Papers

AI systems increasingly search among candidate answers and deploy the highest-scoring one. Increasing search changes which errors matter, so a precise evaluation at one computation budget can leave another budget unresolved. We connect this information gap to the cost of closing it. For independent best-of-n search, aggregate reliability measurements identify deployment only through the directions they observe; we derive an exact ambiguity frontier when only small search widths are audited. Retaining candidate ranks and truth labels enables a constructive alternative: one audit can estimate reliability across all widths up to N. With known score percentiles, the minimax worst-coordinate mean squared error scales as (1 + log N)/T + N/M, capped at a constant, for expected budgets of T truth labels and M candidate observations. Matching lower bounds allow adaptive label acquisition, establishing that the distinct label and candidate costs are intrinsic to this experiment. An explicit design attains this order; a complementary record-based procedure supplies simultaneous guarantees without a known score distribution. Retrospective mathematical-reasoning and code-generation analyses show why search-dependent validation matters. In held-out CodeRM pools, a shared audit reduces the 95th-percentile maximum error across 100 widths by 58% and 40% relative to uniform labeling. These results turn structural ambiguity into a quantitative prescription for reusable validation.

The recent work of Sarkar and Zhang (2025) introduced Positive Tail Dependence Under the Null (PTDN) and developed Generalized Shifted Benjamini-Hochberg (BH) procedures for two-sided Gaussian $z$- and $t$-testing under known covariance structures. This paper develops further consequences of that framework. First, we derive explicit dependence-adaptive lower and upper bounds for the FDR of the original BH procedure in terms of the conditional variance parameters $\tau_i=1-R_i^2$, where $R_i^2$ is the squared multiple correlation between the $i$th statistic and the remaining coordinates. These bounds recover the exact BH FDR under independence and provide finite-sample, covariance-specific information complementary to generic bounds. We also identify conditions under which the coordinate-specific calibration of shifted BH can provide a rejection advantage over the original BH procedure. Second, we consider the practically important setting in which the covariance matrix is unknown but an independent Wishart estimator is available. Using simultaneous lower confidence bounds for the $\tau_i$'s, we construct a confidence-bound shifted BH procedure and establish finite-sample FDR control. To our knowledge, this is the first shifted-BH-type procedure with a finite-sample guarantee for two-sided Gaussian mean testing under a completely unknown covariance matrix estimated independently. Numerical studies illustrate the behavior of the covariance-adaptive bounds, the potential advantage of shifted BH over BH, and the performance of confidence-bound shifting under unknown covariance.

Binary segmentation is widely used for multiple change-point detection because it is fast, simple to describe, and simple to implement. Its validity rests on the requirement that, at each recursive stage, the procedure identifies one of the true change-points when several are present in the current interval. This holds for detecting changes in mean using the CUSUM statistic, but fails in some other settings, in particular in slope change detection for continuous piecewise-linear signals. We propose Look-Ahead Binary Segmentation (LABS), a modification in which the change-points returned by the two child recursions define a narrower interval on which the parent estimate is re-evaluated. LABS inherits the computational speed of standard binary segmentation, but achieves the near-optimal consistency rate of $O\{(n\log n)^{1/2}\}$ in the slope-change signal setting when the LABS model is chosen via either thresholding or a Schwarz-like information criterion. Simulations show that LABS is fast and achieves state-of-the-art performance.

This paper introduces a novel penalized spatial function-on-function quantile regression framework for analyzing spatially indexed functional data, bridging a critical gap between spatial functional models and quantile regression. Our work makes three key contributions. First, we propose the first spatial function-on-function quantile regression model that jointly accounts for spatial correlation across curves through a functional spatial autoregressive structure while allowing inference on arbitrary conditional quantiles of the functional response. Unlike traditional mean-based alternatives, this approach successfully captures state-dependent volatility and distributional dynamics beyond the conditional mean. Second, we develop a two-stage instrumental-variable estimation strategy to address endogeneity induced by the functional spatial lag. By utilizing tensor-product B-spline expansions with tensor-product roughness penalties, our method ensures optimal smoothness without the destructive information loss inherent in principal component truncation. Third, for fixed spline dimensions, we establish $\sqrt n$-asymptotic normality of the spline coefficient estimators and the induced finite-rank Gaussian-process limits for the reconstructed coefficient surfaces. Extensive Monte Carlo experiments and a high-resolution analysis of Italian PM$_{2.5}$ air quality data demonstrate that spatial function-on-function quantile regression significantly outperforms non-spatial and mean-based competitors, providing a robust and informative tool for environmental risk management and complex functional data analysis. Our method has been implemented in the SpatialFoFReg R package.

