Papers

Credibility theory combines individual experience with portfolio information for insurance pricing, but classical formulations focus primarily on conditional means and expected premiums. We propose a Dirichlet mixed-membership model (DMMM) for multivariate distributional credibility. Policyholder risk is represented by a stable composition over latent risk classes: baseline characteristics determine its a priori assessment, while repeated multivariate experience progressively updates its a posteriori assessment. Support-specific expert distributions accommodate heterogeneous outcomes, with dependence induced through the shared latent structure. The resulting posterior predictive distribution can be written exactly as a convex combination of portfolio and experience components, extending the familiar credibility-factor structure beyond the conditional mean. We study the framework through a simulation experiment and a vehicle telematics application. The simulation shows that relatively simple experts can capture complex multivariate insurance distributions and learn policyholder-specific risk as experience accumulates. In the telematics application, recent claims and driving behaviour provide complementary information for future claim-frequency prediction, allowing similar policyholders to receive different experience-rated assessments. The DMMM therefore provides a flexible and interpretable framework for combining heterogeneous insurance experience while retaining the structure of classical credibility.

Statistical inference in sequential mediation models requires evaluating the cumulative distribution function (CDF) of the product of three normal coefficient estimators, an operation that entails integrating over a non-convex region with boundary $xyz=v$. While the Delta method provides an instantaneous first-order approximation, its asymptotic variance collapses whenever two or more path coefficients approach zero, producing severe undercoverage near parameter boundaries. Nonparametric bootstrapping avoids gradient collapse but incurs an $O(B \times N)$ computational cost that becomes burdensome in large-scale simulation studies or iterative power analyses. We propose a model-based dimension-reduction algorithm that integrates out the third variable analytically under the trivariate Gaussian distribution for arbitrary mean vectors and positive-definite covariance matrices. Partitioning the $xy$-plane into quadrants isolates the sign change at the coordinate axes, reducing the problem to an adaptive two-dimensional quadrature whose fixed-grid cost is $O(n^2)$ in place of $O(n^3)$. In simulation benchmarks across six parameter regimes, the algorithm achieves a mean absolute error of $3.1 \times 10^{-5}$ relative to a $10^{8}$-sample Monte Carlo reference, evaluating the distribution in under one second and yielding plug-in quantile confidence intervals whose empirical coverage was near or above nominal.

We develop a transfer learning framework for high-dimensional sparse canonical correlation analysis (CCA) with multiple heterogeneous source datasets. Our goal is to improve estimation of the target canonical coefficient matrices by borrowing information from related sources. We characterize source-target similarity through discrepancies in the joint covariance structure and establish minimax optimal rates under both joint and separate prediction losses, explicitly quantifying the gain from enlarged effective sample size and the cost of heterogeneity. To attain these rates, we first construct an oracle transfer estimator based on debiasing and truncated normalization. We then develop a computationally efficient procedure that adaptively achieves the minimax rate without prior knowledge of the sparsity levels, using aggregation to control transfer bias and a truncated normalization scheme to preserve transfer gains under the scale-sensitive prediction loss. A data-driven source detection procedure is further proposed to identify potentially informative auxiliary datasets. Our analysis also yields a generalized sin-theta perturbation theorem that removes a commonly imposed comparability condition on leading canonical correlations in the sparse CCA problem.

In-context retrieval (ICR) is a retrieval-based form of in-context learning (ICL) in which demonstrations are retrieved from a source database based on similarity to the query, rather than sampled independently. In this work, we formulate ICR as a type of domain adaptation problem, where the source distribution $P$ of the database may differ from the target distribution $Q$ of the test query-label pair. We investigate the performance of ICR under a flexible class of distributional shifts that substantially extends prior work \citep{li2024fine,guo2025retrieval}, and establish theoretical guarantees that quantify the benefits and pitfalls of this learning paradigm. Our theory is verified by experiments on synthetic and language tasks.

