Papers

Forecasts made at different levels of aggregation are often required to agree—for example, regional forecasts should sum to the national total. Forecast reconciliation imposes such relationships, but the meaning of agreement depends on whether the constraint is applied to a predictive distribution, a reported summary such as a mean or quantile, or a decision based on the forecast. We show that these operations are not interchangeable. Even when every possible outcome from a predictive distribution satisfies an aggregation rule, its separately reported marginal quantiles need not add up; under nonlinear relationships, even the coordinatewise means may violate the constraint. We characterise when linear reconciliation preserves marginal quantiles and show that nonmedian quantiles generally cannot be preserved in a genuine hierarchy. We also explain why conditioning on an exact nonlinear constraint requires specifying how that constraint is observed or approximated, and identify when the predictive mean contains enough information for a constrained decision. In an application to Australian tourism data with 28 rolling forecast origins, the median discrepancy between two natural constructions of the national 95th percentile is 159.9% of the reconciled 90% prediction-interval width under bottom-up reconciliation. Rankings of mean- and quantile-based forecasts also reverse when squared-error loss is replaced by asymmetric inventory loss. These results provide a framework for deciding what should be made coherent and how competing methods should be evaluated.

Long environmental records increasingly combine observations from successive observing or processing systems. Transitions between these systems create temporal seams that may reflect imperfect harmonization or changes in the observed process. We introduce the Temporal Seam Score (S-score, or S), a signed, robust measure for assessing continuity at a known transition time. The complete time series is fitted with a seam-blind model accounting for trend, seasonality, and optional covariates. The residual contrast between symmetric windows around the transition is compared with matched reference contrasts calculated within the contributing segments. The S-score is the standardized departure of the seam contrast from this reference. An S-score near zero indicates that the seam is consistent with ordinary temporal variability. Its sign indicates the direction of the discontinuity, while its magnitude (|S|) indicates how exceptional the discontinuity is. Simulation showed that |S| reflects the imposed discontinuity relative to residual variability: a 20% shift produced a median |S| of 2.58 at a residual coefficient of variation of 0.10, compared with 0.84 at 0.50. In a MODIS-to-Sentinel-3 vegetation record, 121 of 500 sites yielded finite S-scores, of which 90 met all reliability criteria. Among these reliable sites, median S was +0.217 and median |S| was 0.627, indicating little systematic directional displacement and a median seam departure below one reference spread. The S-score provides a general and auditable diagnostic for evaluating temporal continuity across generations of sensors, algorithms, and observing systems without requiring temporal overlap, thereby supporting harmonization quality assurance and more reliable attribution analyses based on long-term environmental records.

Markov chain Monte Carlo is widely used for model assessment and parameter fitting in astronomy and astrophysics, leading to numerous ready-to-use packages implementing various sampling techniques. For large-scale analyses involving many individual targets, computational efficiency becomes paramount, as it dictates the overall runtime. We introduce Nii-MALA, a high-performance C implementation of the Metropolis-Adjusted Langevin Algorithm, which leverages Message Passing Interface for parallelization and incorporates automatic differentiation backends for gradient evaluation. The code offers flexible control via a configuration file, allowing users to specify the proposal step sizes for all model parameters across parallel chains. We validate its effectiveness on 15- and 30-dimensional Gaussian distributions and a bimodal distribution, and further benchmark it against several other codes using radial velocity data from 51 Pegasi b. Our benchmarks confirm the efficiency of Langevin sampling over random-walk sampling in obtaining the effective sample size, particularly as the dimensionality of the target function increases. This algorithmic advantage can be effectively leveraged when paired with an efficient automatic differentiation library. However, for complex functions commonly encountered in astronomy—such as radial velocity orbital fitting—the derivative-dependent Langevin sampler shows lower efficiency than parallel tempering or alternative sampling approaches, as revealed by effective sample size analysis.

Xu, Vardi and Safran (ICML 2026) prove that over-parameterized ridge regression over an unstructured random Gaussian feature map groks, with the delay between memorization and generalization growing as $1/\lambda$ in the weight decay. We show that on a structured feature map the same delay does not appear. For a band-limited Fourier feature map over $\mathbb{Z}_p^2$ carrying a single-character target that lies inside the expressible class, enlarging the band at fixed positive weight decay drives peak held-out accuracy monotonically from $1.00$ to $0.07$, with no memorize-then-generalize regime anywhere along the sweep. The degradation is not an interpolation effect. It sets in at capacity ratio $q/n = 0.638$, far below the interpolation threshold, on separate grounds from the exact null space that appears above it. What does have a sharp boundary is the active support. Holding the nominal dimension fixed and masking the band back to $1089$ active modes restores held-out accuracy of $1.000$ with zero variance across seeds, while the full $4225$-mode band collapses to $0.185$. The number of active modes acts through the teacher-weighted spectrum of the empirical Gram matrix and not through the capacity ratio, which makes this a statement about feature geometry and not a restatement of double descent.

