Papers

Selecting weights and projection directions from the observations used for testing can invalidate a reference distribution. We develop a framework for exact conditional inference under rotation symmetry. A maximal invariant permits adaptation while leaving a uniform common phase for calibration. This yields explicit Cauchy references for invariantly selected weights and projections. Under separate block rotations, the same principle gives exact normal-score combinations with coefficients selected from all blockwise invariants. The calibration results require neither densities nor moments. To illustrate the framework, we study direction learning and power in spherical Gaussian location models, identifying a regime in which consistent detection is possible without consistent direction recovery. For heterogeneous functional data, a matched mean–residual construction produces the required null symmetry under a conditional Gaussian model. Simulations compare Radius and Blend, two representative procedures, with sample splitting, and an electrocardiogram analysis illustrates their application.

Resampling is the standard remedy against weight degeneracy in particle filters, yet its effect on the spectrum of the particle second-moment matrix is poorly understood. We study a single resampling step for a d-dimensional ensemble of N particles with d/N -> c. For any unbiased scheme, the resampled empirical second-moment matrix dominates its pre-resampling counterpart in conditional convex order, while the conditional mean is preserved; the excess second spectral moment equals the Frobenius energy of the resampling perturbation. For multinomial resampling, the empirical count law converges to the mixed-Poisson law determined by the limiting weight law. For residual resampling, the count law is determined by the joint limit of floor-fraction pairs: we classify all residual limits of weight sequences converging to delta_1 in law as a one-parameter family between the deterministic identity and the full multinomial count law. Each count law feeds a common covariance-side spectral map, yielding exact consequences: a closed-form zero-eigenvalue mass, a rank-and-spectral-gap transition with a universal lower edge benchmark, computable regular edges, unbounded right support, and scheme-specific second-moment costs. The effective sample size determines one bulk statistic but cannot determine the limiting spectrum. Pre-registered numerical experiments confirm the theory at every level.

Learning interaction structures from spatiotemporal event data is a key statistical problem in fields such as neuroscience, epidemiology, and seismology. We study this problem for Hawkes processes on a continuous spatial domain, where events may trigger future events at different locations. Following the framework developed in “Spatiotemporal Hawkes processes with a graphon-induced connectivity structure” by Baars, Laeven, and Mandjes (2026), we model the spatial interaction mechanism using an underlying graphon that characterizes how the intensities' excitation strength varies across locations. By expanding the baseline intensity, temporal excitation kernel and graphon in orthogonal bases, we represent the problem as a high-dimensional parametric inference problem. We propose a sparse penalized least-squares estimator that recovers the effective interaction structure while controlling model complexity. Our analysis establishes non-asymptotic error bounds in a high-dimensional regime where the number of parameters may grow with the observation horizon, without requiring the conditional intensity to be almost surely bounded. We further establish sparsistency results, showing that the latent interaction network can be consistently recovered. Simulation studies demonstrate the good finite sample performance of the proposed method and an application to large-scale neuronal spike data, comprising 1.81 million events, showcases its practical effectiveness.

We study the recovery of the hidden vertex correspondence between two correlated dense random geometric graphs. Each graph is generated from $n$ latent unit vectors in $\mathbb R^d$; the vectors $u_i^A$ and $u_i^B$ of corresponding vertices satisfy $\langle u_i^A,u_i^B\rangle=\rho$, and the vertex labels of the second graph are permuted by an unknown permutation $\pi$. For every fixed correlation $\rho\in(0,1]$ and every edge density bounded away from zero and one, we give a randomized polynomial time algorithm that recovers $\pi$ exactly with probability tending to one whenever $d\ge(\log n)^L$, where $L$ depends only on the correlation and the density bounds. The algorithm assigns to each vertex a signature formed by counting decorated paths with a constant number of seeds. The number of such seeded templates grows exponentially with the path length, with an arbitrarily large constant base, so the resulting statistics amplify an arbitrarily small constant correlation. We control the dependencies created by the latent geometry in two ways: when $d$ is larger than $n/(\log n)^{L_0}$ for a suitable constant $L_0$, by comparing local moments with those of correlated Erdős–Rényi graphs; for smaller $d$, by spectral estimation of the latent geometry combined with a harmonic correction. We also provide low-degree evidence for a tradeoff between the correlation and the computational cost.