We develop a unified information-theoretic framework for lower bounding minimax quantiles. The starting point is a loss-adapted Neyman–Pearson metaconverse that bounds the minimax success probability at every loss threshold and confidence level. The bound separates the small-ball behaviour of the prior under the loss from the statistical distinguishability of the observation model, and is optimised over an auxiliary output distribution. Different relaxations of this Neyman–Pearson bound yield converses based on \(f\)-informativity, Sibson mutual information \(I_\alpha\), Maximal Leakage, and Amemiya norms. Classical Fano and Le Cam lower bounds are recovered as special cases. The framework also clarifies why different information measures are suited to different recovery criteria. Maximal Leakage is exact for a class of symmetric exact-recovery problems. We use this identity to derive finite-sample bounds on the full minimax exact-recovery risk in the balanced Gaussian weighted stochastic block model, as well as two-sided finite-sample minimax-quantile bounds for low-rank matrix estimation under isotropic bounded-energy noise. For approximate Hamming recovery, we exhibit a heterogeneous binary model in which an optimised finite Sibson order yields a strong converse while the Maximal Leakage specialisation is trivial. Finally, for one-coordinate Poisson localisation, a Bennett-type Young function used through its Amemiya norm recovers the exact success-probability scale, whereas classical Fano and fixed-power relaxations are strictly weaker. These results show that sharp converses for minimax quantiles require adapting the information measure to the recovery resolution, whether exact or approximate, and to the tail behaviour of the likelihood ratio.

The Bingham distribution is a family of antipodally symmetric distributions on the unit sphere, characterised by an exponential-of-quadratic change of measure with respect to the uniform distribution. Kent, Ganeiber and Mardia proposed a rejection sampler for generating samples from Bingham distributions using proposals from an angular central Gaussian (ACG) distribution. Their empirical results suggest that the least efficient regime is the high-concentration limit, where the acceptance probability is of order $d^{-1/2}$ in dimension $d$, implying a polynomial complexity guarantee. In this note, we verify this dimension-dependent prediction, establishing the uniform guarantee $\inf\{\alpha_D:D=D^\top\in\mathbb{R}^{d\times d}\}\ge c_\star/\sqrt{d}$, where $c_\star=0.759\ldots$. A one-dimensional high-concentration limit demonstrates that the $d^{-1/2}$ rate is unimprovable and that even the constant $c_\star$ cannot be improved beyond $0.857\ldots$. The proof relies on a novel interpretation of the acceptance probability and a comparison principle for weighted sums of chi-squared random variables, which may be of independent interest.

Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the $L^p$ maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process “peeking penalty.” The partition function can be read as a coalescent–a prefix-sharing probability of independent copies–and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.

We study large-sample properties of higher-order Markov chains on a finite alphabet $\Sigma$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $\Sigma^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals $\sum_{t}\! g_n(Y_t^{(n)})$ under natural ergodicity and sparsity conditions. The normalization involves the stationary return time to a suitably chosen state and accommodates triangular arrays with $m_n\!\to\!\infty$ and $m_n/n\!\to\!0$. We further illustrate the assumptions in a binary variable length Markov chain (VLMC), deriving explicit lower bounds on stationary masses that yield a concrete growth regime (e.g., $m_n\log m_n/n \to 0$) ensuring the CLT. These results provide asymptotic foundations for inference in sparse/partitioned higher-order models; including VLMCs and sparse Markov models (SMMs) where the effective dimensionality grows with the sample size.

