Papers

Two-factor ANOVA is widely used in experimental studies but relies on additivity, normality, independence, and homoscedasticity. These assumptions are often violated for nonnegative, positively skewed observations. Although Box–Cox-type transformations are commonly used, they may reduce interpretability and require a subjective choice of transformation. We propose an alternative framework in which nonnegative observations affected by two factors are modeled by gamma distributions with unknown shape and scale parameters that may depend on factor levels. We develop likelihood ratio tests (LRTs) for main and interaction effects. The asymptotic LRT (ALRT) uses the asymptotic chi-square distribution, which may be inaccurate for small to moderate samples. We therefore propose a parametric bootstrap LRT (PBLRT) that determines critical values by simulation. Extensive simulations show that the PBLRT maintains the nominal significance level well. Real-data examples demonstrate its applicability and show that its inferences can differ from those of traditional ANOVA.

Let $k_{1,\varepsilon}(n)$ be the smallest number of real linear measurements needed by a randomized oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pm\varepsilon$, with probability at least $2/3$. For every fixed $0<\varepsilon<1$, we prove \[ \frac{n^2}{(\log n)^{A_\varepsilon}} \le k_{1,\varepsilon}(n) \le C_\varepsilon \frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}. \] Previously, the best unrestricted bounds for general linear sketches of the Schatten–1 norm were $\Omega(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff'19), leaving a polynomial gap. Our bounds close that gap up to polylogarithmic factors and give a nontrivial logarithmic saving below the $n^2$-measurement storage bound. The result extends much further. Write $k_{p,\varepsilon}(n)$ for the analogous sketch dimension for the Schatten–$p$ norm. For every fixed finite $p>0$ that is not a positive even integer, there are positive constants $A_{p,\varepsilon},C_{p,\varepsilon},c_p$ such that \[ \frac{n^2}{(\log n)^{A_{p,\varepsilon}}} \le k_{p,\varepsilon}(n) \le C_{p,\varepsilon}\frac{n^2}{(\log n)^{c_p}}, \] so $k_{p,\varepsilon}(n)=n^{2-o(1)}$ throughout the non-even regime. Together with the known tight bounds $\Theta_{p,\varepsilon}(n^{2-4/p})$ for positive even $p$ and $\Theta_\varepsilon(n^2)$ for $p=\infty$ (Li, Woodruff'16), our results close the remaining polynomial gap across the Schatten family and complete, up to polylogarithmic factors, the polynomial-order classification of general linear sketches for all Schatten-$p$ norms.

Estimating spectral density matrices for future space-based gravitational-wave detectors such as LISA is challenging due to the long duration of the data and the correlated instrumental noise across multiple time-delay interferometry channels. In this work, we investigate a specialized mean-field stochastic gradient variational Bayes (SGVB) procedure for fast posterior approximation in long-duration multivariate spectral density estimation. Based on the existing eigenbasis representation of the blocked Whittle likelihood approach, the posterior model applies a Cholesky factorization to represent the inverse spectral density matrix and models the resulting frequency-dependent entries with cosine basis functions, with a discounted regularized horseshoe prior assigned to the basis coefficients. We test mean-field SGVB as a stand-alone posterior approximation by comparing it with Hamiltonian Monte Carlo (HMC) targeting the same posterior model. In simulation studies based on autoregressive and moving average processes, SGVB produces posterior median spectral estimates close to those obtained from HMC at substantially lower computational cost. We then apply the method to two one-year, fixed-delay, stationary, noise-only LISA simulations and show that the SGVB spectral density estimates are consistent with HMC and Welch estimates across the LISA analysis band, while requiring substantially less computation. These results demonstrate that SGVB provides a scalable Bayesian approach to spectral density estimation for long-duration multivariate LISA noise analysis.