Non-exchangeability and radial asymmetry have got substantial attention in the literature, but relation between the two has not been studied yet. We propose the question: if a copula has a given degree of radial asymmetry, what is the range of possible values of the measure of non-exhangeability. We determine the set of all points whose first coordinate is the radial asymmetry of a copula and the second coordinate is its non-exchangeability. We show that this set is a non-convex hexagon.

We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter $\gamma$ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals $\gamma^2$ times a positive compression error for a deterministic second-chaos operator, up to $o(\gamma^2)$, with designs allowed to depend on $\gamma$. After enlarging the probe dictionary, the optimal $m$-response coefficient is asymptotic to $\|Lg\|_{L^1(D)}^2/(8m)$, where $L=-(2\pi)^{-1}\Delta_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

We study sampling from $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition number $\kappa=\beta/\alpha$. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce reflected proximal dynamics, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant $\chi^2$ divergence, both methods achieve total variation error $\varepsilon$ using $\widetilde O(\sqrt{\kappa}d^{1/4})$ expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields $\widetilde O(\kappa^{6/7}d^{1/4})$ expected oracle queries. In both regimes, the dependence on the target accuracy $\varepsilon$ is $\mathrm{polylog}(1/\varepsilon)$, making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.

A compact metric tree is the natural domain of spatial data recorded along a river network, a root system, or the branched processes of a cell, and a suitable distance between two points is the length of the path joining them. That distance admits almost none of the covariance models of spatial statistics, since only the roughest members of each family stay nonnegative definite, and the resulting fields are nowhere mean square differentiable. The repair proposed here is a change of metric. Comparing two points through their distances to the leaves gives a Hausdorff-type metric built from the shortest-path one. The new metric has a closed form, agrees with the shortest-path one inside every edge, and is strictly smaller across branching points. Any correlation function valid in a Euclidean space of dimension equal to the number of leaves may then be composed with it, returning the Matérn, powered exponential, and generalized Cauchy families at every degree of smoothness, and, on trees with few leaves, also correlations that change sign or have compact support. The same construction yields the multivariate Matérn and space-time models of the Gneiting class. In simulations covering a range of smoothness and of spatial scale, maximum likelihood provides estimates without appreciable bias, and on a reconstructed microglial cell the smooth models reduce the cross-validated prediction error by half.

This paper develops a unified framework for distributional regression with arbitrary response variables. Rather than relying on scalar thresholds or Euclidean structure, we model conditional event probabilities indexed by measurable sets, which extends distributional regression to general response spaces while retaining a direct connection to classical event-based formulations. Within this framework, we study two complementary estimators: a preliminary estimator based on separate regression of event indicators, and a constrained estimator that enforces inclusion-consistent predictions through a directed acyclic graph (DAG) over a finite set family. We establish general error bounds for both estimators and further specialize the results to dense ReLU networks, yielding explicit convergence rates overcoming the curse of dimensionality. Simulation studies and real-data experiments further demonstrate the practical effectiveness of the proposed framework.

The $r$ largest observations of a block form a dependent random vector whose support moves with all three parameters of the $r$-largest generalized extreme-value ($r$GEV) model. For a fixed number $r$ of order statistics, we show that the model is differentiable in quadratic mean if and only if $k_0<1/2$, where $k=-\gamma$ is the shape parameter; the threshold is thus the same as for block maxima, because only the largest or the smallest retained order statistic can reach the moving boundary. In this range the Fisher information is positive definite and decomposes into the block-maximum information plus contributions of the lower order statistics, and it equals the negative expected Hessian, which we prove without using any estimator. The model is then locally asymptotically normal, whereas for $k_0\ge1/2$ root-$n$ local asymptotic normality fails. Numerically, using the five largest order statistics instead of the block maximum alone lowers the efficient asymptotic standard error by about $45\%$ for the shape parameter and by $20$–$24\%$ for the 100-year return level in heavy-tailed cases ($k_0\le-0.2$), but by only about $2\%$ for the return level when $k_0=0.4$.