Data attribution seeks to quantify how individual training examples shape a model's predictions and underpins problems including data valuation, machine unlearning, and model interpretability. Despite having a long line of work, computationally scalable methods often struggle to predict the effect of removing training data in neural networks due to their non-convex nature. To overcome this challenge, metagradient-based methods such as MAGIC (Ilyas and Engstrom, 2025) differentiate each prediction through the entire training run and compute its exact influence with respect to the training data, but require a separate run for every prediction. To reduce this cost, we cast budgeted attribution as estimating a large influence matrix from a small number of measurements. We show that the measurements most appropriate for recovering this matrix differ from those best suited for attribution itself. We then present two algorithms, MAGE and SPELL, suited for reconstruction and attribution respectively, that run on existing metagradient machinery at no extra cost. Empirical studies demonstrate strong performance over existing baselines across training scales and measurement budgets.

We study lazy single-site Metropolis dynamics $P_{\beta}$ at inverse temperature $\beta \geq 0$ on $\{-1,+1\}^d$ for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most $\kappa\le1/2$, the worst-case total-variation mixing time of $P_{\beta}$ is at least of order $d\exp\{c\beta(d-1)\}$ for $\beta\ge1$, with universal $c>0$. Averaging over global spin reversal reduces the mixing time to $\mathcal O(d^2(1+\beta))$ for both $GP_\beta G$ and $(P_\beta+G)/2$, where $G$ is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits $\{-x,x\}$. Partition-function interpolation gives the lower bound for $P_\beta$. The upper bounds follow from Wu's Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.

Regular-grid spatial functional datasets arise naturally in gridded environmental, oceanographic, climate, and remote-sensing applications, where each spatial location is associated with an entire curve. Existing spectral spatial functional principal component analysis methods provide an important frequency-domain approach for such data, but they do not directly target finite-neighbourhood least-squares reconstruction from an estimated latent spatial component field. To address this, we propose Spatial Functional Generalized Dynamic Principal Components (SFGDPC), a reconstruction-based dimension-reduction method for regular-grid spatial functional data. Each function is first represented by basis coefficients, and each coefficient vector is reconstructed from a scalar latent spatial field and its Chebyshev neighbourhood on the rectangular grid. The spatial neighbourhood radius is selected using a Bayesian information criterion (BIC)-type conditional reconstruction criterion. In the local-neighbourhood transfer simulation, SFGDPC reduced mean cumulative normalized mean squared error (NMSE) relative to spatial functional principal component analysis (SFPCA) by approximately 38-53% across the reported component counts and covariance conditions. In the Indian Ocean sea surface temperature (SST) application, one SFGDPC component produced lower whole-grid reconstruction error than both the boundary-safe and high-cap SFPCA benchmarks across all 33 annual fields. The results support SFGDPC as a local reconstruction-based complement to spectral SFPCA for regular-grid spatial functional data.

This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0 \in \sum_{i=1}^n (G_ix + T_ix)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, ND-DFFP and NI-DFFP, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While ND-DFFP utilizes a homogeneous network-dependent stepsize, NI-DFFP reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.

Wildfire spread poses substantial environmental and public-health risks, motivating interpretable models for short-term forecasting. We develop a statistical framework that combines cellular automata with ideas from network epidemiology to model wildfire evolution across a spatial lattice. Each grid cell is classified as available, burning, or consumed. State transitions distinguish spread from burning neighbors, intrinsic ignition, and cessation of burning, with transition rates linked to meteorological and environmental covariates. A likelihood-based estimation procedure yields transition-specific covariate effects and probabilistic forecasts of cell states. We assess forecasting performance in a simulation study and in applications to the 2018 California wildfires and the 2019-2020 Australian wildfires. We also compare the method with a published forecasting approach using the 2017 Haypress fire. The results show strong short-term discrimination in many settings, with reduced accuracy at longer forecast horizons and during abrupt fire expansion. The framework provides an interpretable basis for studying wildfire dynamics and identifies opportunities to improve ignition forecasts through richer spatial and observation models.