Anti-amyloid therapies and blood-based biomarkers are changing Alzheimer disease workups into a two-stage measurement workflow: screen broadly with cheaper information, then spend scarce confirmatory amyloid measurements where they support the decision that will be reported. Amyloid positron-emission tomography (PET) remains one such protocol measurement for amyloid burden, but PET slots, trial budgets, and payer-facing evidence packages are finite. This paper asks a deliberately operational question: when is simple transparent PET validation enough, and when is a fitted residual-uncertainty score worth the added complexity? For a weighted protocol target, the first-order value of validating subject i is the product of target influence and residual protocol uncertainty. Generic uncertainty sampling uses only the second factor and can spend PET measurements on subjects that are hard to predict but weak for the scientific, clinical, or commercial claim. We apply this rule to the A4/LEARN PET archive, treating observed PET as a design laboratory for scarce-confirmation studies. For the primary APOE4 carrier versus non-carrier contrast in Centiloid 24-or-higher PET positivity, simple APOE4-balanced validation recovers nearly all of the target-specific gain: at PET budget 200, the confidence-interval width ratio relative to random validation is 0.923 for APOE4 balancing and 0.914 for target-specific scoring, while generic uncertainty sampling is 0.980. Other targets behave differently: target-specific scoring gives larger gains for an age-slope analysis and for cutoff-indexed PET positivity. The practical message is simple: spend scarce protocol measurements according to the claim being validated, not only according to prediction uncertainty.

Causal discovery aims to understand the relationships between individual random variables. In many applications, such as brain imaging and climate modeling, it is more meaningful to consider interactions among groups of variables. Existing methods assume that knowledge of such groups or clusters is explicitly available when modeling interactions. However, in practice, these clusters as well as the causal relationships among them, are latent. In this paper, we present a novel approach based on variational inference to simultaneously infer both the latent clusters and causal structures. We learn an approximate posterior over clusters and graph-structure by considering variational distributions based on categorical and Bernoulli models respectively. We derive variational lower bounds and estimation techniques to learn variational and model parameters. The effectiveness of our proposed methods for cluster and causal discovery are demonstrated on both synthetic and real data sets.

Positronium lifetime imaging (PLI) provides local annihilation environment data beyond conventional activity imaging. Existing approaches, however, often estimate lifetime parameters over predefined regions, neglecting multiple lifetime populations within the same location. We present a 3D, population-specific Bayesian framework for fast voxel-wise PLI. A partial system matrix describes the spatial probability of detected events, while measured lifetimes provide soft assignments to slow, fast, and noise populations. These event responsibilities are used to estimate a decay-rate posterior independently for each voxel, preserving local lifetime variation and statistical uncertainty. In simulations, our formulation recovered spatially varying slow-population decay rates and a common fast-population rate, whereas a single-population model produced systematic bias. Slow-population two-standard-deviation coverage ranged from 93.8% to 98.1%. Experimental validation using 124I triple-coincidence data from a Siemens Biograph Vision Quadra scanner produced separate slow- and fast-population maps for aluminum, nickel, copper, and quartz. The long-lived quartz component matched ortho-positronium, while the fast population showed material-dependent differences among metals. Fast-population coverage was lower (69.1%), indicating underestimated uncertainty. The method is highly efficient, requiring only seconds to minutes per population on a single CPU core. This framework provides fast, population-specific PLI with Bayesian uncertainty quantification, making spatially resolved statistical inference feasible and practical for volumetric applications.

We introduce a new class of regression models for scan statistics on real-valued signals. These allow for improved fitting of non-stationary signals to contrast with the interval anomalies identified by the scan statistics. Our models can represent generalized likelihood ratio statistics. While these methods naively require $O(n^4)$ for a length $n$ signal, we provide algorithmic improvements which lead to linear time algorithms (with assumptions on max interval width). Our methods, especially ones based on Nadaraya-Watson kernel regression, are demonstrated as especially effective in detecting both synthetically planted anomalies, and for identifying a real “platforming” issue in interferometric astronomy.

