Papers

Constant-step stochastic gradient descent with momentum (SGDM) augments the parameter recursion with a momentum state that carries geometrically decaying memory of past stochastic gradients. We study this coupled recursion as a nonlinear time series and develop a framework for its stationary behavior and distributional limits. The joint parameter-momentum process forgets its initialization geometrically under a finite $p$th moment for $p > 1$, yielding a unique stationary causal solution. Under a second-moment condition, this contraction leads to an invariance principle and quenched Gaussian limit for averaged SGDM, while a small-learning-rate analysis gives a Gaussian limit for the last iterate. In a stochastic-curvature quadratic model, we further study how the rate of initialization forgetting relates to stationary variance and averaged fluctuations.

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.

For discrete outcomes, a valid conditional quantile can have a hit probability different from its nominal level. Errors in different covariate groups can also cancel when residual signs are pooled. We study sequential audits that address both problems by testing the two inequalities that define quantile-set membership. The audit records whether each response falls below, equals, or exceeds its forecast. We derive the information projection for this three-outcome experiment and use continuous betting mixtures within the cells of a covariate tree. Context-tree weighting combines the resulting audits over all prunings, with an explicit finite-path oracle bound. Under iid alternatives with uniformly positive sign probabilities, a countable mixture of refining trees attains the full conditional sign-information rate. A matching upper bound shows that no valid evidence sequence based on the same observations can attain a larger almost-sure exponential rate. We also bound the information lost through covariate coarsening, derive a recurrence for the local mixtures, and discuss quantile selection and confidence-sequence inversion. Simulations show how partition resolution affects detection, and a held-out count-data example illustrates the difference between quantile validity and exact-hit calibration.

We study the LR2 one-parameter lifetime distribution obtained by mixing an exponential component of rate $\theta/2$ and a gamma component with shape four and rate $\theta^2$. The exponential mixing probability is $\theta^2/(1+\theta^2)$ for $\theta>0$. We derive the density, survival function, moments, transforms, truncated moments, quantiles, order-statistic distributions, mean residual life, and stress–strength reliability. Identifiability follows from the density at the origin, and strict monotonicity of the mean yields a unique moment estimator. A complete density-turning-point criterion and analytic initial and tail hazard expansions clarify the possible shapes. The component rates cross at one half. The hazard decreases initially for every positive parameter and eventually increases in the gamma-dominated regime; it eventually decreases in the exponential-dominated regime. We develop likelihood estimation, observed information, constrained expectation–maximization, spacing estimation, and inference under regularity conditions, and state the likelihood for independent right censoring. A reproducible Monte Carlo study with eight settings and 4,000 generated samples compares likelihood and moment estimation and evaluates log-scale confidence intervals. The analysis distinguishes mathematical properties from empirical validation and explains the restrictions associated with a fixed working time unit.

Thompson sampling (TS) achieves logarithmic and asymptotically optimal regret in classical stochastic bandits, but its behavior in multi-agent learning remains less understood. We study TS in an unknown finite two-player zero-sum matrix game with Gaussian bandit feedback. Under centralized self-play, TS samples a single payoff matrix from the posterior and plays a Nash equilibrium of the sampled game. For every fixed zero-sum game with a unique Nash equilibrium, we show that the expected cumulative duality gap is $O(\sqrt T)$. The rate improves to $O(\log T)$ for pure equilibria. We then identify the exact coefficient of the $\sqrt T$ term, which depends only on the equilibrium strategies and the noise level. We also establish a minimax lower bound matching the square-root dependence on $T$ and an instance-dependent lower bound matching the logarithmic dependence on $T$ for a class of games with a unique pure equilibrium. Experiments support the predicted rates and coefficient.

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.