Sparse pursuit after dictionary learning can yield a precise atom support even when its physical interpretation is not justified by the calibration data, especially for highly coherent dictionaries where alternative calibration-compatible dictionaries may assign different physical meanings to the same selected support. We develop resolution-aware physical-support inference that jointly accounts for uncertainty in the learned dictionary and in the representation of a deployment signal. Our cross-dictionary confidence correspondence retains calibration-compatible dictionaries and deployment-compatible sparse representations, then projects the surviving explanations onto physical-support space. For local coherent-atom classes with separation scale s, once the deployment data resolve the coherent-block explanation and its atom support, the minimax physical resolution from N calibration signals satisfies $\delta_{\mathrm{opt}}(N,s)\asymp\min\{s,\frac{1}{\sqrt{N}s^2}\}$, with relative resolution governed by the orientation-information scale $Ns^6$. Deployment replication improves physical localization only when orientation changes cannot be absorbed by adjusting the active coefficients. For computation, we introduce active endpoint bracketing (AEB), an adaptive finite-bank procedure that evaluates only candidates that can still affect the physical report and otherwise safely coarsens or abstains. Finite-bank experiments, including a four-region synthetic application, show that a point-valued plug-in selector can be physically overprecise, whereas AEB avoids unsupported refinement with fewer candidate evaluations.

We propose computationally efficient tests for equality of mean vectors of two or more high-dimensional populations. Central to our approach is an equivalence between equality of means and a zero population logistic regression parameter. We establish this equivalence for independently distributed observations without imposing common distributional assumptions across populations. Our procedure uses logistic Lasso to screen informative variables and an unpenalized logistic refit for inference in the reduced dimension, yielding asymptotically correct size and consistency. For a specified two-sample Gaussian submodel and sparse discriminative class, the test also attains the minimax separation rate. The framework extends to multiple populations through multi-class logistic regression. Simulations demonstrate accurate size control, strong power, and favorable computational scaling compared with existing tests under unbalanced designs and variance heterogeneity. Applications to gene-expression data with more than twenty-two thousand variables illustrate the practical scalability of the proposed procedures.

In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope of modern quantitative convergence theory. For usual gradient-based samplers, this reflects a genuine instability issue, whereby Metropolis acceptance rates can degrade badly. In this work, we study the gradient-free random-walk Metropolis sampler, and show that for a wide range of light-tailed targets, the acceptance probability remains stable for reasonable choices of proposal variance, from which effective and favourable mixing time estimates can be deduced. The analysis relies on a simple relationship between the first and second derivatives of the log-density of the target distribution.

Many estimators are ratios of coprime polynomials in a sample covariance matrix, and their accuracy depends on the relative fluctuation of the sample denominator. Under Gaussian sampling in fixed dimension, we call a nonconstant polynomial denominator self-normalizing if the first-order variance of its relative error does not depend on the population covariance. We prove that these denominators are exactly the flag powers, nonzero constant multiples of products of positive integer powers of nested generalized variances. Equivalently, the denominator's sample-to-population ratio has a covariance-independent finite-sample law, which we determine explicitly. Sufficiency is classical; the new converse shows that a first-order variance condition forces an exact sampling law. We show that relative stability, meaning bounded first-order relative variance, characterizes uniform tightness of scaled relative errors over positive-definite covariances. It permits replacing the sample denominator by its population value in the limit theory of the ratio. Self-normalization is its rigid core. We locate these classes in applications, where regression on predecessors in a fixed order yields only constant or self-normalizing denominators, instrumental-variable formulas yield relatively unstable ones, and nonparametric identifiability does not guarantee relative stability.

We solve the open problem of determining the maximal value of Spearman's rho when Spearman's footrule is prescribed, thereby completing the exact attainable region of these two quantities. To prove this result, we reformulate the underlying copula optimization problem as an optimal transport problem with a linear moment constraint and construct the unique optimal coupling through a matching feasible dual potential and its contact set. Equivalently, this coupling minimizes the variance of $|U-V|$ among all couplings of $U,V\sim\mathcal{U}(0,1)$ with prescribed mean $\mathbb{E}|U-V|$. Our main result admits several applications: First, in the context of finite rankings, we obtain an improved Cauchy–Schwarz inequality between Spearman's footrule distance and the associated quadratic rank difference. Second, in the framework of generalized mixability, we characterize the attainable constant values of $|U'+V'|$, for $U',V'\sim\mathcal{U}(-1/2,1/2)$, and determine the minimal quadratic deviation for a given mean. Third, we derive explicit bounds relating Chatterjee's rank correlation $\xi(X,Y)$, which can detect complex functional dependence of $Y$ on $X$, to the copula correlation ratio–a rank-based fraction of explained variance–by exploiting their conditional i.i.d. representations in terms of Spearman's footrule and Spearman's rho.

