage components. We develop a high-dimensional semiparametric mediation framework for survival outcomes. GC content is treated as the exposure, a high-dimensional baseline damage spectrum (a vector of per-read damage counts stratified by trinucleotide context and error type) as the mediator, and storage-quality failure as the outcome. Nonlinear covariate effects in both the mediator and survival models are approximated using deep neural networks. A three-step procedure combining product-of-coefficients screening, Smoothly Clipped Absolute Deviation (SCAD) penalized estimation, and joint significance testing is developed for mediator selection and inference. Applied to an aging experiment on electrochemically synthesized DNA, the method identifies 14 significant mediators, all corresponding to single-base deletions, with estimated mediated effects concentrated in trinucleotide contexts ending in C. These results reveal deletion-type damage as a major pathway linking sequence composition to reduced archival reliability and suggest candidate sequence features for future optimization and error-control strategies. The proposed framework thus offers a mechanism-oriented statistical approach for understanding and improving the reliability of DNA data storage.