A quickest change detection problem is studied where the decision-maker has access to multiple experiments, but can choose only one experiment at a time to detect the change. The true post-change distribution under each experiment is unknown, but known to belong to a family of distributions. It is assumed that each of these families of distributions has an element that is least favorable in a well-defined sense. An algorithm is proposed that is designed using the least favorable distributions. It is shown that the algorithm is first-order asymptotically optimal, as the rate of false alarms goes to zero. This algorithm minimizes the worst-case average detection delay over all post-change distributions within the assumed families, subject to a constraint on the average run length to false alarm and constraints on the average fraction of time each experiment is performed before change.

Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate $\gamma$ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the $\sqrt{\gamma}$ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to $\gamma^{\beta-1/2}\ell(1/\gamma)$, with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.

A correct conditional quantile at a point mass need not have an exact nominal hit probability, and pooled residual signs can conceal opposing conditional errors. We construct sequential audits of quantile-set membership using strict left-tail, tie, and strict right-tail outcomes. A closed-form information projection identifies the available sign information. Combining local continuous betting mixtures through context-tree weighting yields an anytime-valid audit and a finite-path oracle inequality over all prunings of a fixed covariate tree. A countable mixture over refining trees attains the full conditional information rate under iid alternatives with uniformly positive sign probabilities; no valid sign-based evidence sequence has a larger almost-sure exponential rate. We also bound the information lost at finite partition resolution and give an exact recurrence for evaluating the local mixtures. A closure argument identifies the limits of testing a prescribed quantile selection, and test inversion gives confidence sequences under conditional model correctness. Simulations examine localized errors, subgroup cancellation, and adaptive covariate allocation. A held-out count-response example illustrates why exact-hit calibration and quantile validity are different hypotheses.

In nonparametric density estimation without privacy constraints, it is known that a confidence set whose coverage holds uniformly over a `rough' class of regularity $r\geq0$ can simultaneously attain, in $L^2$-diameter, the minimax rate over a smaller, smoother class of regularity $s>r$ if and only if $s \le 2r$. We study such adaptive honest confidence sets under non-interactive $\alpha$-local differential privacy (LDP) constraints. Our main result is that, for bounded $\alpha$, such `smooth-rate adaptation' is now possible if and only if $s \le 4r/3$. In this smaller range we construct an honest private confidence set whose diameter is of the private minimax order on each of the two classes. As in the non-private case, the threshold arises because an honest set cannot, at a smooth density, be narrower than the separation rate arising in a related testing problem. Under LDP we prove that this private testing rate is of order $(n\alpha^2)^{-2r/(4r+3)}$, which is at most the private estimation rate $(n\alpha^2)^{-s/(2s+2)}$ over the smooth class exactly when $s \le 4r/3$. For $s > 4r/3$, however, we show that smooth-rate adaptation becomes possible once one discards from the rough class the densities that lie outside the smooth class but within an $L^2$-radius of the order of the private testing rate.

The median-of-means (MoM) estimator is a robust alternative to the sample mean for heavy-tailed or contaminated data. While its non-asymptotic properties are well understood for i.i.d.\ samples, much less is known for dependent sequences. We establish non-asymptotic deviation bounds for the MoM estimator computed from consecutive blocks of an identically distributed $\alpha$-mixing sequence, both for a single mean and uniformly over a class of functions (the median-of-means process). All bounds remain valid, with explicitly adjusted constants, when a sufficiently small fraction of the blocks is adversarially corrupted. As applications, we obtain deviation bounds for a median-of-means Nadaraya–Watson regression estimator, for the MoM kernel density estimator, and for empirical and Bahadur-type estimators of Value-at-Risk and Conditional Value-at-Risk. Finally, we show that the sampling of bulk materials, where the measurements form a spatially dependent random field, reduces to the one-dimensional setting if the samples are taken in well-separated groups.

