Papers

Approximate Bayesian computation (ABC) replaces likelihood evaluation by simulation and comparison of observed and synthetic data. We establish minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates. The contraction theorem uses local prior mass, bounds on ABC acceptance probabilities, and control of prior mass outside a sieve. In fixed-design orthogonal-series regression with centered $g$-and-$k$ errors, an infinite Gaussian series prior with a compact scale hyperprior yields minimax-rate contraction and a minimax-rate clipped posterior mean. In compound Poisson decompounding, only random sums are observed and the target is the underlying jump density. With an unknown count intensity in a fixed compact subinterval of $(0,\pi/2)$, we prove stability of the zero-count-augmented trigonometric population summaries and use a square-root Gaussian series prior on the space of probability density functions. Over bounded periodic Sobolev classes of smoothness $\alpha>d/2$, a polynomially enlarged synthetic sample yields ABC contraction at rate $n^{-\alpha/(2\alpha+d)}$ and posterior mean squared risk of order $n^{-2\alpha/(2\alpha+d)}$, matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.

Clinical studies generally assume that group-level averages are useful quantities for guiding individual-level decisions in the clinical care of each individual patient. Precision medicine has notably closed the gap towards truly individualized care through highly refined subgrouping. Today, digital health technologies and other modern sources of dense, personal "small data" enable a different approach to treatment individualization—one that seeks to characterize a single person's own recurring health patterns first and foremost, rather than identifying the best subgroup to which they might belong. In this chapter, we review the key concepts underlying n-of-1 studies, single-case designs, and other "multitudinal" approaches for digital health applications, and explore their relationships to other digital health methods. We also share some promising future directions for "esametry", the statistics of the digitized multitudes within each of us.

Randomized sketching reduces the computational cost of large ridge-regression problems, but the coefficient error depends on the realized sketch. We extend the paired-row bootstrap from randomized least squares to ridge regression, enabling coefficient-error estimation using only compressed data. Under a fixed coefficient dimension and increasing data and sketch sizes, we derive asymptotic linear representations and Gaussian limits for the sketched estimator and the conditional bootstrap distribution. These results establish uniform consistency of the bootstrap error distribution and asymptotically exact coverage when the estimator and error bound are computed from the same sketch. For sketches with independent, mean-zero, variance-one entries, an explicit covariance formula separates the effects of the residual, regularization, and fourth moment of the sketch entries, and shows that Rademacher entries minimize the leading covariance matrix in the Loewner order. We also develop a fast linearized bootstrap, an order-statistic correction for finitely many bootstrap replicates, and a Bonferroni rule for selecting from a fixed set of sketch sizes. Experiments on two real and two synthetic data sets support the proposed methods. With a sketch size 15 times the number of coefficients, 199 bootstrap replicates, and nominal coverage of 95%, the bootstrap with refitting attains coverage between 92.0% and 95.3%; after the order-statistic correction, coverage ranges from 94.7% to 97.7%.

In this paper we show how to extend quantile-stratified sampling to produce simulations from various multivariate distributions. These simulations have desirable space-filling and coverage properties relative to simulation using IID sampling. We examine the coverage performance of these simulations against IID sampling by looking at plots of ordered log-density values from the simulations.

Diffusion models learn to reverse a predefined corruption process, but sampling still requires a costly time discretization and depends on the chosen noise schedule. We study these two issues for variance-preserving diffusions with matrix-valued schedules. Our analysis transfers reverse-time discretization errors to the forward corruption law and treats two numerical schemes within a common framework. The first freezes the score and yields, through a matrix-sensitive local comparison and forward information dissipation, an ambient-dimensional step complexity with leading factor $d/\varepsilon^2$ for KL accuracy $\varepsilon^2$. The second keeps the known Gaussian drift exact and freezes the posterior mean. For data of metric-entropy dimension $k$, a forward Markov identity, an anisotropic covering estimate, and Stieltjes integration by parts give the corresponding factor $k\log k/\varepsilon^2$. In both cases, the proof identifies a local error, accumulates it through the forward evolution, and inserts the result into a common KL decomposition. The local errors further provide directional criteria for matrix schedules and an asymptotically optimal square-root adaptive grid. A high-dimensional Gaussian-mixture experiment illustrates the resulting schedule and grid improvements.