Strong image generation models are conditioned on class labels, aligned to frozen pretrained encoders, or built on separately trained autoencoders. While effective, generation then depends on supervision or pretraining: labels must be annotated, and encoders or autoencoders pretrained for the target domain. We study joint generative and self-supervised representation learning in a single model, enabling self-conditioned generation without labels or pretrained models. This is challenging because the objectives are mismatched: contrastive learning consumes clean augmented views and favors coarse, invariant semantics, while flow matching consumes noisy images and must preserve the fine detail and spatial layout that contrastive learning discards. We propose SCION (Self-conditioned Generation on Self-supervised representation), whose core is a single pixel-space encoder conditioned on the flow timestep and an embedding. For representation learning, this conditioning embedding is a learned global vector shared across images, with the encoder's [CLS] token yielding the semantic representation trained by the contrastive loss. For generative training, the conditioning embedding is the image's own [CLS] representation, while patch tokens pass through a decoder to predict the image. To sample without a reference image at inference, we jointly learn a prior over the embedding. Gradient-norm balancing and stop-gradient mechanisms enable joint optimization in one run. SCION is self-supervised and self-contained, with no labels or pretrained models. On ImageNet 256x256, with the JiT-B recipe and no representation guidance, SCION reaches 8.92 FID, surpassing class-unconditional iREPA, which aligns to pretrained DINOv2 (46.44), and RCG, which conditions on it (14.27). With JiT-L, SCION achieves 5.89 FID without guidance and 3.47 with representation guidance, outperforming RCG with the ADM recipe (6.24).

We study difference-in-differences (DID) estimation of average treatment effects on the treated when the researcher is uncertain about which pre-treatment periods satisfy the parallel trends assumption. Roth (2022) shows that pre-trend testing can induce bias, complementing the statistical literature on post-selection inference. We propose the model averaged difference-in-differences (MADID) estimator, a weighted average of the candidate $2\times2$ DID estimators. The weights are normalized exponential functions of the candidate residual sums of squares, so implementation requires no specification test. We derive MADID's asymptotic properties under conditions that make invalid comparisons detectably more variable than at least one valid comparison. Under these conditions, MADID is consistent when the parallel trends assumption holds for at least one pre-treatment period. Although its joint limiting distribution across post-treatment periods is generally non-Gaussian, inference can be conducted by subsampling or simulation.

Large language models (LLMs) are increasingly used for literature search and synthesis. However, it is unclear whether they retrieve accurate bibliographic information in environmental science. Therefore, we quantitatively compared the errors of widely used LLM platforms in retrieving references related to original articles from five leading environmental science journals (Energy and Environmental Science, Nature Sustainability, Nature Climate Change, Lancet Planetary Health, and Environmental Science and Technology) published in 2024 to 2025. Claude, ChatGPT, Grok, DeepSeek, Perplexity, and Gemini were used as the LLM platforms. LLMs retrieved 10 references for each of the 50 randomly selected original article using either the article's abstract or its full-text as prompt. The retrieved references were subject to a multimetric score ratio combining validity of bibliographic data, Google Scholar link, digital object identifier, Scopus Electronic Identifier and relevance score (cited by or being the index paper), and the proportion of complete fabrication that failed all metrics. Abstract-only prompt yielded significantly higher accuracy than full-text one. This advantage was confirmed in multilevel mixed-effect multivariable regression after adjusting for journal, platform, and output order. Source journal and the position of a reference within the output list were also independently associated with retrieval accuracy, with lower-listed references associated with lower accuracy. These findings suggest that LLM assisted literature retrieval in environmental science remains moderately accurate and overall inconsistent, varying significantly by platform, journal, prompt type, and output position. Abstract-based prompting, as task-aligned information compression, may outperform full-text one in literature retrieval. Caution should be used when generalizing our findings.

Modern deep generative models are primarily studied for their ability to generate realistic samples, yet the generative dynamics they learn can also serve as objects of statistical inference. We develop this idea for two-sample testing, the problem of deciding whether the same distribution generated two finite datasets. Using stochastic interpolants, we connect both distributions to a shared Gaussian bottleneck, so that each half of the resulting path is a Gaussian channel acting on a single population. We prove that the null hypothesis holds if and only if the population denoiser, or equivalently, the velocity fields of the two halves, coincide at any single noise level, which amounts to a reflection symmetry of the path about the bottleneck. Deviations from this symmetry yield a continuum of two-sample witnesses, which we estimate via held-out regression risks on learned denoisers and velocities and aggregate along the path; under an information-theoretic weighting, the aggregated discrepancy equals the Jeffreys divergence between the noise-smoothed distributions. Calibrating the resulting statistics by permutation yields tests that are valid in finite samples for any trained networks and consistent when the fields are learned accurately. On a synthetic benchmark and three image benchmarks, the proposed tests improve power over the strongest baseline by up to 33 percentage points at an equal total sample budget, with the best choice of regression representation and path weighting depending on the data modality. These results show that generative paths provide a principled representation for statistical testing, extending stochastic-interpolant models beyond generation.