Heap-leaching hydraulics vary between regions of an ore bed, motivating fast prediction models that accommodate different material properties. This study develops a parameter-conditioned physics-guided machine-learning (PGML) surrogate and demonstrates its use in simulated saturation control. A dataset of 4096 numerical Richards-equation trajectories was divided into 2867 training, 614 validation and 615 evaluation cases. The network predicts hourly saturation profiles from the current profile, irrigation rate and four hydraulic parameters, and is trained with an approximate Gardner mass-balance penalty. A daily predictive controller compared 200 constant irrigation rates over a 24-hour horizon. Across the evaluation cases, the median full-cycle tracking deviation was 4.9% of a 0.70 saturation target, equal to that of a tuned proportional-integral (PI) controller, while the PGML controller reached the target without overshoot in the median and applied approximately 4% less irrigation. Moderate physics guidance reduced long-horizon prediction error by approximately 30% relative to data-only training and reduced irrigation-bound activity, but slightly worsened closed-loop tracking. For a matched prediction workload, the surrogate was approximately 70 times faster than the numerical solver, and batched evaluation cost under 2 ms per column for 5000 columns. The results support a reusable physics-guided surrogate for large-scale saturation control in simulation.

We study eigenpair estimation for the tensorized, density-weighted Laplace–Beltrami operator from $n$ independent data points sampled from a multi-manifold mixture distribution where manifolds may intersect. Let $d$ denote the largest intrinsic dimension of the manifold components and $s\geq2$ the smoothness of the sampling densities. For simple, nontrivial eigenpairs, we establish the minimax rate $n^{-s/(d+2s)}$ for approximating the eigenvalue and eigenfunction of the tensorized Laplace–Beltrami operator, and show that a continuum plug-in estimator based on kernel density estimation attains the same rate when the supporting manifolds are known. We further analyze a manifold-agnostic graph Laplacian estimator and prove that its eigenpairs converge to those of the tensorized Laplace–Beltrami operator at rate $n^{-1/(2.5d-1)}$, up to logarithmic factors, improving upon previously known spectral convergence rates for multi-manifold learning. Numerical experiments on synthetic multi-manifold data corroborate the rates guaranteed by our theorems. These results characterize the information-theoretic difficulty of multi-manifold spectral estimation and quantify the additional cost of learning without oracle component information.

Split Gibbs sampling (SGS) is a versatile tool to sample from complex posterior distributions, such as those arising in imaging inverse problems. However, existing SGS instances are limited to quadratic variable couplings, restricting their applicability to Gaussian likelihoods. This letter genearalizes SGS to non-Euclidean geometry by introducing a coupling kernel defined through a Jensen-Bregman divergence. This framework not only enables posterior sampling in these settings but also unifies several ad hoc sampling schemes from the literature under a principled theoretical foundation.

Sparse autoencoder (SAE) features are typically discovered from how their activations co-vary with behaviour, yet they are often used through interventions such as steering or ablation. We take intervention attribution, the first-order effect of a specified intervention on the model output, as our estimand. For a broad class of response functions, we show that observational feature-discovery evidence generally does not identify this attribution, and derive an exact gradient-based decomposition of the gap between linear-probe predictions and intervention attribution. Across four model-SAE pairs, median probe-to-attribution magnitude ratios range from tens to hundreds, while attributions closely track small-step steering. For single-latent ablations with nonzero effects, gradient predictions match the realized sign in 91-97% of cases, compared with 17-35% for probe-implied changes. Attribution-selected sets also produce larger absolute mean ablation effects on held-out prompts than sets selected by correlation, rank tests or probe coefficients. Together, these results establish intervention attribution as a principled, gradient-estimable target for selecting SAE features for intervention.

Large language models (LLMs) raise new questions for instructors, among them how to integrate LLMs productively into coursework and what to evaluate in students' work when an LLM can produce much of it directly. LLM-guided homework assignments offer one answer. Instructors configure the LLM to guide students in ways that achieve specific pedagogical goals, such as recalling material, discovering new concepts, or critically evaluating deliberately flawed reasoning, and grade the full student–LLM conversation, and with it the student's reasoning, rather than only the final answer. In this paper, we report on the large-scale deployment of such assignments, implemented through LLteacher, an open-source tool that supports recall, discovery, and critical-thinking assignments, in a large introductory statistics course serving mostly pre-major undergraduates. Results on 134 students show meaningful gains in self-reported confidence, with over 80% agreeing that the assignments improved their comprehension of course material and their ability to interpret statistical results; gains were largest among students who started with lower confidence, suggesting this approach may help narrow gaps in preparation. Interaction patterns varied widely, from long single messages to rapid iterative exchanges, and hallucinations, when present, were rarely disruptive to learning.