In this paper, we study the effects of employing multiplier bootstrap to analyze the asymptotic distributions of the largest eigenvalues of high-dimensional sample covariance matrices in both spiked and non-spiked models. Our findings demonstrate that the multiplier bootstrap establishes several phase transitions in the limiting edge distributions of both unconditional and conditional bootstrapped covariance matrices, provided the different classes of multipliers. In the nonspiked setting, unbounded multipliers lead to Frechet or Gumbel limits for the largest eigenvalue of the bootstrapped covariance matrix, both conditionally on the observed data and unconditionally. For bounded multipliers, the unconditional model exhibits transitions among Tracy-Widom, Gaussian, or Weibull limits, determined jointly by the aspect ratio p/n, the upper-endpoint behavior of the multipliers, and the population covariance matrix. The conditional model displays analogous Gaussian and Weibull regimes; in contrast, the conditional counterpart of the unconditional Tracy-Widom regime collapses to a point mass. In the spiked setting, under suitable signal-strength conditions, the leading eigenvalues of both the unconditional and conditional bootstrapped sample covariance matrices are asymptotically Gaussian for bounded as well as unbounded multipliers, under some mild assumptions. Our theoretical results also clarify the feasibility and adaptability of the multiplier bootstrap for spectral inference in high-dimensional sample covariance models. Numerical simulations confirm the accuracy of our results and the effectiveness of the proposed spectral inference procedures, which may be of independent interest.

We prove a quantitative central limit theorem for linear functionals of regularized empirical-risk minimizers in the proportional-dimensional regime \(p=O(n)\). The data columns are independent, not necessarily identically distributed, and satisfy a uniform columnwise Poincaré inequality. Under uniform curvature and smoothness assumptions, and for a quadratic regularizer, we show that every nondegenerate statistic \(\sqrt n\,u^\top\hat\theta\), centered by its expectation and normalized by its standard deviation, converges to a standard normal random variable in Wasserstein distance, with rate \(O((\log n)^7n^{-1/4})\). The proof is based on moment and stability bounds for the minimizer, a second-order leave-one-out expansion, and a perturbative normal-approximation argument for functions of independent variables. We also prove the variance upper bound \(\Var(u^\top\hat\theta)\le C\norm{u}_2^2/n\), identifying the \(\sqrt n\) fluctuation scale.

This study investigates the methodological and theoretical properties of session handover in applications that use large language models. A task may continue in a new session when the context reaches the model's input limit, when the application restarts, or when another agent is asked to finish the task. The application must then decide which information from the earlier session to pass on. We formulate handover as the transfer of a task-relative in-context learning (ICL) state and distinguish exact recovery of earlier material from preservation of the target distribution. Under an exogeneity condition, predictive equivalence characterizes the coarsest deterministic sufficient handover and gives a fixed-length bit requirement. The analysis isolates the effects of the memory constraint, the writer, and the continuation procedure, and quantifies the cost of writing before the realized downstream query is known. We propose a three-part record that stores decisions and constraints exactly, uses task-justified statistics for repeated evidence, and retains original observations whose effect is not preserved by those statistics. Gaussian linear regression gives an exact finite-dimensional handover and finite-bit perturbation bounds, while nonparametric regression gives upper and lower bounds that relate memory to squared prediction error. These results provide a theory and method for deciding what a handover must retain and how its memory requirement depends on the continuation task.

We propose a measure of lottery complexity that combines probabilistic dispersion with dissimilarity between outcomes. Taking degenerate lotteries, which yield a single outcome with certainty, as the simplest alternatives, we measure complexity by proximity to this class. Specifically, for each degenerate lottery, we compute the expected distance between its certain outcome and the lottery's outcomes, and take the minimum across all degenerate lotteries. We provide an axiomatic characterization establishing uniqueness up to positive rescaling, study how complexity changes under outcome compression, and characterize maximally complex lotteries. Finally, we introduce Complexity-Adjusted Expected Utility preferences and characterize adherence to stochastic dominance and strong risk aversion.

The paper focuses on information theoretic converse bounds for the alignment of $m$ correlated graphs and for the detection of correlation among $m$ graphs. A simple idea for $m\geq 3$ is that if the alignment of $m-1$ of the graphs is revealed as extra information (by a genie for example) then it is still necessary to produce the alignment between the one remaining graph and the others, i.e. the last matching must be accomplished. For both Gaussian and Erdos-Renyi models, the last-matching problem is equivalent to one with two observed graphs, providing a path to extend converse bounds for $m=2$ to larger $m$. While the method is rather obvious for alignment, we show that the method can also be used to derive converse bounds for weak detection of correlation.