Small area estimation (SAE) traditionally borrows strength across domains through a prespecified working model. We introduce small area learning (SAL), a learner-agnostic framework that formulates SAE as a multi-target semi-supervised learning problem and treats outcome prediction as a nuisance task. Areas may differ through unrestricted covariate shift and structured conditional shift, with information shared through common and area-specific components. We develop standard and cross-fitted estimators for low-complexity and flexible learners, unifying synthetic, model-based, and fully residual-corrected estimation. For area means, we establish consistency and asymptotic normality and provide variance estimators and confidence intervals. We extend the theory to settings with only aggregated population covariates and characterize the resulting aggregation bias. SAL reveals a bias–variance trade-off: partial residual correction can reduce mean squared error, whereas full correction yields estimators robust to working-model misspecification, potentially at higher variance. Simulations yield RMSE reductions of 34%–49% relative to direct estimation, with the best configuration varying across settings. An empirical application favours a cross-fitted estimator using a flexible learner under the chosen validation criterion. The numerical studies also illustrate how shrinkage can produce overly optimistic confidence intervals at the expense of strong modeling assumptions.

Sequential statistical procedures are increasingly embedded in scientific systems in which observations arrive incrementally from heterogeneous sources. In such systems the statistical update is only one part of the computation: observations must also be acquired, aligned across inputs, ordered, delivered, persisted, and replayed, and results must be passed to systems that display them or act on them. Sequential methods are nevertheless missing from the general software toolkit. They run inside proprietary experimentation platforms and commercial statistical packages, or exist as offline functions over complete data, and the batch fit interface that dominates machine-learning libraries has no place for a procedure that listens, keeps state, and decides when to stop. We propose an architecture that separates statistical inference from data topology, execution runtime, run lifecycle, persistence, and external action. Its central abstraction is the Sequential Procedure: a state-transition function that maps an inferential state, a valid statistical input, and a statistical random state to a new state and a scientific output. The procedure does not define how observations are acquired, paired, scheduled, stored, or acted upon. We specify the architecture semantically rather than through an object model, derive seven conformance invariants that any implementation can be tested against (among them synchronous/asynchronous, checkpoint, replay, and batch equivalence), and describe a dependency-free reference implementation in which all seven hold bitwise. Lorden's 2-SPRT for the Kiefer-Weiss problem serves as the reference procedure; the same contract accommodates sequential estimators, Bayesian filters, change-point detectors, and stateful predictive models.

We investigate the problem of uncertainty quantification in the mean estimation framework. Given an i.i.d. sample from a distribution with unknown mean $\mu$ and variance $\sigma^2$, we want to construct an estimator $\wh\mu_n$ and provide a computable and size-optimal upper bound for the error $|\wh \mu_n - \mu|$. While estimators possessing strong deviation guarantees are well known, no data-dependent non-asymptotic upper bounds for their performance exist in general. We show that this gap can be bridged by introducing a parameter that characterizes the “effective sample size” available for uncertainty quantification. This parameter captures the difficulty of the problem, interpolating between “easy” cases where sub-Gaussian confidence intervals exist and “hard" cases where their construction is impossible. Using this characterization, we design confidence intervals of optimal length that are fully adaptive to the unknown variance. Numerical experiments confirm that our approach maintains nominal coverage even in asymmetric and heavy-tailed regimes where other existing methods fail.

In this paper, we develop an asymptotically normal estimator for the value of policy improvement (VPI) under adaptively collected data. This value, which is defined as the gap between the value of the best treatment policy in a class and that of a baseline policy, gauges whether or not it is worth deploying a personalized treatment policy. Building an estimator for the VPI is naturally more difficult than estimating the value of a fixed policy. This is because, in addition to contending with the correlated data and decaying propensities associated with adaptive experiments, any estimator must also handle the non-regularity, or lack of pathwise differentiability, of the underlying estimand when the optimal policy is non-unique. Our proposed estimator works by sequentially fitting a weighted empirical welfare maximizing policy on historical data and then evaluating it using inverse propensity weighting on future samples. We neither assume that the optimal policy is unique nor make a margin assumption on the underlying heterogeneous treatment effects. A key difficulty in analyzing our estimator is showing that the regret incurred through policy learning is asymptotically smaller than the standard error of the estimator. We accomplish this by showing faster regret rates for policy learning under decaying propensities, which may be of independent interest. We not only evaluate the coverage of our estimator on a mix of synthetic and semi-synthetic data, but also employ it to estimate the VPI in an adaptive experiment evaluating the efficacy of job assistance programs in Jordan due to Caria et al. Although our estimates are generally more conservative than their hierarchical Bayesian analysis, which assumes that the optimal policy is unique, we still corroborate the benefits of personalization amongst Syrian refugees.