The problem of partial identification concerns bounding causal quantities that remain unidentifiable given the observed distribution and the causal diagram of the underlying structural causal model (SCM). While bounds have been developed for certain causal quantities with continuous outcomes, they are necessarily univariate and are unable to reflect differing levels of confidence. Building on previous work in partial identification, we propose a simple Bayesian method for deriving probabilistic bounds on a large class of causal quantities with potentially multivariate and/or continuous outcome variables.

For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $\alpha.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $\alpha \downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate $O(\sqrt{\alpha})$ in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order $\sqrt{\alpha}$, in contrast to the $\alpha$-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.

Shapley value (SV)-based methods are the prevailing framework for feature attribution in machine learning, yet existing population-level Shapley estimators generally assume that observations used to evaluate the coalitional game are fully observed under a common feature space. This assumption is routinely violated in multi-site studies across biomedicine, social science, and environmental monitoring, where institutions record different features under different protocols, producing systematic blockwise missingness across sources. We first show that the standard remedy of imputing missing features before computing Shapley values introduces systematic, coalition-dependent bias into the resulting attributions. We then propose FUSHAP (Fusion Shapley Attribution from Partially-observed data), a method that leverages partially-observed auxiliary sites to reduce the variance of a preliminary single-site Shapley estimate without imputation. A permutation-based screening step detects and excludes sites whose data distributions are incompatible with the target population. In synthetic experiments, FUSHAP achieves $3$–$8\times$ lower MSE than the single-site estimator and $2$–$3\times$ lower MSE than imputation baselines without incurring imputation-induced bias, and the screening procedure identifies misaligned sites with $82\%$ power at moderate misalignment and $100\%$ for strong misalignment. On multi-site air quality and multi-center clinical data, FUSHAP reduces MSE by approximately $3$–$7\times$ relative to the single-site estimator; in the clinical application, standard imputation can increase MSE above the single-site baseline.

We consider nonparametric regression with a Banach-space-valued predictor where the covariate takes values in a general separable Banach space. Our primary objective is to develop a unified framework for estimating the regression operator without relying on an inner-product structure, thereby extending classical and functional nonparametric regression beyond Euclidean and Hilbert spaces in the presence of measurement error. We establish the fundamental properties of the proposed estimator and develop its large-sample theory under conditions formulated in terms of the geometry and local concentration of the Banach-valued predictor. In particular, we study pointwise estimation and inference and discuss the construction of pointwise confidence intervals and uniform confidence bands. Our proposed framework is further extended to accommodate error-in-variables settings, where the predictor is observed with contamination. We also investigate related inverse problems arising from the construction of inverse weighting functions and consider partial contamination models. These developments provide a general framework for nonparametric regression and inference with complex infinite-dimensional predictors and identify several statistical and mathematical challenges that arise when the usual Hilbert-space structure is unavailable.

We study sampling from a distribution supported on an unknown compact $d$-dimensional $C^2$ manifold $M\subset\mathbb{R}^D$, observed only through i.i.d. uniform points from $M$. We reconstruct the constraint using an adaptive local-convex-hull estimator and target an ambient distribution penalized by squared distance to the reconstruction. Although the reconstructed set may be nonsmooth or fail to be a manifold, its Hausdorff accuracy alone suffices to control the Wasserstein error. We quantify the tradeoff between reconstruction accuracy and penalty strength and show that optimal tuning gives an error proportional to the square root of the Hausdorff error. A planar example proves that this dependence is sharp for the proposed scheme. For a $d$ dimensional constraint reconstructed from $N$ observations, the error becomes $\mathcal{O}( (N/N)^{1/d})$. Finally, we prove uniform geometric ergodicity of a Gaussian random-walk Metropolis–Hastings sampler and combine reconstruction, approximation, and mixing into an explicit finite-time guarantee.

This paper introduces a novel framework for Regularized Multivariate Functional Principal Component Analysis (ReMFPCA) via Functional Singular Value Decomposition (SVD). The proposed method extends existing MFPCA approaches by incorporating a generalized functional SVD within a Hilbert space framework, enabling simultaneous regularization of both functional principal components (PCs) and their associated PC scores. A key innovation of this framework is the inclusion of a sparsity penalty on the PC scores, which enhances interpretability by filtering out irrelevant subject-specific variations. This dual-penalization strategy represents a significant advancement beyond existing covariance-based eigen decomposition methods, which penalize only the functional PCs. Two power algorithm implementations, sequential and joint, are proposed, together with a cross-validation approach based on iterative regression for optimal smoothing parameter selection. Comprehensive simulation studies and real data applications demonstrate that the proposed framework substantially improves the extraction of informative and interpretable components, offering methodological and practical benefits for analyzing multivariate functional data across diverse domains.

