Papers

We study distribution-free sequential changepoint detection for independent observations with unknown and unrestricted pre- and post-change laws. We build on the conformal test martingales and associated e-detectors of Vovk(2021), which control the probability of false alarm (PFA) and the average run length (ARL) respectively. The majority of these works focus on validity, with statistical efficiency usually left for simulations. We develop a comprehensive theory of how conformal p-values behave under non-exchangeable data with a changepoint at an unknown time $T$. We use this to analyze the post-change growth and resulting detection delay of conformal martingale methods, and prove that the standard existing methods are suboptimal for PFA and ARL control, and can lead to delays that are $\Omega(T)$ and $\Omega(\sqrt{\text{ARL}})$ respectively. We propose different conformal e-processes and e-detectors that are provably minimax optimal, with delays $\Theta(\log T)$ and $\Theta(\log \text{ARL})$ respectively, and have much shorter delays in simulations.

Many inference problems are constrained from two sides: forward information and a second constraint on outcomes. Dual Boundary Condition Inference (DBCI) multiplies them and renormalises. The rule is not new: it is the product-of-experts form and the unit-exponent member of the logarithmic-pooling family. The paper contributes a classification and reducibility characterisation. The algebra does not fix the exponent at which the inputs combine: given the statistic log p_fwd + log p_bwd, which already builds in the product form, maximum entropy supplies the family (p_fwd p_bwd)^lambda and selects no member. If the inputs are read as two equally weighted opinions, three requirements select lambda = 1/2: unanimity preservation, consistency under a common Bayesian update, and minimal symmetric Kullback-Leibler divergence. If they are read as separately applied factors, two requirements select lambda = 1: a neutral second input must leave the first unchanged, and an update to one input must pass through unchanged. For a rank-one intermediate projective measurement, the Aharonov-Bergmann-Lebowitz (ABL) rule realises this form with no extra parameter. Every positive exponent orders outcomes identically, so the choice is invisible to picking the likeliest outcome, but not to scoring a class by total mass. Under the relevant identifications, the same product form gives Bayes' rule, while a final effect proportional to the identity returns the ordinary forward Born probabilities. The second result is a Two-Boundary Reducibility Criterion: DBCI factors through the forward boundary exactly when the effective backward boundary, up to positive rescaling, does. So a fixed prior or model-fixed structural constraint adds no distinctions between cases sharing the same forward boundary, though it may encode substantial information and still materially affect the result.

We introduce Rolling Conformal Prediction (rolling-CP), a distribution-free predictive inference method for the setting of sequential model training. Specifically, given a data stream $(X_1,Y_1),(X_2,Y_2),\dots$, at each time $n$ the trained model may depend on the observed history $\{(X_i,Y_i)\}_{i<n}$. This setting arises naturally in modern sequential training, including one-pass training over massive datasets and continual fine-tuning or test-time adaptation of language models during deployment. Rolling-CP first calibrates each incoming observation against the current predictor and then rolls it into future training. In this way, we avoid the need for data splitting. Remarkably, although the models at times $n=1,2,\dots$ may have entirely different properties and accuracy levels, for exchangeable data it is nonetheless possible to establish a guarantee of marginal coverage, with a familiar universal factor-two guarantee (a worst case guarantee of $1-2\alpha$ coverage, as compared to the target level $1-\alpha$), without any assumptions of stability or any restrictions on the model training process. For i.i.d. data streams, we further prove high-probability training-conditional validity uniformly over time; under stability conditions, coverage guarantees sharpen towards $1-\alpha$. Numerical experiments on sequential regression, multiclass SGD, and one-pass neural-network training further demonstrate the practical effectiveness of rolling-CP.

Birnbaum's theorem states that the sufficiency and conditionality principles jointly imply, and are implied by, the likelihood principle. To enable formal certification of the results in Lean, we make the conditionality relation explicit through a finite-sample formulation based on \cite{Evans2013}. We reformulate the theorem as a statement about classes of statistical procedures that preserve statistical relations, and show that the proof reduces to the propagation of evidential equality along chains of conditionality-related inference bases (experiment–observation pairs). The main results are illustrated by Venn diagrams, and a worked finite example exhibits two pairs of inference bases: one related by conditionality but not by sufficiency, and the other related by sufficiency but not by conditionality. For finite parameter spaces with at least two points, the class of likelihood-invariant procedures with codomain $[0,1]$ is strictly contained in the class of sufficiency-invariant procedures.