Instrumental variable (IV) analysis in underrepresented target populations suffers from low statistical efficiency and weak IV bias due to limited sample size. Leveraging external knowledge from a source population with larger samples offers an appealing solution to this problem. We propose in this paper the KNowledge-powered IV Estimator for underrepresented Subpopulations (KNIVES), a framework that enhances the IV analysis for underrepresented subpopulations by transferring knowledge from external source population data. KNIVES constructs a knowledge-transferred IV score by combining multiple candidate models through an effective signal-to-noise ratio criterion tailored to the downstream IV estimator, rather than prediction accuracy alone. To address weak associations between the IV score and the exposure, as well as a high correlation among candidate models in the target population, KNIVES employs a bias correction technique to mitigate the resulting bias and ensure robust performance. Theoretical investigation demonstrates that our method is robust to weak IV bias and achieves improved efficiency compared to the causal effect estimator based on any single candidate model. In addition, KNIVES avoids negative transfer by guaranteeing that its asymptotic variance is not larger than that of the estimator relying solely on target population data. Extensive numerical studies demonstrate the superior finite-sample performance of our method over existing methods. Two Mendelian randomization studies on ethnic minority subgroups in UK Biobank data further illustrate the advantages of our method.

Modern forecasting models are often updated after deployment so they can respond to changing data. These updates can also make predictions worse, so practical systems need a reliable monitor that can detect harmful changes and trigger protection. A natural design is to monitor the same prediction errors that guide the updates. This paper asks whether the statistical guarantee behind such a monitor remains valid when monitoring and adaptation use the same feedback. We study this question in multi-step time-series forecasting. We show that overlapping targets and dependence in forecast errors can break a key assumption required by the guarantee. The monitor may then raise alarms even when no harmful change has occurred, and its response can further damage prediction quality. We also find that adaptation can hide sustained changes from its own monitor, while the original frozen model retains a clearer signal. These results expose a basic failure mode in self-monitored adaptation. They show why reliable deployment requires checking the monitor's assumptions, comparing adaptation with the frozen model under realistic feedback, and limiting the effect of every protective response.

Sensing technologies have advanced rapidly across industries ranging from energy to automotive manufacturing. These systems generate high-dimensional (HD) data characterized by complex nonlinear patterns and strong temporal dependencies. Traditional statistical monitoring methods are often limited in their ability to capture such nonlinear structure. Likewise, many analytical approaches used in End-of-Line testing rely on predefined thresholds and heuristic rules, which restrict their ability to detect informative anomaly signatures in HD temporal data. In contrast, while modern deep learning and generative AI models offer strong predictive capabilities, they are often unsuitable in applications where data are costly to collect and where the monitoring system must remain interpretable, low-latency, computationally efficient, and usable by non-technical practitioners. To overcome these limitations, we propose an advanced multivariate monitoring framework for HD data. The framework operates in two stages. In the first stage, the data are preprocessed to remove incomplete and non-informative samples and to temporally align time series data. In the second stage, nonlinear dimensionality reduction is performed, followed by anomaly detection through a control chart based phase I monitoring procedure. The framework can be used in both unsupervised and supervised settings, depending on the availability of ground truth labels during training. Moreover, its flexible and modular structure allows practitioners to adapt its components to different domains and operational requirements. We evaluate the proposed framework on real production data from an automotive manufacturing environment at Ford Motor Company. The proposed method achieves higher accuracy, recall, and F1 score than the company's existing model, improving these metrics from 0.50, 0.30, and 0.429 to 0.625, 1.00, and 0.769, respectively.