Importance weights are essential in domain adaptation under label shift, yet their utility is often undermined by the finite sample uncertainty associated with their estimation. Existing methods typically analyze this uncertainty through Gaussian elimination on interval-valued linear systems, which leads to overly conservative confidence regions and inefficient downstream applications. We propose a paradigm shift from inversion-based inference to a direct matrix constraint framework. We use this framework to define a joint confidence region and extract marginal intervals via linear programming, deriving provably tighter bounds for importance weights while maintaining exact finite-sample validity. Furthermore, we analyze the confidence region's geometry and provide the theoretical results for its diameter bounds. Evaluated across text, image, multimodal benchmarks, including AGNews, MNIST, CIFAR-10, N24News, and a real-world autonomous driving dataset, nuImages, our approach consistently yields shorter confidence intervals and smaller prediction sets than inversion-based methods.

High dimensional spatio-temporal models can quickly run up against computational limitations. Recursive or multi-stage Bayesian algorithms are one way to address this computational challenge. Here we consider an increasingly used two-stage algorithm. The first stage models locations independently and in parallel across space. Resampling particles from this stage-one model with Metropolis-within-Gibbs methods and suitably computed acceptance ratios can impose spatial dependence across locations at a low computational cost, while targeting the same posterior distribution as the single-stage MCMC algorithm at a fraction of the overall computational cost. In this paper we show three complete examples of how two-stage MCMC algorithms can be used to efficiently fit Bayesian spatio-temporal models to large datasets. Specifically, we consider: a spatio-temporal self-exciting count model on areal data utilizing intrinsic conditional autoregressive (ICAR) prior distributions; a spatio-temporal model of point-referenced data using a stationary, isotropic distance-based covariance function; and a binary model fit to data on a spatial lattice utilizing a latent Gaussian process to capture all spatio-temporal dependence. In each example, we define the model and describe the corresponding two-stage MCMC approach. We then compare the resulting posterior samples and computational efficiency with those obtained using a standard single-stage MCMC algorithm. While varying across the three examples and numerous parameters, the two-stage approach often achieves roughly an order of magnitude higher computational efficiency, with the two-stage and single-stage algorithms producing closely agreeing posterior samples.

Count data often contain a large number of zeros in ecological and environmental research, arising from factors such as true species absence, imperfect detection, or the rarity of certain events. Choosing and interpreting suitable statistical models, like Poisson, Negative Binomial, zero-inflated, or hurdle models, can be difficult for non-experts because of the models' assumptions and diagnostic requirements. To address this, we introduce an interactive web application built in R with the Shiny framework that helps users analyse, compare, and interpret count-data models that may exhibit zero inflation. The application enables users to upload datasets, explore distributions and relationships among variables, and fit a variety of models including Poisson, Negative Binomial, zero-inflated, and hurdle approaches. It incorporates tools for model comparison using information criteria, visualisation of coefficient estimates and confidence intervals, assessment of variable importance, and diagnostic checks based on simulated residuals. Importantly, the app provides clear visualisations of both the count-generating and zero-generating processes, making it easier to communicate components that are typically challenging to interpret. To improve accessibility, the tool offers guided explanations of outputs and diagnostics, making it valuable for both research applications and teaching. A case study using environmental count data illustrates how the application helps users identify zero inflation, choose appropriate models, and gain insight into the ecological processes underlying the data. Overall, the application narrows the gap between advanced statistical methods and practical analysis, supporting reproducible workflows and enhancing the clarity and interpretability of models for zero-inflated count data.

Climate change is a global threat to the long-term sustainable development of energy systems. Recent works have explored the emerging opportunity of coordinating different energy carriers and sectors (e.g. electricity, natural gas, heating, hydrogen, transportation, and water sectors) to unlock the cross-sector flexibility against climate change. This review has established a holistic framework for resilient multi-energy systems through the lens of nested coupling. It covers the most recent progress in resilience resources, resilience evaluation, resilience-oriented operation & planning, resilience pricing & investment, and real-world implementation. This work differs from prior studies through a full investigation on climate change impacts (distribution shifts), resilience pricing, and global projects. Within this area, we advocate for a unique and interdisciplinary perspective spanning across energy systems, climate science, sociology, economics, and data science. At the end, seven major challenges and opportunities are identified, including data deficiency, distributed coordination & privacy, high-fidelity simulation, and machine learning techniques. Researchers, industrial experts, and policy makers can follow this review to capture the emerging trend and future opportunities in this growing area.