We study a function-on-function linear regression model in which the response at time $t$ depends on both the past trajectory of a predictor and its concurrent value. The model combines an $L^2$-historical effect with a point-evaluation effect, and these two coefficient functions are not automatically identifiable. We characterize the resulting non-identifiability and show that the concurrent and historical effects are separately identifiable whenever the covariance eigenfunctions of the covariance operator of the predictor are not pointwise square-summable. This mild and novel condition prevents the concurrent point evaluation from being represented by an $L^2$-historical effect. Building on an orthogonalized representation of the predictor process, we propose a smoothing-spline estimator for both coefficient functions and establish consistency rates. The rates reveal an interesting trade-off between path regularity and eigenvalue decay: smoother predictor trajectories lead to faster convergence of the historical-effect estimator, whereas rougher trajectories lead to faster convergence of the concurrent-effect estimator. Simulation studies and two real-data applications demonstrate the practical importance of disentangling concurrent and historical effects.

Reliability set estimation identifies input regions where a response probability exceeds a target level, bridging estimation and safety-critical decisions. Practitioners typically start with a working model, an imperfect approximation of the true response surface. Relying on this imperfect model may incur decision risk, potentially certifying unsafe regions as safe. We develop a unified framework that turns such a working model into a trustworthy decision rule. First, a modeling-then-calibration procedure decouples estimation from decision. Since the true set is unobservable, we introduce an asymmetric, observable surrogate loss and use a separate calibration set to select a bias-correcting threshold, reducing decision risk and achieving $O_P(1/n)$ volume convergence. Second, we leverage conformal risk control with the surrogate loss to control false inclusion risk, which is the most safety-critical error, at a pre-specified level regardless of working model quality. Together, these calibration procedures show that a separate calibration set is necessary for risk control. Third, an adaptive design concentrates observations on the reliability set and its boundary, improving model quality where errors most affect decisions while controlling budget elsewhere. Numerical studies show not only more accurate set estimates but also calibrated finite-sample risk control that classical plug-in methods lack.

Moment restrictions provide a flexible basis for quasi-Bayesian inference when a full likelihood is unavailable, but the weighting matrix in a quadratic moment criterion determines both the relative importance of the moments and the information scale of posterior updating. We propose curvature-calibrated quasi-Bayesian updating, which uses the inverse of the covariance (or long-run covariance) of the moment conditions evaluated at a self-consistent quasi-posterior center. The resulting fixed-point procedure alternates between covariance estimation and simulation from a fixed-weight quasi-posterior, thereby avoiding parameter-dependent weighting during each simulation run. Under a Bernstein-von Mises condition for the fixed-weight quasi-posterior at the efficient population weight, we show that the calibration map is locally contractive, that its fixed point is consistent at the standard parametric rate, and that the Gaussian approximation continues to hold under the calibrated data-dependent weight, with covariance given by the inverse Godambe information matrix. Under a uniform fourth-moment condition, the scaled quasi-posterior covariance converges to the same matrix, so quasi-posterior and repeated-sampling uncertainty agree to first order. Simulations show improved covariance calibration and interval coverage after a few updates. An application to longitudinal binary-response data illustrates the method with within-subject dependence and overidentified residual moments.

Kendall's tau and Spearman's rho, two widely used dependence measures in statistics and risk management, are often treated as interchangeable, yet can disagree sharply: Schreyer et al.~(2017) established the exact region of attainable $(\tau,\rho)$ pairs. We study the complementary question of equality, namely, identifying nontrivial families of copulas $C$ satisfying $\tau_C=\rho_C$. We prove that this equality holds for every factorizable copula of the form $C_{e,\alpha}\ast C_{\beta,e}$, where $\alpha$ and $\beta$ are piecewise linear monotonic surjections (PLMS). For these PLMS-generated copulas, we study further dependence measures, including Chatterjee's rank correlation coefficient and tail dependence coefficients, revealing some useful algebraic formulas and unexpected phenomena. In particular, Chatterjee's coefficient can exhibit extreme asymmetry.