We consider simultaneously testing hypotheses about thousands of units, e.g., genes or proteins, where each unit yields a handful of replicate measurements and we test whether its mean is zero. A widely used approach in genomics, implemented in the limma software, borrows strength across units to learn the distribution of the unit-specific variances, then computes moderated t-statistics. Here we develop the first procedures using moderated t-statistics that (i) control the false discovery rate (FDR) in finite samples under independence across units and null group invariance, without assuming limma's hierarchical model, and (ii) match the power of an oracle local false discovery rate procedure in a sparse asymptotic regime under limma's working model. Benjamini-Hochberg (BH) with standard t-test p-values has asymptotically zero power in the same regime. Our approach learns the variance distribution from statistics that depend on each unit's data only through its orbit under a compact group of transformations that preserves the null distributions, such as sign flips or orthogonal rotations. We then calibrate the resulting moderated t-statistics against group-transformed statistics pooled across units to obtain compound p-values, which we use with BH and a close variant. For small finite groups, we also construct a Selective SeqStep+ procedure using the same learned scores. Our approach extends to two-sample tests and tests of linear model coefficients.

Commonly, classifiers and monitoring procedures are trained from labeled data by optimizing an objective such as the misclassification rate. This may lead to unrepresentative conditional distributions of the outcome (the labels) given important external variables, different from the conditional laws in the population. We show how to modify any given threshold-type classifier resp. monitoring rule to achieve representative conditional label prediction by using adapting the threshold to a covariate $Z$ (the context) to distribute sensitivity while maintaining the false alarm rate. In case that the alarm event is unknown, this approach also allows to (approximately) infer the event in terms of a thresholding rule. The approach is implemented by a computationally cheap nonparametric estimation procedure, and its properties are studied in terms of nonasymptotic error bounds and asymptotic distribution theory including empirical process theory. These results allow to construct uniform confidence bands, functional hypothesis tests and change-detection procedures. For the well known FICOS credit scoring example, often used in interpretable machine learning, threshold adaptation leads to an easily interpretable decision rule which can compete with state of the art methods including transformers, in terms of common classification metrics.

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate $n^{-1}$, matching the fully compact case and improving over the less favorable $n^{-1/2}$ rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove $\sqrt{n}$-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a $\sqrt{\log n}$ factor. Simulations support the theory.

In sparse random matrices, spectral outliers (eigenvalues and singular values located away from the bulk) emerge due to degree fluctuations: high degrees inflate the operator norm, while low column degrees reduce the least singular value. As proved by Feige and Ofek (2005) and Le, Levina, and Vershynin (2017), degree regularization enforces concentration at the expected norm scale. However, precise bounds incorporating the cutoffs remain unexplored and challenging since regularization introduces dependencies among entries. For the first time in the literature, we provide variance- and cutoff-dependent bounds for extreme singular values and eigenvalues of regularized inhomogeneous random matrices. In the absence of regularization, our lower bound for the least singular value matches the same leading constant obtained by Brailovskaya and van Handel (2024). Moreover, our error term vanishes under the milder condition $d/\log N\to\infty$, as opposed to their stronger requirement $d/(\log N)^4\to\infty$. A key ingredient is to extend spectral radius bounds for nonbacktracking matrices to the dependent setting. We build on approaches for independent cases established by Benaych-Georges, Bordenave, and Knowles (2020), as well as Dumitriu and Zhu (2024), and carefully handle edges traversed only once. Our proof framework separates deterministic spectral comparisons from probabilistic estimates: once Loewner inequalities and columnwise variance controls are established, the remaining probabilistic analysis boils down to verifying the graph moment conditions formulated in this paper. We hope this framework can be extended to handle general random matrices with more complex dependencies.