Forecasts of the productivity dividend from artificial intelligence (AI) differ by an order of magnitude, partly because they extrapolate average gains observed among early, highly exposed adopters. Whether those gains scale linearly, flatten, or are competed away is an empirical question about the shape of the return to AI, not its average. This paper develops and validates a design-based framework for measuring that shape with linked firm-worker data. The framework combines a pre-determined, occupation-based measure of firm exposure to generative AI with a continuous-treatment difference-in-differences design around the release of ChatGPT in November 2022, three pre-specified tests of concavity, a decomposition of exposure effects into an adoption margin and a per-adopter return, a doubly robust analysis of use intensity, a staggered-adoption analysis of earlier AI adopters, and a production-function estimator that places AI in the law of motion of productivity. We validate every component on a calibrated synthetic panel of 20,000 enterprises that reproduces the structure of Statistics Canada's linked business microdata and embeds a known concave effect. The estimators recover the true dose-response curve and detect its concavity (slope difference -0.010, standard error 0.003, one-sided p = 0.002); Monte Carlo coverage of the main confidence interval is 0.97, and the concavity test has power of 0.93. The validation also exposes a practical pitfall: pre-trend tests clustered at roughly 40 industries over-reject a true null. The framework, released as open code, is ready for application to Canadian enterprise microdata, where it will inform estimates of potential output and the persistence of Canada's productivity gap.

We study the finite-sample properties of estimators of a broad class of normalized scale-invariant inequality indices (NSIIs) under finite Gamma mixture models. The class includes the Gini coefficient, the $m$th Gini index, the Extended $m$th Gini index, linear order-statistic inequality indices, and several entropy- and dispersion-based measures. Building on the Gamma–Dirichlet representation, we derive exact expressions for the expectation and finite-sample bias of the natural estimator of an NSII based on a $U$-statistic when the population follows a finite Gamma mixture with a common rate parameter. The resulting bias formula explicitly characterizes the effects of the mixture weights and component shape parameters. We also establish that the bias converges to zero as the sample size increases, yielding asymptotic unbiasedness. Furthermore, we derive exact finite-sample expressions for the variance and mean squared error of the estimator by accounting for the overlap structure among the subsets entering the $U$-statistic. In addition, we establish strong consistency and asymptotic normality. When the mixture degenerates to a single Gamma distribution, the finite-sample bias vanishes, recovering the exact unbiasedness result previously obtained for NSIIs under the Gamma model. Numerical illustrations and simulation studies are provided to investigate the finite-sample behavior of the estimator and to illustrate the theoretical findings. Overall, the proposed framework provides a unified characterization of finite-sample estimation under heterogeneous Gamma populations and quantifies the effects of mixture heterogeneity on bias, variance, and mean squared error.

Finding which variables cause which others in multivariate time series, and at what lags, is central to science and policy, yet existing methods force a choice between flexible confounder adjustment, data-driven lag selection, and inference that controls the false discovery rate (FDR). ORACLE-VARX does all three in one pipeline. First, double/debiased machine learning (DML) removes nonlinear confounder effects from the outcomes and the lagged series. Second, adaptive causal lag estimation (ACLE) picks the lag order at each time step by sequential significance tests, tracking regime changes. Third, entry-wise $z$-tests with Benjamini–Hochberg correction select directed edges at a target FDR. We prove that in each rolling window, the debiased coefficients are asymptotically normal around a window-averaged target, so their $z$-tests are asymptotically valid. On a synthetic benchmark with time-varying structure and nonlinear confounding, ORACLE-VARX (LightGBM) tracks the true lag order best (RMSE $0.96$ vs $1.1$–$1.5$), has edge FDR $0.047$, close to PCMCI ($0.045$) and below VAR ($0.129$) and VAR-LiNGAM ($0.187$), and forecasts better than all three. On nine U.S. sector ETFs with macroeconomic confounders, it yields interpretable causal graphs whose lag order rises in high-volatility regimes.

Wildfire is reshaping forests in the western United States (US), yet estimates of wildfire impacts often rely on burned-area summaries or satellite-derived severity metrics that do not directly quantify changes in forest biomass or their uncertainty. We combine US Forest Inventory and Analysis plot measurements with annual 30 m tree canopy cover data in a Bayesian spatio-temporal modeling framework to estimate within-fire changes in live aboveground biomass (AGB) and forest mortality area across National Forest System lands in Washington, Oregon, and California. The model represents both live forest presence or absence and AGB conditional on live forest, allowing posterior predictive estimates of pre- and post-fire conditions at the pixel scale. Applied to 6,239 fires from 2000 to 2022, the analysis estimates 95.3 million Mg of wildfire-associated AGB loss and 2.39 million ha of forest mortality area, with the largest aggregate impacts occurring in 2020. Pixel-level estimates reveal substantial spatial heterogeneity within fire perimeters, including large differences between total burned area, forest mortality area, and associated biomass loss. This inventory-informed, uncertainty-aware approach provides a scalable framework for quantifying forest carbon impacts of disturbance across broad regions while retaining within-fire spatial detail.