Membership inference asks whether a text was used to train a language model, whereas AI-generated text detection asks whether it was generated by a language model rather than written by a human. Existing likelihood-based methods typically compress token-level probabilities into a few prespecified scores, most often using only probabilities conditioned on the full preceding context. We propose likelihood-array regression (LAR), which evaluates each target token under nested left-context windows and organizes the resulting likelihood-derived features into a structured array. After aligning arrays across texts of different lengths, LAR learns how detection information varies with context scale, token position, and likelihood features. LAR-1 aggregates learned contributions from individual aligned cells, while LAR-2 adds second-order features formed from pairs of evaluations of the same target token across context lengths. For within-path quadratic model, we establish matching minimax lower and upper bounds, characterize errors from finite-dimensional approximation and random squared projections, and derive conditions under which an oracle spectral sieve attains the minimax rate. Across multiple scoring language models, LAR substantially improves membership inference and AI-generated text detection over likelihood-based baselines. The analyses further show that shorter-context likelihoods contain information beyond conventional full-context probabilities, while second-order features provide additional gains for membership inference.

We develop a population-level formulation of functional multiple-set canonical correlation analysis (P-FMCCA) for multivariate functional data in an infinite-dimensional Hilbert space. Since covariance operators for functional data are typically compact, the inverse covariance operators that appear in the formal extension of multiple-set canonical correlation analysis (MCCA) are generally unbounded and are not defined on the whole Hilbert space. We therefore provide sufficient conditions under which the proposed population formulation is well-defined and show that the resulting constrained maximization problem is characterized by an eigenvalue problem for a Hilbert-Schmidt extension of the relevant correlation operator. We further establish a canonical decomposition induced by the P-FMCCA components and introduce the associated truncated canonical representation. In addition, we formulate functional homogeneity analysis at the population level and show that the finite-dimensional equivalence between homogeneity analysis and MCCA does not generally carry over to the infinite-dimensional setting. Finally, we prove that, for the finite-rank truncated canonical representation, functional homogeneity analysis admits an explicit characterization in terms of the P-FMCCA components, thereby providing a population-level counterpart of the classical finite-dimensional correspondence.

This manuscript develops a non-parametric and robust framework for estimating the scale of additive noise in weakly sparse systems. The method does not require independence, prescribed dependence, or temporal regularity of the noise sequence. We introduce a class of order-statistic estimators based on comparing the sorted observations with deterministic or random proxies generated from a reference noise distribution. This purely spatial approach avoids preliminary filtering or temporal decorrelation, and therefore preserves the sparsity structure of the latent signal. We establish non-asymptotic concentration inequalities for weighted loss functions, with bounds that separate the contribution of the signal from the discrepancy between the ordered noise and the proxy. We then control this proxy discrepancy in independent and correlated regimes, including heavy-tailed reference laws. Finally, we apply the method to high-frequency observations of continuous-time stochastic processes, obtaining scale estimators for fractional Brownian motion and stable Lévy noise in the presence of lower-variation additive perturbations.

In recent years, there has been a growing interest in comparing the reliability of systems with dependent components. Most of the recent works consider Archimedean copulas to define the dependency structure of systems and examine specific familiy of distributions. In this work, we provide distribution-free results to compare, in the usual stochastic order under some majorization conditions, coherent systems with heterogeneous and dependent components where the dependency structure can be defined by any copula. Applications of our main results for specific copulas and different families of marginal distributions are also given.

Staggered distributional difference-in-differences produces cohort-specific potential-outcome distributions, but applied work typically wants one overall quantile treatment effect. Two natural summaries—averaging cohort quantile treatment effects (QTTs) and mixing cohort distributions before inversion—use the same policy weights yet answer different target-population questions and can disagree even in sign. We derive the exact sharp interval for their gap conditional on cohort quantiles and weights, a globally sharp range-only envelope, and a locally sharp density-tilt representation. We develop joint smooth and mass-point-safe inference and show that neither estimator is uniformly more precise, even when the estimands coincide. In a same-object reconstruction of a public staggered-QTT application, holding data, identification, distributions, and weights fixed while changing only aggregation order reverses reported signs at several quantiles. Aggregation order is therefore part of the estimand and must be chosen before inversion.