We present new inference tools for change point detection in high-dimensional time series. We discuss two distinct statistical applications: First, sequential change point testing in an incoming data-stream. Second, retrospective localization of multiple changes, with confidence intervals at a globally controlled error level. Test statistics are built on the maximum norm to generate power against sparse and asynchronous changes. Both problems are tackled by related multiscale statistics that search for changes in the data at many different levels of resolution. For fixed dimension, our statistical approaches can be validated using traditional Hölderian invariance principles. In this paper, we present the high-dimensional analogue: Hölder-Gauss-approximations, which can be (roughly) interpreted as the Gaussian approximation for a Hölder-norm of the high-dimensional partial sum process. Such approximations are of interest beyond change point detection and can be used for other problems such as for stationarity testing in high dimensions. We evaluate finite-sample performance in a simulation study and give an application to air contamination due to wildfires in California, which occurs asynchronously across a panel of measuring stations.

Entropy functionals and their associated divergences underlie many statistical methods, including maximum entropy inference, minimum divergence estimation, and goodness-of-fit testing, yet choosing among Shannon, Rényi, Tsallis, and more general entropies is often a matter of convention rather than structural principle. We introduce a measure theoretic framework in which admissibility requires the entropy of an input measure to be bounded above by that of its reference measure whenever the former is absolutely continuous with respect to the latter. Under generalized mean-value composition, we characterize all such entropy functionals and obtain a four-level hierarchy determined successively by the mean generator, entropy scale, and additivity assumptions. A continuous strictly monotone generator $g$ is admissible exactly when $t\mapsto g(1/t)$ is strictly convex for increasing $g$, or strictly concave for decreasing $g$. This resolves a question posed by Rényi (Proc. 4th Berkeley Sympos. Math. Statist. Prob., 1961) concerning which generalized means may replace the arithmetic mean in his entropy axiomatization. The same criterion is equivalent to strict convexity of an associated Csiszár $f$-divergence generator and therefore yields data processing under Markov kernels with an exact equality condition. Within this hierarchy, product additivity singles out the Rényi family, while internal additivity, or product additivity together with arithmetic mean-value composition, singles out Shannon entropy. The characterization is constructive and yields new admissible entropy and divergence families, including integral-transform examples.

This paper studies trajectory-wise estimation of generalization error for primal–dual algorithms in non-smooth regression. Motivating examples include \(\ell_1\)-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle–Pock algorithm and related primal–dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.

Three-player Auction Bridge is a finite imperfect-information game in which dynamic partnerships create a distinctive interaction between scoring, strategic incentives, and payoff distribution. This paper develops a unified statistical and game-theoretic framework to evaluate a traditional scoring rule against a modified mechanism designed to improve strategic incentives. We first identify a structural defect in the original scheme: a bid-invariant slam bonus can make lower contracts strictly more attractive than higher ones. A bid-dependent correction removes this incentive distortion. We then compare the two schemes using higher-moment analysis, fairness measures, Nash equilibrium analysis, and General-Sum Counterfactual Regret Minimization (GS-CFR). The results show significant differences in the shape of the payoff distributions and reveal that, although both schemes remain highly balanced across physical seats, the modified scheme substantially increases inequality across strategic roles. The game is shown to be neither zero-sum nor constant-sum, motivating a general-sum rather than minimax analysis. Full-game GS-CFR further indicates a substantial increase in the bidder's payoff under the modified scheme, accompanied by a reduction in defender payoff and a shift from separating to partially pooling bidding behavior. These findings demonstrate that scoring-rule design can fundamentally reshape incentives, payoff distribution, and information transmission in imperfect-information games. The proposed framework provides a systematic approach to evaluating such mechanisms from both statistical and strategic perspectives.

We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $\rho_n^*$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion $\rho_n^* = n^{-1} - Cn^{-4/3} + o_n(n^{-4/3})$ for an explicit constant $C$ related to the Airy function. The minimax risk is attained by Bernoulli randomization with a nonlinear shrinkage estimator. Our results show that standard procedures such as complete randomization with difference in means are only minimax optimal up to first order in $n.$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.