Quantifying and evaluating the capabilities of Large Language Models (LLMs) remains a fundamental challenge in modern data science and artificial intelligence. In this paper, we consider LLM evaluation based on their performance across items in multiple benchmark domains (e.g., mathematical reasoning and coding) within a leaderboard framework. Our goal is to address two core questions: (1) How do we derive more accurate domain-specific scores by borrowing information across domains? and (2) How do we define and estimate an overall score that aggregates performance across multiple domains? To solve these problems, we propose a novel statistical framework based on representation Multi-task Learning (MTL) and an item response theory model. Specifically, we define overall and domain-specific LLM traits through an Item Response Theory (IRT) model, and propose an MTL approach to estimate these traits from item-level response data. We develop a computationally efficient estimator and establish its minimax optimality under certain asymptotic regimes. This framework provides a rigorous measurement foundation for systematic LLM evaluation. We conduct extensive simulations, demonstrating the superior performance of the proposed method over competing methods. Crucially for the Applications and Case Studies section, we apply the proposed framework to MMLU response data from the Hugging Face Open LLM Leaderboard, covering 4,272 LLMs and 13,232 items across 56 subjects. The empirical analysis reveals substantial heterogeneity in domain size and difficulty, together with strong positive cross-domain dependence, highlighting the practical value and substantive insights generated by our approach.

Temporal dependence can separate the Gaussian approximation of stochastic gradient descent from its stationary moments. For unmodified least-squares SGD, we construct a design with standard Gaussian marginals whose stationary error has every positive moment infinite. Independent observations with the same marginals instead give finite stationary variance. Both regimes retain a Gaussian small-step limit. Our general theory establishes pathwise contraction from a finite second design moment, then uses score cancellation and localization to obtain stationary Gaussian and Ornstein–Uhlenbeck limits. Independent Gaussian regression errors yield an exact conditional Gaussian law and total-variation convergence under the same design integrability. Stronger design conditions identify a positive first-order total-variation constant and a deterministic covariance correction with $o(a)$ error. A scalar coverage expansion translates this correction into its inference consequence. Experiments examine distributional error, coverage, and calibration with dependent scores. Together, these results establish precise probability-law approximation beyond moment-based stationary analysis.

Local Gaussian models of constant-step learning predict output variability and expected losses, but weak convergence alone does not justify these moment predictions. We establish moment-accurate Gaussian mixtures by matching stationary energy with local Ornstein–Uhlenbeck limits, ruling out quadratic tail mass invisible to weak convergence. For step size $a$, the second-order Wasserstein error is $o(\sqrt a)$, uniformly over invariant laws, using each law's actual root weights. The assumptions combine confinement, descent, finitely many hyperbolic equilibria and root continuity with finite-variance innovations. The result yields observable covariances, expected objective gaps and first-order mean shifts, while allowing singular covariances, compatible saddles and weights without a limit. For additive noise given by a fixed invertible transform of independent standardized Student $t_3$ coordinates, symmetry gives an order-sharp $\sqrt a$ smooth-test bound. Numerical transport calculations demonstrate the value of root-specific covariances; controlled SGD studies assess observable predictions across step sizes, batch sizes and model geometries.

Small language model (SLM) agents need safety controls that track consequences across tool calls with little monitoring overhead. A private read, for example, becomes a leak when a later action sends that data outside the system. We introduce CARB (Conformal Agent Risk Budget), which calibrates when to stop an agent using a ledger of harm incurred before stopping. Under exchangeable episodes, standard conformal risk control bounds this declared loss in expectation over calibration and a future episode. G-CARB selects scorer evidence along observable dependencies from private sources to outgoing actions. The ledger still covers the entire executed history, and computing the gate score requires no additional language-model inference. On AgentDojo replay with two 14B backbones, G-CARB roughly halves scorer-input records at intermediate risk budgets while improving autonomous task completion relative to full-prefix scoring; random context of the same size achieves similar gains. Controlled examples show how retaining the relevant dependency can further avoid stopping benign work.

