construct size-invariant bounds

To address scenario-based decision-making under uncertainty, this paper proposes a risk-controllable decision-making method based on scenario compression. To overcome the looseness and strong distributional assumptions inherent in existing risk upper bounds, we derive the first compression-size-dependent risk bound that requires no additional assumptions—integrating stochastic geometric analysis, compression set theory, and refined probabilistic inequalities with optimization. This bound significantly improves tightness: for identical numbers of scenarios and prescribed risk tolerance levels, the upper bound on decision failure probability is reduced by 20–40% on average. The method ensures theoretical rigor while maintaining broad applicability across diverse data-driven robust decision-making settings, thereby providing a more reliable risk-quantification framework for robust optimization under uncertainty.

This work addresses a critical limitation in existing multiple testing procedures, which control only the expected false discovery proportion (FDP) and lack high-probability guarantees for the realized FDP, particularly when data-driven thresholds are employed, thereby compromising statistical validity. The authors propose a distribution-free, finite-sample valid framework that constructs a high-probability simultaneous envelope around the empirical distribution function of conformal p-values under the null hypothesis. This approach yields, for the first time, a uniform high-probability upper bound on the FDP that holds simultaneously over all possible rejection thresholds. The method accommodates arbitrary post-hoc threshold selection and allows users to tailor the envelope’s shape to obtain tighter bounds in regions of interest. Empirical evaluations on both synthetic and real-world data demonstrate that the resulting bounds are not only valid but also substantially less conservative than those from existing methods.

Existing two-sample location tests for high-dimensional data rely on restrictive assumptions—such as fixed dimensionality or specific dimension-to-sample-size asymptotics (e.g., (p/n o ext{const}))—whose validity is often unverifiable in practice. Method: We propose a dimension-uniform asymptotic test, grounded in a novel uniform-over-dimension central limit theorem, integrating functional CLT and adaptive standardization to construct a test statistic independent of the relative growth rate of (p) and (n). Contribution/Results: The proposed test rigorously controls Type I error uniformly across all dimensions and retains asymptotic power. Simulation studies and real-data analyses demonstrate its substantial superiority over Hotelling’s (T^2) and state-of-the-art high-dimensional tests—particularly in small-sample, high-dimensional settings—thereby offering a robust, assumption-lean alternative for modern multivariate inference.

This work investigates the minimax detection boundary for uniformity testing of multinomial distributions under ℓₚ deviations, focusing on the intermediate regime where the sample size satisfies \( N = o(n^2) \) and the signal-to-noise ratio converges to a finite positive constant. By introducing a Poissonized model, constructing a Poisson mixture prior, and applying a conditional central limit theorem for weighted sums, the authors establish—for the first time—a matching lower bound on the minimax risk in this regime. This lower bound precisely coincides with the known upper bound, thereby fully characterizing the asymptotic behavior of the minimax risk at the level of sharp constants: the risk converges to the nontrivial constant \( 2\Phi(-u^*/2) \).

Existing fairness evaluation methods rely on point estimates compared against fixed thresholds, ignoring sampling uncertainty and applying uniform criteria across subgroup sizes—particularly failing in intersectional settings where sparse subgroup samples yield overly wide confidence intervals and unreliable statistical inference. This paper proposes the first adaptive, bimodal statistical testing framework for fairness assessment: Wald tests with theoretical guarantees for large subgroups, and calibrated Bayesian Dirichlet–multinomial estimation for small subgroups. The framework ensures interpretable and verifiable fairness decisions across the full spectrum of subgroup sizes. Through central limit theorem analysis, Monte Carlo–based confidence interval estimation, and extensive experiments on multiple benchmarks, we demonstrate that our method significantly improves statistical power and discriminative robustness—especially under data scarcity and high-dimensional intersectional scenarios.

This study addresses a theoretical gap concerning order statistics under non-independent and identically distributed (non-i.i.d.) data by extending classical probability inequalities to settings involving weak dependence or approximate identical distributions. Under mild regularity conditions, the authors establish that the relevant probabilities still converge to 1/2. Building on this result, they develop a unified finite-sample theoretical framework applicable to bootstrap methods, permutation tests, and conformal prediction. The work provides rigorous probabilistic guarantees for high-dimensional resampling-based inference and delivers the first finite-sample validity proof for conformal prediction under non-i.i.d. assumptions, thereby substantially broadening the applicability of these widely used statistical procedures.

This work addresses the generally infeasible problem of nonparametrically inferring the power trade-off function for testing two unknown distributions from finite samples. To render the problem tractable, the authors introduce a structural assumption of “exact realizability,” stipulating that the rejection region belongs to a specific class of sets. Within this framework, they establish that the set class having finite VC dimension is both necessary and sufficient for finite-sample testability. Leveraging this insight, they construct the first simultaneous confidence band with non-asymptotic error control and without reliance on asymptotic approximations. Their approach integrates the Neyman–Pearson lemma, VC dimension theory, and set-class approximation techniques, yielding tests that strictly control Type I error and achieve uniform power under realizable alternatives. In monotone likelihood ratio models, the method attains nearly tight local separation rates and extends to one-dimensional log-concave distributions.

This work addresses the challenge of distribution identity testing in high-dimensional or continuous domains, where traditional total variation distance becomes ineffective. The authors propose a fooling distance framework based on bounded discriminator classes, integrating integral probability metrics with Boolean function classes to achieve sample-efficient testing algorithms. By unifying three directions—testable learning, verification of learning, and structured distribution testing—the study extends the Ak-testing framework and establishes theoretical connections among them. Technically, the approach combines Rademacher complexity analysis, membership query mechanisms, and polynomial density modeling over the hypercube, yielding testable proper learners for halfspaces and decision trees, deriving verification lower bounds, and designing efficient identity testers for decision tree distributions and low-degree polynomial densities.