We propose a Bayesian nonparametric mixture of regression trees with a Dirichlet process prior over tree-parameter pairs, enabling data-driven selection of ensemble size and unifying CART, BART, random forests, and boosting. A novel splitting rule driven by the posterior predictive of a Gaussian process within each terminal node generates flexible, smooth decision boundaries; remarkably, the GP density cancels exactly in the Metropolis–Hastings ratio for GROW/PRUNE moves, ensuring computational feasibility. An exact Gibbs sampler for posterior predictive inference propagates uncertainty through random tree traversal. A parallel MPI implementation distributes independent tree updates across processors, achieving adequate speedups. We prove posterior consistency at rate $n^{-1/4}$ in Hellinger distance under only continuity of the true regression function, allowing misspecification, via the identity $h(\Theta)=0$. Simulations on Friedman benchmark show near-nominal coverage (0.94 Gaussian, 0.92 Cauchy), robust to high-dimensional noise and heavy tails, outperforming BART and bagged CART. Applications to QSAR toxicity, crime, riboflavin, wheat genomics, and air quality confirm reliable credible intervals and automatic sparsity. The DP mixture offers a principled, robust, theoretically justified alternative for challenging regression with honest uncertainty quantification.

This note revisits the convergence of the Parikh–Boyd two-way block-splitting algorithm for large-scale distributed learning through the He–Yuan prediction–correction framework. Simultaneous row–column partitioning is also relevant to hybrid federated learning, where data may be heterogeneous in both samples and features. We lift the reduced iteration to an equal-dimensional product space and reconstruct the primal and dual coordinates omitted by its implementation. The induced orthogonal-complement structure establishes exact iteration-by-iteration equivalence with the published updates. A mixed variational-inequality representation then yields a fundamental descent inequality, global convergence, an ergodic complexity bound, and current-iterate residual estimates under standard convexity, solvability, exact-subproblem, and invariant-initialization assumptions. The analysis also shows that the reduced state recursion is a metric proximal point iteration. No strong convexity, differentiability, or full-rank condition is imposed. The derivation clarifies which algebraic initialization conditions allow the reduced implementation to inherit the full-space convergence and complexity guarantees without modifying its local updates.

A common question about networked time series is whether outcomes changed because shocks transmit more strongly or because the pattern of connections changed. Standard practice inserts a recorded network into an outcome regression and reads movements of the fitted coefficient as changes in transmission strength. When the network is latent, time varying, and measured with error, this reading fails: changes in strength and changes in composition can produce the same outcome distribution at a single date, and the population coefficient moves under composition changes alone. The question becomes answerable when outcomes are analyzed jointly with repeated noisy measurements of the network, such as paired reports of bilateral trade flows. For the joint model we establish necessary and sufficient conditions for local identification, estimators of the strength and composition paths, a simultaneous confidence band for the strength path, confidence sets that remain exact under weak identification, breakdown bounds under common reporting bias, and an exactly sized test that detects changes on the observed path and attributes them to strength or to composition. Simulations assess each procedure at its stated boundary. On a mirror-reported trade panel of eighteen economies over 1995 to 2020, the diagnostics flag exactly the crisis years and the composition coordinate attached to European Union membership declines by roughly two thirds. The estimand is predictive dependence, not a causal effect.

This paper describes an approach to variance reduction in randomized experiments using side information (covariates of the response unaffected by treatment). As with Double ML[Chernozhukov 2018], models of arbitrary complexity may be employed without concern for bias due to overfitting or regularization. For additive treatment metrics, the approach minimizes variance optimally. For more complex treatment metrics (e.g., multiplicative and ratios) the Delta Method estimate of variance is minimized. Through a variety of examples, the paper also explores modeling considerations.