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.

Real-world datasets often exhibit severe class imbalance, degrading the performance of deep learning models, particularly on rare classes. While imbalance is well studied in binary and multiclass settings, it is substantially more challenging in multilabel problems due to complex label co-occurrence. In clinical applications such as ECG classification, this challenge is coupled with a second critical challenge: the need to prioritize high-risk conditions, regardless of their prevalence, to mitigate the high cost of false negatives. We introduce the Multilabel Class-Prioritized Probabilistic Sampler (ML-CLAPPS), a learnable sampler that optimizes a distribution over training instances to control the model's exposure to different classes. This approach addresses multilabel imbalance while enabling prioritization of significant conditions via a user-defined importance vector (IV). Empirical evaluation on a real-world multilabel ECG image dataset demonstrates that ML-CLAPPS achieves state-of-the-art performance. We show that it aligns class exposure with the specified IV, and that increasing a class's importance leads to corresponding improvements in its classification performance, enabling explicit prioritization of clinically critical conditions.

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.

We establish uniform score bounds and Lipschitz continuity for truncated Gaussian, Cauchy, Boltzmann, and softmax policies under explicitly stated parameterizations.

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.

A goodness-of-fit test tells whether a logistic risk model fits, but the analyst also needs to know what is wrong: an overall risk that is too high is repaired by updating the intercept, risks that are too extreme by recalibration, a curved relation between score and risk by a flexible recalibration, and misfit within groups of patients only by revising the model. We split any misfit into these four parts, which are orthogonal and match the calibration hierarchy. Each part gets a group of score tests that sees that part only, and the failing parts are named by closed testing. In external validation, in large samples and for small departures, a part is wrongly named with probability at most the chosen level, and the highest part named points to the lightest update that removes the misfit. Checking a model on its development data never sees an error in the overall risk or the calibration slope, and with binary or skewed covariates it can mistake misfit within patient groups for a curved link. External validation sees all four parts, for any prediction model. In simulations it kept its error rate when the model was right and for small departures, and usually pointed to the right update. On intensive-care data its verdicts for one model differed in-sample, on similar patients and after transport.

Recursive synthetic-data training can degrade a fitted model, but its effects on the variable-screening stage that precedes model fitting are less understood. When each generation labels data for the next, a marginal screener ranks variables against the current pseudo-label target rather than the original response. Across generations, this target drift can erode or reverse the active–inactive score margin and cause screening to discard variables that clean-label screening would retain. We call this process screening collapse. In a Gaussian linear model with minimum-norm recursive generators, we establish finite-sample screening bounds, characterize a late-generation information-loss limit, and derive a minimax impossibility result. Screening collapse can precede and compound ordinary model collapse: omitted variables incur a covariance-weighted loss that no refit restricted to the selected variables can eliminate, while recursive labels further degrade estimation on the retained variables. We also examine two approaches to mitigating screening collapse. Fixed-fraction real-label retention preserves a clean-label component but leaves resolution fixed, whereas data accumulation can avoid screening collapse if the pooled target preserves a positive margin and the growing sample resolves it. Numerical experiments on gene-expression data support the theoretical findings.

Effective unsupervised domain adaptation (UDA) requires a representation that captures transferable structure from both the labeled source domain and the unlabeled target domain. We introduce Joint Contrastive-Adversarial Representation Learning (JCAR), a two-stage framework that learns a common representation from the union of unlabeled source and target samples by combining contrastive learning with adversarial domain alignment, and then freezes the representation to fit a source-supervised predictor. We establish a non-asymptotic excess-risk bound for this procedure that couples representation and domain-discriminator estimation while separating downstream prediction, with explicit neural-network approximation errors. Under the stated assumptions and common Hölder smoothness $\beta\geq1$, the bound identifies a sufficiently large unlabeled-sample regime in which the downstream rate is $\widetilde{\mathcal{O}}(m^{-\beta/(2\beta+d)})$, where $m$ counts labeled source samples and logarithmic factors are suppressed. The labeled-sample exponent is governed by the latent dimension $d$ rather than the ambient dimension $p$, which enters the pretraining cost. Experiments on Office-31, Office-Home, and DomainNet demonstrate clear transfer gains across diverse domain shifts, with particularly strong performance when representations are learned from scratch.