We study adaptive confidence sets for mean vectors of multivariate Gaussians with identity covariance under Huber contamination when the contamination proportion is unknown. We show that an information-theoretically optimal adaptive confidence set can be constructed by simultaneously quantifying uncertainty across all one-dimensional projections. The resulting confidence set achieves the optimal diameter and reveals a substantial adaptation cost compared with the minimax estimation error. Although statistically optimal, this construction is computationally infeasible. We establish a Statistical Query (SQ) lower bound showing that achieving diameter $O(m^{-1/2})$ requires either exponentially many queries or query tolerance $d^{-\Omega(m)}$. Thus, in contrast to robust mean estimation, adaptation to the unknown contamination proportion incurs both a statistical cost and a computational barrier within the SQ framework. Finally, we give a low-degree moment-matching algorithm that constructs an adaptive confidence set using $d^{O(m)}$ samples and time. Its diameter bound contains an additional $O(m^{-1/2})$ term beyond the information-theoretic rate, matching the dependence on $m$ in the SQ lower bound. Together, these results characterize the statistical limits of adaptive robust confidence sets under Huber contamination and quantify a computational barrier within the SQ framework.

Flow matching can learn structure directly from data, raising the question of when additional constraints improve generation. We study penalties that encode such constraints, with physics-informed flow matching as a motivating example. Satisfying a constraint more closely need not make the generated distribution more accurate. We therefore compare penalised and unpenalised generative modelling through their expected squared $2$-Wasserstein error. Our analysis covers both vector-valued and functional data in a Hilbert-space setting. We first derive geometric criteria that account for constraint violations and the distribution of samples after projection onto the constraint set. We then solve the statistical comparison explicitly for specific velocity-field classes and constraints via flow matching. In this model, penalisation improves accuracy exactly when the reduction in estimation variance outweighs the increase in squared bias. The resulting formula shows how this trade-off depends on sample size, the model class, the noise schedule, and the penalty weight.

When can combining source and target data yield much better performance than using either alone? From a minimax viewpoint, writing $R(n,n_T)$ for the minimax risk from $n$ source and $n_T$ target observations, we ask when $R(n,n_T)$ decays strictly faster than both $R(n,0)$ and $R(0,n_T)$. We call this the Synergistic Learning Phenomenon (SLP) and quantify it by the Synergistic Advantage Ratio $\mathrm{SAR}=\min(R(n,0),R(0,n_T))/R(n,n_T)$. We study domain adaptation with a known link: source regression $f$, target regression $g=\mathcal{A}f$ for a known operator $\mathcal{A}$, and possibly different designs. Three links—identity, integration, and differentiation—and the density ratio $\rho=h_T/h_S$ determine whether synergy is possible, at what rate, and in which sample-size window. For the identity link (covariate shift) with a Beta$(a,1)$ source and uniform target, a known necessary condition for SLP is $a\geq2+1/(2\beta)$, where $\beta$ is the smoothness of $f$ (Result I). An integral link $g(x)=\int_0^xf(t)\,dt$ can yield the joint rate $(nn_T)^{-1/2}$ even for a bounded ratio (Result II), and under a Beta$(a,1)$ source has a phase diagram with cuts at $a=2$ and $a=3$: integral SLP for $1<a<2$, an extra level-split rate $n_T^{-1}$ for $2<a<3$, and covariate-shift-type spatial SLP for $a>3$ (Result III). A derivative link $g=\dot f$ synergizes only under a stronger singularity (Result IV), giving a sharp integration–differentiation asymmetry under a bounded ratio. Across all three links, synergy arises from role separation: each dataset resolves a distinct bottleneck associated with the link and designs. We characterize the sample-size windows for synergy and how SAR varies with $n$ and $n_T$. Simulations support the predicted phase structure, and a mirrored integration–differentiation analysis of real vehicle and smartphone signals illustrates the asymmetry.