A normal–inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For $\sigma^2\sim\mathrm{InvGamma}(\alpha,\beta)$, $\mu\mid\sigma^2\sim\mathcal{N}(\gamma,\sigma^2/\nu)$, and $z\mid\mu,\sigma^2\sim\mathcal{N}(\mu,\sigma^2)$, the marginal law of $z$ depends on $(\nu,\beta)$ only through $c=\beta(1+1/\nu)$. Hence the reconstruction-visible parameter space is the three-dimensional quotient $(\gamma,\alpha,c)$, with a one-dimensional fiber degree of freedom. For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective. Writing $\rho_0=2\beta_0/\nu_0$ and $T=c/\{\alpha[(\gamma-\gamma_0)^2+\rho_0]\}$, we show that inverse canonical allocation $1/\nu_{\rm can}$ is an explicit strictly increasing function of $T$. For $\alpha>1$, the ratio $u_{\rm epi}/u_{\rm var}=1/\nu_{\rm can}$ is therefore determined by the quotient state and prior; for rank-only use under a common calibration, $T$ contains the same coordinatewise ordinal information. The residual prior gauge is characterized rather than eliminated: $(\gamma_0,\rho_0)$ govern ordinal dependence, while $(\nu_0,\alpha_0)$ determine numerical calibration and the analytic ceiling of $1/\nu_{\rm can}$.

Let $P$ be an irreducible reversible Markov kernel on an $m$-state space $\Omega$, and denote its right spectral gap by $\Gap(P) =1-\lambda_2(P)$. From a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For each fixed $m\ge2$, define \[ c_m^\star= \inf_{\substack{|\Omega|=m,\ P\ \mathrm{irreducible\ and\ reversible} n\ge2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=\Theta(m^{-1})$. Thus, the reciprocal optimal comparison cost $C_m^{\mathrm{opt}}=(c_m^\star)^{-1}$ has order $m$. In the two-state case, we determine the exact values $c_2^\star=1/4$ and $C_2^{\mathrm{opt}}=4$. %the exact constants for $m\ge3$ remain open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson–Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=\Theta_P(n^{-1})$.

In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}{\alpha}$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $\alpha$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}{\alpha}$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/\alpha$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $\alpha$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $\Lambda(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E \Lambda(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^{\beta}/\mathbb{E}_{P_0}[R^{\beta}]$: the ceiling as $\beta\downarrow0$, $R$ at $\beta=1$. Which $\beta$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/\alpha)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(\beta>1)$ below it and by flattening above.

Covariate adjustment improves estimation efficiency in randomized experiments, but standard calibration and augmentation methods, when applied to distribution or survival functions, do not preserve monotonicity—a fundamental property of the estimand. We propose using empirical likelihood with covariate-balancing constraints to construct a covariate-adjusted empirical measure for each treatment arm. Estimators of a broad class of distributional functionals, including cumulative distribution functions, survival functions, quantiles, and restricted mean survival times, are then derived as plug-in functionals of this measure, automatically inheriting proper shape constraints. We establish asymptotic normality with an explicit, guaranteed efficiency gain over unadjusted estimators. The asymptotic distributions are invariant to the randomization scheme, providing a unified inference procedure under simple randomization and all commonly used covariate-adaptive designs satisfying a mild balancing condition. This unified construction, adjusting the empirical measure once and deriving all estimators from it, offers a principled reconciliation of covariate adjustment with shape preservation. Simulations and an application to the SURPASS-4 trial confirm the theoretical gains.

This paper investigates the estimation of the regression operator in function-on-function regression models. While traditional research has predominantly focused on linear models or their immediate nonlinear extensions, we propose a neural operator approach to accommodate general regression operators under mild smoothness assumptions. Operator learning has emerged as an active area of machine learning, particularly for solving physical models governed by partial differential equations. Using this paradigm, our methodology introduces the separable neural operator, a neural-operator architecture that represents the regression operator through input-dependent coefficient functions and output-dependent basis functions. Beyond adapting this architecture to the regression operator estimation problem, we establish the consistency of the estimator under relatively mild smoothness and sampling conditions, allowing functional data to be observed on dense, possibly irregular, discrete grids. We also apply the proposed approach to the BGC Argo data and demonstrate its potential for oceanographic research.

The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $\Theta (\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.

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

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

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

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

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

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