We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.

Estimating heterogeneous single and interaction treatment effects from observational data under multiple simultaneous treatments is crucial for decision-making. To mitigate estimation variance, previous studies balance representation distributions between every pair of treatment patterns. However, such pairwise balancing scales quadratically with the number of treatment patterns and fails to preserve consistent local proximity structures across patterns, which degrades counterfactual estimation. To address these challenges, we propose the Causal Inference for Heterogeneous Single and Interaction Treatment Effects Network (CIHSI-Net), a deep learning framework built on a novel Barycentric Fused Gromov-Wasserstein Balancing (BFG-WB) objective. BFG-WB aligns the representation distribution of each treatment pattern with a shared Wasserstein barycenter, achieving global alignment while reducing the computational complexity from quadratic to linear, and its Fused Gromov-Wasserstein discrepancy preserves the local proximity structures essential for reliable heterogeneous effect estimation. Simulation studies show that CIHSI-Net consistently outperforms state-of-the-art baselines, and an application to real-world marketing data demonstrates its practical utility in complex multi-treatment scenarios.

Asymptotic normality approximations often fail to hold for extremum estimators when the true value of the parameter is at or close to the boundary of a parameter space. I analyze and develop tests using a quasi-unconstrained estimator, which is asymptotically normal even when the true parameter vector is near or at the boundary. These results generalize previous work with this estimator by allowing for more types of constraints and showing how the method can naturally be modified when a nuisance parameter is also high-dimensional. I show that variations of Wald, Likelihood Ratio, and Lagrange Multiplier tests can control size in a uniform sense, provided the initial constrained estimator is sufficiently accurate. Lastly, I apply the method to an application involving network estimation with panel data.

We propose Variance Driven Exploration (VarDE), a principled approach for pure exploration in highly stochastic environments, where the exploration process is dominated by stochastic variance. VarDE is built on a fundamental principle: sampling effort should be allocated to minimize the uncertainty of the final decision. We formalize the uncertainty of the final decision through a smooth decision function and derive allocation rules that explicitly capture how stochastic noise in individual components affects the reliability of the final output. We apply this methodology to three core problems of pure exploration – Best Arm Identification (BAI), Monte Carlo Tree Search (MCTS), and Best-Policy Identification (BPI) – with theoretical guarantees on variance decay and simple regret. Empirically, we demonstrate consistent and significant improvements of VarDE over existing methods, with especially strong gains in highly stochastic environments.

The sampling process of Denoising Diffusion Probabilistic Models (DDPMs) can be accelerated by leveraging second-order information in the form of approximations to the denoising posterior covariance – allowing samples of acceptable quality to be produced in fewer but larger sampling steps. Previous attempts at using such information have used drastic (e.g.\ diagonal) simplifications of the covariance. These do not do justice to the peculiar statistical structure of natural images, which exhibit strong non-diagonal correlations between pixels and color channels, and a slow-decaying power-law frequency spectrum. Here, we develop a novel covariance model that captures these features. Our Kronecker-DCT (K-DCT) model uses a Kronecker-factored decomposition of inter-color covariances and spatial covariances modeled in the frequency domain using the Discrete Cosine Transform (DCT). The use of the DCT reduces the computational complexity from quadratic to log-linear, resulting in negligible computational and memory overhead in each denoising step. By learning K-DCT-structured amortizations of the denoising posterior covariance using pre-trained score models on CIFAR-10, Celeb-A, ImageNet and LSUN datasets, we show improved performance compared to previous SOTA denoising samplers, both in terms of FID and likelihoods, especially in the regime of few denoising steps.