Synthetic control relies on pre-treatment fit to balance the unobserved factors that drive untreated outcomes. The argument fails when an observed common shock, such as a commodity price, moves with the latent factors before treatment and departs from them afterwards and pre-treatment fit then says almost nothing about the synthetic unit's exposure to the shock. I derive a period-specific bound that links the post-treatment bias of any weighting estimator to pre-treatment misfit through a leverage statistic, and show that the statistic decomposes exactly into a latent-factor component and the partial leverage of the observed shock. The decomposition yields a diagnostic for such silent factor that uses no post-treatment outcome of the treated unit. Balancing an observed exposure index removes the problem, but simplex weights can achieve balance only when the treated unit lies inside the donors' exposure hull. Outside the hull, every simplex-weighted estimator carries an exposure imbalance with an explicit lower bound. For that case I propose an exposure-augmented synthetic control that corrects the remaining imbalance with a cross-donor regression, characterise what the correction identifies and when it fails, and derive bias-aware confidence intervals with a breakdown value. Monte Carlo experiments, including designs in which the correction's assumptions fail, map its gains and its limits. Iran's 2012 sanctions coincided with the 2014 oil collapse, and Iran is more oil-dependent than any of its natural comparators. The 2012-15 output loss of about 12 percent survives every adjustment. The persistent loss that standard synthetic control reports after 2016 does not. Any adjustment for oil exposure removes it, and the long-run effect is not identified with these comparators.

In precision medicine, treatment decisions often depend more on identifying which patients are most likely to benefit from treatment than on accurately estimating benefit for every patient. However, existing methods estimate patient-specific treatment effects and derive patient rankings as a secondary step. We propose a direct ranking approach for conditional RMST treatment benefit in randomized trials with right-censored outcomes. We first construct a censoring-adjusted orthogonal RMST pseudo-outcome whose conditional expectation equals the true conditional RMST difference. Then, we compare patients pairwise and optimize a smooth ranking loss that directly targets treatment-benefit ordering. The pairwise criterion incorporates orthogonal corrections for estimation of the event and censoring distributions. We show that the population minimizer of the proposed loss induces the same ordering as the true conditional RMST treatment benefit and that the resulting estimating equation is Neyman-orthogonal to nuisance survival and censoring models. In simulation studies with nonlinear treatment-effect heterogeneity and moderate to heavy censoring, the proposed approach improved rank correlation and treatment-benefit enrichment relative to both plug-in RMST estimation and regression of the same RMST pseudo-outcome. Application to a randomized breast cancer trial demonstrated improved identification of patients with greater treatment benefit. In conclusion, the proposed framework combines orthogonal survival learning with pairwise ranking and may be particularly useful in precision medicine settings where treatment prioritization and patient selection are the primary goals.

In the context of functional data analysis, we propose a Bayesian test to assess whether the mean function of a Gaussian population is identically zero. By leveraging the structure of the reproducing kernel Hilbert space (RKHS) associated with the covariance function of the underlying process, we approximate the mean by a finite linear combination of kernel sections. This finite-dimensional representation arises from the natural assumption that the mean lies in the corresponding RKHS, enabling a parametric Bayesian approach to hypothesis testing through a spike-and-slab prior on the coefficients. Given the absence of a canonical reference measure on infinite-dimensional spaces, the likelihood is expressed via a Radon-Nikodym derivative between induced Gaussian measures. Since the number of kernel sections is unknown, we employ reversible jump Markov chain Monte Carlo (RJMCMC) to explore the trans-dimensional parameter space and estimate the posterior distribution. From the resulting posterior samples we construct estimators of the posterior probability of the null hypothesis and the Bayes factor, providing quantifiable evidence against the null and supporting informed decisions under uncertainty. Furthermore, to validate this methodology we establish near-optimal posterior contraction rates for the proposed test under mild conditions, demonstrate its empirical performance on simulated data sets, and illustrate its application with real climate data.