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.

In this work, we propose a transfer learning algorithm for covariate shift, integrating sample reweighting with representation learning. A key feature of our approach is the use of a lower-dimensional latent space for modeling, in contrast to conventional methods that operate directly in the high-dimensional ambient space. Our method estimates the density ratio of learned latent features for downstream weighted prediction. We establish explicit target-domain excess-risk rates with independent data folds for strictly sufficient learned representations with regular learned-space prediction scores and density ratios. We give dimension comparisons under stated smoothness conditions and a variance comparison for true weights and a fixed predictor. General finite-sample inequalities retain representation-quality errors. Experiments on simulated and real-world datasets demonstrate gains from combining learned representations and latent weighting, and reveal sensitivity to initialization and domain choice.

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.

Parallel cluster-randomized trials can identify mean effects by site characteristics and risk differences between fixed predictors, while leaving residual heterogeneity and absolute prediction risk (mean squared error of site-effect predictions) unidentified. We acquire missing same-site cross-arm information by randomizing participants within selected sites. Under randomized site selection with compatible participant reference distributions and intervention versions, we characterize necessary and sufficient observation patterns for quadratic targets in a model with unrestricted joint distributions of site response means. From finite independent participant samples, we construct unbiased moments and exact covariances. Under regularity conditions, we prove consistency as informative paired site counts grow, allowing fixed within-site sample sizes. For a fixed predictor, we optimize paired-site selection and unequal cell counts using corrected risk variances, pilot error, and costs. In an International Stroke Trial information-deletion experiment, pilot allocation reduces the conditional variance of reconstructing the complete-information corrected risk estimate by 34.3% relative to uniform allocation with equal expected cost. In 40 Honduras villages, correction reduces estimated residual variation from 0.04081 to 0.00282 in squared log-consumption units for the same fixed explanatory basis; the explained share rises despite a smaller explained component, illustrating how participant sampling noise can inflate estimated residual variation.

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.

Longitudinal health data often exhibit within-subject correlation, subject-specific heterogeneity, and complex nonlinear predictor–response relationships that may not be adequately captured by conventional mixed models. We propose a Deep Generalized Neural Network Mixed Model (DGNMM) that extends the GLMM framework by using a deep neural network to learn nonlinear population-level relationships while incorporating directly trainable, regularized subject-specific effects to capture individual heterogeneity. The model is estimated using prediction-oriented stochastic optimization with separate regularization of the neural-network parameters and subject-specific effects. This formulation allows us to investigate when increased nonlinear representation capacity provides additional predictive benefit in the presence of subject-specific heterogeneity. A simulation study varies population-level nonlinearity and subject-specific heterogeneity and compares DGNMM with GLMM, artificial neural network (ANN), Maity–Pal (MP), and GNMM alternatives for continuous and binary longitudinal outcomes. In the real-data application, DGNMM achieved the lowest MSPE for continuous depression and anxiety prediction, while alternative models remained competitive for binary outcomes. Exploratory subject-level analyses showed heterogeneous predictive gains across individuals, but the magnitude of the estimated subject-specific effects was not significantly associated with predictive gain. Overall, increased nonlinear representation capacity was most beneficial for continuous prediction under nonlinear response structures, while greater subject-specific heterogeneity did not systematically yield larger predictive gains.

A deception detector is only as useful as the trust its alarms deserve. Like a polygraph, detectors that read a language model's internal activations can catch deceptive responses that look honest. Yet current detectors depend on which layers they read, weaken on broader tests, and do not state how certain their alarms are. This white paper proposes the Machine Deception Detector, which targets both detection accuracy and the uncertainty of detector scores. The detector reads every layer of the language model, learns which tokens carry evidence of deception using only response labels, and calibrates its alarm threshold so that its false–positive rate stays below a prescribed level. An ensemble of such detectors measures the uncertainty of each score and raises an alarm only when its members agree, with a confidence level that sets how certain each alarm must be. The proposed evaluation trains on instructed honest and deceptive statements, tests on five unseen deceptive tasks of Liars' Bench read by six Qwen3 models from 0.6B to 32B parameters, and is designed to also show which layers carry the deception feature and how this changes with model size.