In the era of modern data science, the rapid proliferation of high-dimensional and functional datasets has fostered increasing interest in the investigation of paradigm shifts and structural breaks. Unlike classical univariate time series, structural changes in functional data need not occur simultaneously across the entire domain; instead, they may emerge locally, producing heterogeneous distortions across the underlying functional structure. The patterns of instability often exhibit sparsity, where it is not known a priori which specific parameters are undergoing a transition. However, in functional contexts, these shifts are often "localised". The difficulty lies in the high dimensionality of the parameter space, where the signal-to-noise ratio may be low for individual components, necessitating the aggregation of information across dimensions to detect a global change. This paper addresses the problem of detecting structural shifts in a functional time series from a Bayesian perspective. We have developed various novel methodologies that capture the inherent structural distortion in a sequence of random functions, both individually and simultaneously. The formulation of the problem is based on the state-space representation of a functional time series. Efficient Blocked Gibbs Sampling algorithms have been proposed to identify these locations accurately. Further, we demonstrate the effectiveness of our methods on several financial and temperature datasets.

We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the unmasking growth complexity ({UGC\xspace}). Its local increments directly control Kullback–Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {UGC\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to certified-optimal samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {UGC\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {UGC\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetilde{\Omega}(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cramér-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.

Covariate-adaptive randomization procedures are widely used in clinical trials to improve covariate balance. In modern applications, experimenters often have access to many covariates, motivating the need for a theory of covariate-adaptive randomization procedures with a diverging number of covariates. This paper studies two classes of covariate-adaptive randomization procedures, referred to as imbalance-efficient covariate-adaptive randomization and imbalance-robust covariate-adaptive randomization, when the feature dimension diverges. We derive convergence rates for the imbalance of the covariates used in randomization. For both procedures, the imbalance is of a smaller order than that under complete randomization when the feature dimension is $o(n)$, whereas it is of the same order when the feature dimension is $\Omega(n)$. For imbalance-robust covariate-adaptive randomization, we further establish the asymptotic properties of the imbalance of additional covariates and use these results to derive the asymptotic distribution of the difference-in-means estimator for the average treatment effect and construct asymptotically valid confidence intervals. Furthermore, we provide extensive numerical and empirical studies to illustrate the practical relevance of our theoretical results.

The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with $d$ the dimension and $N$ the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order $(d\log N/N)^{1/2}+d^2\log N/N$ with probability at least $1-10N^{-1}$. For single Gaussian marginals estimated by their sample moments, it is of order $d^2/N$ in expectation. For the regularized empirical Monge map based on an $\epsilon$-inflation of the sample covariance, the mean squared error is of order $d^2\{(1+\epsilon^{-1})/N+\epsilon^2\}$; as soon as $N$ is a sufficiently large multiple of $d$, it reduces to $d^2/N+d\,\epsilon^2$, up to an exponentially small term. These estimates are illustrated numerically.

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent~$\frac{2}{d+2}$ obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

This study develops a weighted framework for measuring the discrepancy between two nonnegative lifetime distributions through cumulative past extropy. We propose two measures, referred to as the weighted cumulative past extropy inaccuracy (WCPEI) and the weighted cumulative past extropy Kullback-Leibler divergence (WCPED). The generalized weight function is considered in this study. We investigate a number of theoretical properties of these measures. Empirical distribution function-based nonparametric estimator is subsequently constructed for the weighted cumulative past extropy inaccuracy ration (WCPEIR). Its finite-sample behavior is studied through Monte Carlo simulation experiments for different sample sizes. To illustrate the practical relevance of the WCPED, two applications are considered. First, an extropy-based goodness-of-fit procedure for testing uniformity is developed using the proposed divergence measure. Its power is then compared with that of several established uniformity tests under a variety of alternatives. Second, an image analysis application is presented in which the proposed measure is employed to assess changes in the distributions of pixel intensities when the image resolution is altered. The framework is further extended to a dynamic setting by conditioning on the lifetime information available up to a specified time point. This leads to the dynamic weighted cumulative past extropy inaccuracy (DWCPEI) and dynamic weighted cumulative past extropy divergence (DWCPED). Their theoretical properties are derived, and the corresponding nonparametric estimation procedures are proposed. The finite-sample performance of these estimators is evaluated through simulation studies using R software.