Model averaging emerges as an attractive ensemble technique that strategically synthesizes all available information from potential candidate models to enhance estimation accuracy. Despite of having been widely practiced in various areas in biostatistics and economics, its application to causal inference is rather limited. We propose to estimate the average treatment effect (ATE) by forming the weighted combination of the robust rank-based estimators obtained from a series of candidate models, where the model weights, selected via the growing window cross-validation, dynamically adjust the importance of candidate models. The proposed robust rank-based model averaging (RRMA) is not only robust to handle potential outliers and heavy-tailed error distributions but also capable to reduce the risk of model mis-specification and enhance the flexibility and the accuracy of estimation. Theoretically, we establish the desirable asymptotic properties of RRMA, including the asymptotic optimality, consistency and asymptotic distribution. Moreover, motivated by the asymptotic distribution of RRMA, an easy-to-implement confidence interval is also developed for the ATE. Simulation experiments and an empirical application demonstrate the superiority of RRMA over existing competitors in estimation accuracy, coverage probabilities and widths of confidence intervals.

Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension $d \ge 2$, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.

Detection of geometry in random graphs and matrices has been extensively studied in recent years. For testing a dense random geometric graph on $n$ vertices with latent points in dimension $d$ against an Erdős–Rényi graph, the detection threshold $d\asymp n^3$ is well known since Bubeck, Ding, Eldan, and Rácz (2016). We sharpen this result by establishing the optimal asymptotic testing error in the critical regime $d/n^3\to\lambda\in(0,\infty)$. The log-likelihood ratio is shown to be asymptotically normal and dominated by the signed triangle count. Moreover, we consider testing the rank statistics of inner products between latent points against uniformly random ranks. We show that ranks preserve all asymptotic testing information available in the full matrix models, and a cubic statistic based on rank statistics attains the optimal asymptotic error. This then yields an asymptotically optimal test for random kernel matrices with a possibly unknown, strictly monotone kernel. The proofs combine the classical framework of Davies (1985) and Le Cam and Yang (1988) for asymptotic normality under information loss with conditional limit theorems for the models in consideration.

For a fixed conformal score function, standard split conformal prediction calibrates a prediction set using the empirical distribution of the conformal scores, without using available side information about the data. We propose a moment-calibrated approach that uses such information to reduce variability across calibration data sets. Side information involving the covariates, the response, or both is expressed as moment restrictions, which are enforced through empirical likelihood weights on the calibration data points. The resulting weighted empirical distribution of the scores determines the weighted empirical quantile without changing the score function. We establish finite-sample bounds on training-conditional and marginal coverage of the moment-calibrated prediction set. We derive asymptotic distributions for training-conditional coverage and average prediction-set size, show that their asymptotic variances are no larger than the corresponding variances under standard split conformal prediction, and characterize when the reductions are strict. In the semi-supervised setting, we identify an optimal moment function of a given covariate representation, develop implementations using prespecified moment functions or a function learned by cross-fitting, and quantify the effect of the unlabeled sample size. We also extend the method to coverage conditions indexed by a class of functions of the covariates through weighted quantile regression. Numerical studies demonstrate the effectiveness of the proposed methods.