To address the overdispersion problem in matrix-variate integer-valued time series data arising in sociology, medicine, and related fields, this paper proposes a matrix integer-valued autoregressive model based on the negative binomial thinning operator. By introducing left and right matricial negative binomial thinning operators, the proposed model not only preserves the integer-valued nature of the data but also effectively handles overdispersion. The probabilistic and statistical properties of the proposed model are systematically investigated. Two estimation methods, namely projection estimation and iterative conditional least squares estimation are developed, and the corresponding asymptotic theories are established. Simulation studies provide concrete numerical results to evaluate the finite-sample performance of the estimators. Real data analysis demonstrates that the proposed model outperforms both the continuous matrix autoregressive model and the multivariate integer-valued autoregressive model in fitting matrix-variate integer-valued time series, while also proving effective in accommodating overdispersed count data.

Assessing fine-scale spatio-temporal air pollution contrasts in urban contexts is a major challenge for environmental epidemiology. We propose a Bayesian spatio-temporal model to predict daily concentrations of NO$_2$, PM$_{10}$, and PM$_{2.5}$ in Rome (Italy), over 2011-2022, on a fine grid scale (1 km), using data from 8 to 13 monitoring stations, depending on pollutant. The model includes meteorological and temporal (working day/weekend) fixed effects together with a lag-1 autoregressive spatio-temporal random effect aimed at capturing spatial and daily dependence. Predictive performance was assessed by leave-one-site-out cross-validation. Estimated exposures were then linked to geolocated cause-specific mortality data for Rome (2012-2019), and a case-crossover time-stratified approach was adopted to investigate acute effects. Cross-validation showed good overall predictive performance, with larger errors at high-traffic sites. Estimated exposure (mean lag 0-5) was positively associated with natural-cause mortality, with percent increase in risk of 1.3 (95% CI: 0.6-2.0) for PM$_{10}$, 2.1 for NO$_2$ (1.4-2.9), and 2.4 for PM$_{2.5}$ (1.4-3.3) per 10 $\mu$g/m$^3$ increase. Corresponding estimates when using the city-specific daily average exposure, instead of our 1 km resolution model, were of comparable magnitude. The proposed Bayesian spatio-temporal framework provides reliable fine-scale exposure estimates for epidemiological use, with results consistent across independent exposure estimates, supporting its application in urban air-pollution health studies.

Digital health measures increasingly inform treatment evaluation, risk classification, clinical monitoring, and regulatory decisions. Existing validity concepts concern a measure's technical soundness, clinical meaningfulness, usability, and scalability. Inferential validity concerns how well a measure supports a scientific, clinical, or regulatory conclusion. We propose assessing inferential validity as a downstream evaluation layer that links measurement system evidence to trustworthy conclusions.

Simulation-Based Inference (SBI) serves as a vital framework for parameter inference in scientific fields where simulators involve intractable likelihoods, yet while amortized generative models offer rapid posterior estimation, they are often restricted by the specific priors used during training, thereby limiting their flexibility as prior knowledge evolves. To address this prior dependency, PriorGuide was introduced as an inference-time guidance method, but due to its intractable formulation, it relies on Gaussian approximations of the reverse transition kernel and Gaussian mixture model fitting for the prior ratio, both of which introduce systematic bias. Motivated by these limitations, we propose an unbiased test-time guidance framework that leverages Density Ratio Estimation (DRE) to learn a score guidance term, effectively decoupling the inference process from the prior training. Moreover, our framework remains agnostic to the specific density ratio estimators, making it a general and flexible framework for handling prior changes. Experimental results across multiple tasks demonstrate that our method matches or outperforms PriorGuide on C2ST and MMD in most tasks while maintaining robustness even under limited overlap between the training and target priors. Furthermore, we apply our method to Bayesian updating for parameter inference from planetary light-curve data, where it also demonstrates strong effectiveness and robustness. Code is available at https://github.com/a-chenchen/dre-based-prior-guidance .