This study addresses the PU classification problem under violations of the SCAR assumption. We investigate logistic regression-based approaches, namely the cluster method and its extensions with strict and non-strict Lasso regularization. The primary contribution of this work is the integration of the SMOTE technique to alleviate class imbalance and systematically assess its impact on the performance of the considered algorithms. SMOTE is first applied to rebalance the training dataset. Next, cleaning labels are derived via 2-means clustering. Logistic regression is then trained on the cleaned data, where identified positive instances are augmented with additional true positives and the remaining observations are treated as negative. The experimental evaluation is conducted on 13 real benchmark datasets and one synthetic dataset. For comparison, we include the naive approach and the Spy-EM method. The results demonstrate that incorporating SMOTE improves classification performance when the SCAR condition is violated and indicate moderate robustness of the LassoJoint method in this setting.

Let $P_d$ be the elliptope of $d \times d$ correlation matrices, and, for a standardised law $F$, let $S_d^F$ be the set of correlation matrices of random vectors with all margins $F$. We give two short proofs. First, an eleven-term representation of $|x|^4$ as a sum of fourth powers of linear forms on $\mathbb{R}^4$ yields $S_{11} \neq P_{11}$ for uniform margins. Second, the six diagonals of the regular icosahedron yield $S_6^F \neq P_6$ for arcsine margins. Together with a result of Devroye and Letac, this gives $S_d^F = P_d$ if and only if $d \le 5$, settling a conjecture of theirs.

Heavy-tailed response variables are routinely modeled as a function of covariates, but the asymptotic theory available for the resulting maximum likelihood estimators sidesteps certain technical difficulties by restricting the parameters to a compact set, which is not the natural parameter space. We consider a Fréchet model whose scale parameter depends log-linearly on a vector of covariates, while the tail parameter is constant, and we study the conditional maximum likelihood estimator of the tail parameter and the regression coefficients jointly. We show that this estimator is consistent over the full, noncompact parameter space, and that it is asymptotically normal, with the inverse Fisher information as limiting covariance matrix. Consistency is obtained by a specific compactification of the parameter space relying on the cosmic closure of Euclidean space. The extension of Wald's consistency theorem needed for that step, from single observations to blocks of observations, is stated in an abstract setting and is possibly of independent interest.

Training-free diffusion inverse solvers typically choose between local measurement guidance and costly clean-space posterior updates. Independent posterior refresh can improve global correction by sampling a clean conditional and re-noising it, but its practical realization requires probability-flow ODE integration and clean-space Markov chain Monte Carlo (MCMC). We propose Direct Conditional Transition Sampling (DCTS), a direct stochastic-flow approximation to the same ideal refresh target. Rather than explicitly drawing a clean sample, DCTS estimates the measurement-conditioned clean mean along a short inner path and transports Gaussian source noise directly to the next noisy state. A denoiser-compatible sufficient statistic and a covariance-scaled operator update enable this conditional-mean estimation. Experiments on four inverse problems demonstrate that DCTS achieves competitive reconstruction quality with up to $16.8\times$ speedups over competing methods.

Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.

We introduce an equivalence relation on exponential families, and prove that two exponential families are equivalent in this sense if and only if the corresponding induced Hessian manifolds are isomorphic. Moreover, we classify exponential families of order 1 with constant Hessian sectional curvature. To this end, we show that for an exponential family of order 1, it has constant Hessian sectional curvature if and only if the natural exponential family equivalent to the given family has a quadratic variance function (NEF-QVF). The classification coincides with that of NEF-QVF by Morris (1982) essentially.

Many modern prediction tasks involve data from multiple heterogeneous sources, while the test distribution may differ substantially from any individual source. Although heterogeneity poses challenges, it also offers an opportunity: different sources may provide complementary information, with some regions of the feature space better represented in one source than another. We propose Multi-Source Randomly Localized Conformal Prediction (MS-RLCP), which builds on the local coverage properties of randomly localized conformal prediction (RLCP) (Hore and Barber, 2025) and extends it to multiple sources through data-adaptive source selection. Under the widely adopted assumption of a shared response distribution conditional on the features across sources and the test population, we establish finite-sample coverage bounds using an interpretable notion of envelope distribution that captures their aggregate feature-space representation. Our analysis allows the test feature distribution to be absolutely continuous with respect to the envelope, extending beyond mixtures of source distributions. Under additional regularity conditions, we also establish asymptotic test-conditional coverage. Simulations and real-world experiments demonstrate the effectiveness of MS-RLCP across varying levels of data heterogeneity.