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization – the implicit-function sensitivity of the entropic OT plan to the cost – together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent. The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation. T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).

We study the laws of large numbers for the largest eigenvalues among all principal minors of Wishart matrices and deformed GOE matrices. We propose a new method based on identifying the deterministic sets to which the random sets formed by suitably normalized principal minors converge in Hausdorff distance, thereby reducing the original extreme-value problems to finite-dimensional convex optimization problems. We demonstrate the effectiveness of this method in regimes not covered by the existing second-moment arguments in \cite{cai2021asymptotic,hu2023extreme}. For deformed GOE matrices with fixed minor size \(k\), we determine the limit for every diagonal variance \(a>0\) and identify a phase transition at \(a=2\). Above the transition, the limiting constant satisfies an explicit recursion with no close-form expression, and the optimizers exhibit a nested hierarchical structure, thereby resolving the case left open in \cite{cai2021asymptotic}. For Wishart matrices with general sub-Gaussian entries and fixed \(k\), we characterize the limit through an entropy-constrained deterministic convex set. When the entries are standard Gaussian, we solve the resulting optimization problem explicitly and obtain the exact value of the limiting constant.

We consider the classical additive measurement-error model $X=Y+Z$, where the latent random variable $Y$ has unknown distribution $F_Y$ and the error $Z$ has a known distribution. We develop direct estimators for three functionals of $F_Y$: (i) $F_Y(x)$ at continuity points; (ii) interval probabilities $F_Y(y)-F_Y(x)$ when $x<y$ are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require $F_Y$ to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.

Bayesian inference for A/B testing is a family of prior and stopping-rule configurations with fundamentally different statistical properties, but it is often discussed as a single method, and no systematic overview exists. This paper organizes common configurations into a three-tier hierarchy: 1) posterior coherence with no error control, 2) false positive rates bounded under continuous monitoring via Bayes factor stopping, and 3) false discovery rate control and calibrated shrinkage via empirical Bayes. Many commercial platforms operate at the lowest tier by default. We show that Bayes factor stopping is near-optimal for a broad class of cost functions, including most proposed in the A/B testing literature; because the same rule also controls the false positive rate, the choice between a decision-theoretic and a frequentist formulation is largely one of parameterization. Furthermore, the empirical Bayes prior is the only path to the third tier, but winner-selected corpora, pooled programs, and heterogeneous metrics can each prevent calibration regardless of corpus size. Simulations against group-sequential and always-valid frequentist baselines show that flat-prior posterior stopping exactly reproduces naive peeking, that a well-calibrated empirical Bayes prior achieves the lowest estimation error, and that expected-loss stopping minimizes regret only when shipping a null-effect variant is nearly free. Error rates, estimation accuracy, and regret are all different risks, and the appropriate method follows from the risks an experimentation program needs to control, not the other way around.

We study estimation of the drift matrix in a continuously observed high-dimensional Ornstein-Uhlenbeck process when the drift is exactly or approximately low rank. In this setting, exact low rank induces non-stable directions and hence a non-ergodic regime, resulting in a poorly conditioned empirical covariance matrix. To address this difficulty, we introduce a Weighted Nuclear Elastic Net Estimator that combines ridge regularization with a nuclear-norm penalty expressed in the empirical likelihood geometry. Under a general diagonalizable spectral framework, we establish oracle inequalities relative to arbitrary low-rank comparison matrices. For near low-rank drifts, the approximation error is naturally measured through the singular-value decay of the drift after weighting by the regularized empirical covariance. The stochastic term is controlled by self-normalized martingale arguments under appropriate choice of the tuning parameter. For a symmetric positive-semidefinite exact low-rank model, we verify the empirical-curvature condition required to translate the weighted bound into a Frobenius-norm bound. With an appropriate choice of tuning parameters, the resulting estimator satisfies, up to a logarithmic factor, the standard rank-$r$ matrix-estimation scaling $r d/T$: specifically, its squared Frobenius error is of order $r d\log(T)/T$ with high probability, under an explicit dimension-horizon condition.

We develop a stable convergence theorem for partial sum processes on sample-size dependent stochastic bases. The result allows multidimensional semimartingale limits that have conditionally independent increments and both a continuous and discontinuous martingale part. Motivated by infill asymptotics, it complements classical Gaussian stable limit theorems and supports applications to likelihood based statistical inference.