Given $[0,1]$-valued random variables $X_1,\dots,X_n$ such that $\mathbb{E}[X_i | X_1,\dots,X_{i-1}]= \mu$ for all $i$, we propose a new nonasymptotic confidence interval for $\mu$ that is obtained by inverting terminal e-values generated by a novel betting strategy. When the data are iid, its limiting width matches that of the central limit theorem (“Gaussian-efficient”), finally surpassing the inefficient limits of previous betting intervals. Our main conceptual advance involves designing betting fractions that track the conditional rejection probability of the most powerful terminal test in a limiting Gaussian experiment. When one predictable variance estimator is shared across candidate means, the deterministic inversion is an interval for every data sequence and its two endpoints can be found easily. The width can be improved further with external randomization. In simulations, our method yields the tightest intervals to date; for every distribution tested and all sufficiently large $n$, our deterministic version beats STaR-Bets and is competitive with Gaffke, while the randomized improvement beats both. It thus combines finite-sample validity under martingale dependence, easy endpoint computation, Gaussian-efficient inference for iid data, and excellent empirical performance. We also extend the construction and its efficiency theory to sampling without replacement, where it again achieves state-of-the-art empirical performance.

Modern sequential monitoring problems often involve multiple metrics, where we monitor several data streams simultaneously and may act once the evidence is strong enough. Confidence sequences (CSs) are a natural tool for such continuous monitoring. However, for bounded vector means, existing multivariate CSs are either tight but computationally intractable, or fast to compute but conservative. To address this, we study three lifts of one-dimensional betting-based CSs to higher dimensions: a weighted Bonferroni region, an equivalent max-wealth form, and a portfolio region. The portfolio is typically much tighter, especially in higher dimensions, but its boundary and properties such as volume are not available in closed form. To make this tighter construction usable, we propose tractable outer approximations of the portfolio region that preserve statistical validity: a bounding box, an $\ell_p$-ellipsoid, and their intersection. We prove set relations among all constructions and show empirically that these approximations (i) achieve regions close to the intractable portfolio, (ii) substantially outperform existing tractable multivariate CSs, and (iii) enable practical use cases such as multi-metric A/B testing and model comparison.

Consider testing a finite composite null $\cP={P_1,\ldots,P_L}$ against a simple alternative $Q$ using $n$ i.i.d. observations. Let $\ell_n$ denote the supremum of the expected log e-value over e-variables that are exact under every $P_i$ and pivotal across the nulls. Zhang, Ramdas, and Wang (2024) showed that $\ell_n$ is superadditive and asked whether $\ell_n/n$ converges to the upper bound $\min_i D(Q|P_i)$ under their joint atomlessness and absolute continuity assumptions. We prove that it does. The proof draws on techniques developed by Zhang, Ramdas, and Wang (2024) and by Farooq, Fritz, Haapasalo, and Tomamichel (2024).

Studies enrolling participants on the basis of an extreme baseline value are susceptible to regression to the mean (RTM), such that some observed pre-post change is expected even without treatment. Existing methods estimate and decompose RTM retrospectively. We extend this framework to prospective study design by deriving a closed-form sample-size method for continuous, cutoff-selected, single-arm pre-post studies. The method partitions anticipated total change into that expected from RTM and the residual treatment effect, and powers the study to detect the latter. It uses the general bivariate-normal RTM expression, allowing baseline and follow-up variances to differ, and incorporates the corresponding conditional variance of the pre-post change. To our knowledge, no published method or software implements this cutoff-based RTM framework for prospective closed-form sample-size determination in this setting. The Stata command power onemean_rtm provides solutions for sample size and minimum detectable effect, normal-theory power evaluation, attrition adjustment, and sensitivity analysis. Monte Carlo simulation across 58 scenarios demonstrated accurate Type I error control and showed that achieved power rapidly approached nominal power as sample size increased, with the closed-form calculation generally conservative at very small sample sizes.