Hoffman constants quantify the distance to the solution set of a polyhedral system in terms of its residual; they control error bounds, geometric conditioning, the convergence rate of first-order iterative algorithms, and the stability of regression estimators to target perturbations. Despite their importance, explicit values of Hoffman constants are rarely available due to their computational complexity. In this paper, we derive explicit bounds on the Hoffman constant for a class of structured polyhedral systems arising in sparse regularization. We show that the constants admit a decomposition into spectral, geometric, and interaction components, which together determine their scaling behavior. This framework explains the gap between worst-case dimension-dependent bounds and the favorable conditioning observed empirically in structured problems. As a primary application, we analyze $\ell_1$-regularized least squares, demonstrating that the effective constants depend on low-dimensional active subspaces rather than the ambient dimension. We then extend to the elastic net and adaptive $\ell_1$-regularized formulations, demonstrating how the underlying optimization geometry is altered by the $\ell_2$ penalty and reweighting scheme. Furthermore, we relate our results to classical statistical assumptions and show that the restricted eigenvalue condition bounds only the primal curvature of the active set, leaving the dual interaction geometry unconstrained and is thus insufficient alone for fast theoretical convergence rates. We further show standard $\ell_\infty$ coherence conditions do not yield dimension-independent Euclidean bounds to close this gap. Our theoretical results show that Hoffman constants admit explicit, interpretable forms in structured regression settings and, in practice, provide actionable tools for post-hoc feature pruning and a priori optimal hyperparameter selection.

Royen's Gaussian Correlation Inequality (GCI) is a fundamental result in statistics and probability, but unfortunately, at least by now, the result is not very well known in the statistical community. In this paper, we demonstrate the potential of GCI in high-dimensional testing problems that involve multivariate Gaussian variables. Leveraging the GCI, we establish sharp Type I error bounds under arbitrary Gaussian covariance structures. We also formulate a conjecture and relate it to the well-known Spherical Cap Covering Conjecture in discrete geometry and Weak Simplex Conjecture in information theory.

The distribution function $F(y)=\Pr(S^2\le y)$ of the sample variance $S^2$ of four independent observations is elementary on its entire support for every polynomial parent density on $[0,1]$ (without loss of generality), and we give it in finite closed form for the uniform, Beta$(2,1)$ and Beta$(2,2)$ parents. The support splits at $y = 1/6$, $2/9$ and $1/4$, with the ceiling at $1/3$, all fixed by geometry: in coordinates aligned with the cube diagonal the event $S^2\le y$ is a ball of radius $\sqrt{3y}$, the cube projects to a rhombic dodecahedron, and these are the levels at which the ball reaches its faces, edges and two kinds of vertices. The divergence theorem reduces every moment of the clipped ball to integrals along circular arcs, so each regime closes in algebraic functions of $\sqrt{y}$ and inverse trigonometric functions. For the uniform parent the only non-obvious constant is $K_g = 12\sqrt2\arctan\sqrt2$, proportional to the mean width of the cuboctahedron; the Beta constants are rational expressions in $\pi$, $\sqrt2$, $\sqrt3$ and $\arctan\sqrt2$, as in the uniform case. At $n = 4$ the normal-theory $\chi^2$ recipe used in acceptance testing misplaces the $99$th percentile of $S^2$ by $50\%$ for the uniform parent. For three observations, polynomial parents also give elementary closed forms (arXiv:2609.25333), but two elementary cases in a row do not guarantee a third, and we make no claim beyond $n = 4$.

Data and the governing partial differential equations (PDEs) of continuous-time dynamical systems provide rich structure for statistical learning. A central challenge is how to integrate data with PDE constraints in a principled way to improve generalization. In this paper, we characterize the statistical estimation and generalization properties of learning solutions to the heat equation with a noisy source function from finite, randomly sampled data. We first derive a novel variational formulation, which we characterize as the minimizer of loss function for an energy regularized and ellipticity debiased Sobolev regression. By formulating the learning task as a supervised problem over space-time domains under physically meaningful norms, our approach goes beyond the traditional supervised learning framework by effectively leveraging auxiliary data—samples of the source function of the heat equation—to design a physics-informed energy regularization. Our analysis reveals that learning in linear parabolic PDEs exhibits intrinsic anisotropy between spatial and temporal regularity, in sharp contrast to the static elliptic problems, leading to fundamentally different estimation and generalization behaviors. Specifically, we derive non-asymptotically rate-optimal estimation errors, as well as generalization error, which explicitly capture the interplay among temporal sampling, spatial resolution, and solution smoothness. Our numerical experiments demonstrate substantial orders of magnitude of reduction in testing errors against various physics-informed neural network methods on various parabolic and hyperbolic PDEs.