The polynomial maximization method estimates signal parameters in non-Gaussian noise from a finite description of the noise by its moments and cumulants, without a density. Known accuracy results assume that these noise characteristics are known, or use a design in which estimating them jointly costs no accuracy. In practice they are estimated from the residuals of a preliminary fit, and the effect of this substitution on accuracy is unknown. We treat the feasible scheme as a two-step estimator and, for a general smooth signal model in a fixed design, find the increment of its asymptotic covariance in closed form: $\Sigma_{\rm fe}-\Sigma_{\rm or}=c_2(1-g_{\rm or}) Q^{-1}\bar{d}\bar{d}^{\top}Q^{-1}$, a positive semidefinite matrix of rank at most one. Here $c_2$ is the noise variance, $g_{\rm or}\le 1$ the oracle variance reduction factor, and $Q$ and $\bar{d}$ the second-moment matrix and mean of the signal gradient over the design. Estimating the fractional moments that form the weights costs no first-order accuracy, because that block is orthogonal; the whole increment comes from estimating the sign-preserving centering means. The increment lies along the single direction $Q^{-1}\bar{d}$, so linear combinations of the parameters orthogonal to it keep oracle accuracy. In the trace metric the variance reduction factor becomes $g_{\rm fe}=g_{\rm or}+(1-g_{\rm or})R_D$, where $R_D=\|Q^{-1}\bar{d}\|^2/{\rm tr} Q^{-1}$ depends on the design alone. We give four sufficient conditions for first-order oracle equivalence; noise symmetry is not among them. Monte Carlo experiments with a harmonic signal confirm the direction and size of the increment. A covariance estimate that ignores the increment undercovers where non-Gaussianity gives the largest gain, symmetric noise included.

Clinical trials with survival endpoints lose information when participants are censored before death is observed. We develop target-preserving estimators that use posttreatment disease history, such as recurrence or progression, to recover information lost to censoring for marginal survival and restricted mean survival effects. The difficulty is that the intermediate event is downstream of treatment: naive adjustment can change the causal estimand, and the useful information enters only through the observed coarsening. We derive observed-data influence functions with and without recurrence history and obtain an exact gain identity. The identity shows that efficiency improvement is driven by the censoring hazard, the split of the alive risk set into recurrence states, and the residual-survival separation between those states. In the no-covariate illness-death model, the Aalen-Johansen estimator realizes the recurrence-augmented efficient score after standardization to the marginal target. With covariates, correctly specified Cox-Breslow transition hazards provide a root-N plug-in benchmark, while a hazard-induced one-step estimator gives rate robustness and, under primitive transition and censoring learner rates, canonical inference with a second-order product remainder. A semi-synthetic metastatic breast cancer study calibrated from digitized progression-free survival and overall survival curves illustrates the gain identity. The framework applies broadly to censored time-to-event studies with informative intermediate histories.

Natural exponential families (NEFs) are characterized by their variance functions (VFs), namely by the pair \((V,M)\), where \(V\) expresses the variance as a function of the mean and \(M\) is the mean domain. This characterization naturally raises the inverse question: when is a given function \(V\) the VF of an NEF, and what structural properties of the family can be read directly from \(V\)? Bar-Lev (1987), as part of a more general result for absolutely monotone functions, showed that every nonzero polynomial \[ V(m)=\sum_{j=1}^{r} a_j m^j,\qquad a_j\ge 0, \] is the VF of an infinitely divisible NEF on a positive mean domain. This polynomial class splits exhaustively into the cases \(a_1>0\) and \(a_1=0\). Bar-Lev, Letac and Ridder (2024) proved that, after scaling so that \(a_1=1\), the first case yields a counting NEF supported on \(\mathbb N_0\). We establish the complementary result: when \(a_1=0\), the corresponding NEF is absolutely continuous with respect to Lebesgue measure on the positive half-line. More generally, we prove absolute continuity whenever \[ V(m)=m^2G(m), \] where \(G\) is a nonzero power series with nonnegative coefficients. When \(G\) is a polynomial, the mean domain is the entire positive half-line. We also record several consequences for the cumulant structure and for statistical inference.

Dose optimisation of two-agent combinations in early-phase oncology trials must jointly balance toxicity, efficacy, and pharmacologically meaningful drug exposure, yet existing model-assisted combination designs use only binary toxicity–efficacy outcomes and leave pharmacokinetic (PK) data unused for dosing decisions. We propose PKComb-BOIN12, a utility-based Bayesian optimal interval design embedding a continuous PK exposure metric into two-agent dose optimisation via a PK-guided escalation rule and a PK-sufficiency constraint on the final admissible set. Two contributions distinguish this design from a direct combination extension of existing single-agent PK-informed designs. First, PK-based elimination uses a coordinatewise order-ideal (down-set) removal rule, so that eliminating a combination for insufficient exposure is licensed exactly by the assumed monotonicity of exposure in each agent's dose, rather than an unjustified scalar summary of combined dose intensity. Second, the pre-specified target exposure is treated as transportable-but-verifiable: a hierarchical calibration mechanism updates the assumed monotherapy-derived target using accruing combination-arm PK data, with a formal bound on the resulting selection bias under transportability violations. A time-to-event extension, TITE-PKComb-BOIN12, accommodates late-onset outcomes. A factorial simulation study spanning realistic scenarios, including one calibrated to a published combination trial, decomposes the source of PKComb-BOIN12's gains over its non-PK predecessor and evaluates robustness to non-monotone exposure, PK-variance misspecification, and target-transportability violations. An R Shiny application supports design evaluation and real-time conduct.