A covariance matrix rarely distorts in a single direction: a shift can expand all variances simultaneously, shift variance along one or two principal directions, displace spectral mass without changing marginal means, or rotate the dependence structure. Default mean-shift detectors are blind to these effects, and no existing online procedure calibrates a family of spectral deviation tests with provable false alarm control across both time and tracking window selection; this paper fills that gap. We consider four spectral deviations: $D_{KL}(P_{1}\|P_{0})$, $D_{KL}(P_{0}\|P_{1})$, Jeffreys, and Bhattacharyya, each evaluated on the eigenvalues of the empirical relative covariance operator $\widehat{\Sigma}_{0}^{-1/2}\widehat{\Sigma}_{t}\widehat{\Sigma}_{0}^{-1/2}$. Critical values come from a conditional parametric bootstrap that accounts for estimation uncertainty in both the past and tracking windows, an aspect asymptotic approaches typically overlook. The procedure controls the false alarm rate family-by-family over a predefined monitoring period and a range of candidate window sizes; when a single operational window is needed, a power-based criterion selects it. We prove consistency under constant alternatives via local spectral expansions with second-order sensitivity near the null. Simulations are conservative under the null and show detection power depends largely on spectral shape rather than magnitude: $D_{KL}(P_{1}\|P_{0})$ excels under global inflation, while Jeffreys and Bhattacharyya cover a broader range of alternatives. We illustrate the approach on three financial applications: European stock indices, Fama-French sector portfolios, and large-cap technology stocks.

Many interventions act on an entire spatial exposure field. We study when their induced laws satisfy policy positivity and what its failure implies for causal evaluation through weighting and structural modelling. We show that, for post-transformation fields that are Gaussian, the only admissible translation that satisfy positivity lie on the Cameron-Martin space of the observed exposure field. Under fine-scale variation and regularity assumptions, these are the only admissible non-decreasing pointwise policies on that scale. Common rules, including binding caps and proportional reductions on the Gaussian scale can therefore violate positivity. The Cameron-Martin norm controls both weight variability and the amplification of disagreement between candidate outcome models into disagreement between policy estimates. Without positivity, bounded continuous response models can make arbitrarily similar observational predictions while their policy estimates remain separated. We also give a policy-specific criterion for structural identification conditional on a realised exposure state. Finally, when a policy violating positivity admits weights in every finite representation, their second moments diverge under nested refinement that recovers the full field. These results motivate assessing identification and sensitivity for the specified policy effect across plausible structural models and under refinement of finite representations.

Retrieval-Guided Fine-Tuning (RAG-FT) incorporates retrieved data directly into the training objective, but the statistical consequences of noisy retrieval during training remain theoretically undercharacterized. We study this question by modeling RAG-FT as an estimation problem in a multi-task linear regression framework, using an OLS proxy for single-layer linear self-attention to obtain finite-sample risk bounds. Under homoscedastic retrieval noise, we show that retrieval failure decays exponentially with task separation relative to noise, and derive explicit finite-sample conditions under which RAG-FT achieves lower risk than both target-only and full-corpus training. We then introduce a Distance-Proportional Noise (DPN) model, in which retrieval quality degrades with rank, and compare two estimators under the same retrieval process: the OLS proxy and the literal, uniform-weight forward pass of linear self-attention. We prove that the attention estimator's bias diverges as $\Theta(n^{2q})$ even under exact retrieval, while OLS risk remains $\Theta(d/n)$ for every noise exponent $q>0$. These results locate the instability not in noisy retrieval itself, but in the fixed, unweighted aggregation of the literal LSA forward pass, which reweighting by reliability empirically removes. We validate the predicted rate separation through direct simulation of the DPN model.

In this study, we propose a novel multivariate Nakagami-$m$ distribution whose joint probability density function admits a fully closed-form expression. The proposed density is derived using a Mathai-type construction based on the partial sums of independent gamma random variables. For this distribution, the maximum likelihood estimator is not available in the closed form. To address this limitation, an asymptotically efficient closed-form estimator is developed by applying a one-step refinement to a $\sqrt{n}$-consistent initial estimator. Monte Carlo simulations demonstrate that the proposed estimator achieves a performance nearly identical to that of the numerically computed maximum likelihood estimator, while consistently outperforming the closed-form initial estimators. A real-data application further illustrates its practical utility.