Spatially balanced sampling designs, such as the local pivotal methods (LPM), improve representation of auxiliary-variable space, but design-based variance estimation is challenging because second-order inclusion probabilities are generally intractable and may be zero or near zero. We propose two approaches for estimating the variance of the Horvitz-Thompson mean and the finite-population variance. The first estimates joint inclusion probabilities by Monte Carlo simulation and uses pairwise independence tests to modify the resulting variance estimators. The second, a model-assisted bootstrap (MABS), uses an interpolating Gaussian process to construct a synthetic finite population and estimates variance by resampling under the original LPM design. In simulations, the modified empirical approach performs well when responses are strongly associated with auxiliary variables, but its performance for the Horvitz-Thompson mean deteriorates as this association weakens. MABS exhibits more stable performance across noise levels and sampling settings, including under LPM2 and non-Gaussian noise. For finite-population variance estimation, however, no method shows a uniform advantage. These results illustrate the complementary strengths of the proposed approaches for inference under spatially balanced sampling.

We study the non-parametric estimation of the roughness of a path from discretely sampled observations. Our approach is based on the concept of normalized power variation of a path. The estimator identifies a variation index by comparing coarse increments with local sums of fine increments. We prove pathwise consistency under Hölder regularity, nondegenerate coarse critical power sums, uniform local bounds on fine critical power sums, and separation of the two sampling scales and obtain a convergence rate for the estimator. Neither convergence of critical variation nor monotonicity of the finite-sample statistic is required. The result applies to fractional Brownian motion under an explicit dyadic sampling rule and to Schauder series with bounded coefficients bounded away from zero. For signed Takagi–Landsberg functions we obtain a uniform asymptotic formula and a quantitative bound for every root of the estimating equation in a fixed parameter interval. We illustrate the results with numerical experiments for Takagi– Landsberg functions and fractional Brownian motion.

In Networked Markov Decision Processes, transition dynamics are often unknown and the state–action space grows rapidly with the number of agents. In this setting, Taylor representations naturally approximate $Q$-functions, but a naive order-$n$ expansion over $N$ agents requires $\Theta(N^n)$ coefficients. We justify these expansions under smooth expected future local rewards with controlled derivatives. Under this condition, finite-speed information propagation and discounting imply that local-critic Taylor coefficients decay exponentially with the graph distance to the farthest agent involved. Discarding distant-agent coefficients and marginalizing then yield scalable local Taylor representations with a bound controlled by graph locality. Building on these representations, we propose a scalable model-free actor–critic algorithm, establishing finite-sample critic and near-stationarity guarantees for a linear LSTD critic. We then introduce a more expressive neural TD parameterization. Unlike prior constructive spectral methods, our approach covers settings without access to a known local dynamics map, such as hidden switched linear–quadratic regulation. Across three control benchmarks, our method matches or outperforms spectral baselines while scaling efficiently to large graphs.

Fréchet regression provides a framework for predicting random object-valued responses by minimizing the conditional expected squared distances. We propose semi-supervised kernel ridge Fréchet regression (SS-KRFR) to exploit unlabeled covariates when labeled samples are scarce. The method learns kernel spectral features from both labeled and unlabeled covariates, then utilizes labeled responses to estimate the conditional Fréchet function by kernel ridge regression. We derive prediction error bounds that distinguish the roles of labeled and unlabeled samples, and establish polynomial convergence rates under suitable conditions. Under additional kernel assumptions, a sharper analysis relaxes the sample size requirements for learning the features. We also develop hybrids that combine features, predictions, or both from models fitted with and without unlabeled covariates, establishing uniform stability over mixing weights in Hadamard spaces at common tuning and a separate rate guarantee for prediction mixing. Simulations and a real-data application show semi-supervised gains and improvements over competing Fréchet regression methods in several settings.