Estimating the Hüsler-Reiss precision matrix is a fundamental problem for statistical inference in multivariate extremes. In high-dimensional settings, the number of unknown parameters grows quadratically with the dimension, making regularisation indispensable. Existing approaches regularise the estimation problem by exploiting sparsity. In this paper, we consider an alternative structural assumption, namely that the Hüsler-Reiss precision matrix is block-structured. To estimate such a matrix, we introduce a new regularisation framework based on a convex fusion penalty. By encouraging rows and columns to merge, this approach provides a parsimonious representation of the precision matrix, allowing for the simultaneous estimation of its coefficients and the underlying partition of the variables. The resulting convex optimisation problem is solved by an efficient algorithm combining gradient-based updates with progressive fusion steps. We establish non-asymptotic concentration bounds for the empirical weights entering the penalty and prove consistency of both block recovery and precision matrix estimation under suitable regularity conditions. Numerical experiments demonstrate that our methodology accurately recovers the latent block structure while accurately estimating the Hüsler-Reiss precision matrix across various configurations, illustrating the practical benefits of fusion-based regularisation for multivariate extremes. These benefits are also demonstrated by applying the proposed method to foreign exchange data.

A functional central limit theorem for locally stationary time series taking values in a separable Banach space $B$ is established. The result does not require type-2 or cotype assumptions and therefore covers spaces central to functional data analysis, including $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients, and a bracketing entropy condition controlling the infinite-dimensional tails, the centered and rescaled partial sum process converges weakly in $D([0,1],B)$. The limit is a centered $B$-valued Gaussian process whose covariance is given by the integral of the local long-run covariance, interpreted as an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion, connecting the Banach-space limit to the familiar scalar locally stationary structure. As an application, we derive a self-normalized CUSUM procedure for detecting changes in the mean of linear projections of Banach-valued observations, yielding a pivotal asymptotic null distribution. The finite-sample behavior is illustrated through Monte Carlo experiments and exploratory applications to EEG recordings and daily temperature curves. Examples based on the Faber-Schauder system in $C([0,1])$ and on a $p$-Laplacian model in $W^{1,p}_0([0,1])$ demonstrate how the entropy condition can be verified.

Estimating two-sample comparison functionals within heterogeneous groups is challenging when outcomes are observed only for a small labelled subset while covariates are widely available. We develop semi-supervised estimation of group-specific two-sample comparison functionals and their prespecified aggregates. The proposed estimator combines single-index conditional imputation from labelled cross-arm pairs with averaging over all available covariate pairs. To handle index-score densities that may vanish at attainable endpoints, we use centred stabilization, with a smooth stabilizer in the profile criterion and a shrinking lower truncation of the kernel denominator (the hard floor) in the point estimator. The outer average retains all covariate pairs. Under a correctly specified single-index conditional-mean model and the stated regularity conditions, we establish consistency and asymptotic normality for bounded comparison functionals. A four-role original-observation influence representation accounts for sample overlap and identifies the variance component that unlabelled covariates can reduce relative to supervised estimation. We further establish conditional validity of observation-level multinomial perturbation resampling with complete refitting and shared labelled counts, distinguishing ideal variance consistency from finite-replicate variance estimation. Simulations show small bias and lower mean squared error than supervised estimation across the point-estimation settings, together with near-nominal coverage in the perturbation experiment. A descriptive NHANES 2011-2018 illustration comparing serum creatinine by hypertension history across age groups yields point estimates similar to their supervised counterparts and smaller semi-supervised standard errors for all reported comparisons.

Memory constraints in artificial intelligence motivate accurate function approximation with fewer parameters. We study looping, which repeatedly composes one update function with shared parameters; each output becomes the next input. A looped Transformer, for example, reuses one block, whereas its conventional untied counterpart uses separately parameterized blocks. We compare their parameter requirements for a given worst-case approximation accuracy, or equivalently, their approximation accuracy under a common budget limiting distinct trainable coefficients. We then ask whether this representational parsimony improves statistical accuracy. For general likelihood models, we establish an upper squared Hellinger risk bound for looped sieve maximum likelihood and a minimax lower bound for the jointly tuned untied family. Further loop iterations improve the approximation bound without adding parameters, while increasing computation and the fitted-class complexity bound. For targets of known Hölder smoothness, looped residual feedforward networks and post-layer-normalized Transformers attain the minimax polynomial rate up to logarithmic factors with a fixed number of bounded real parameters. At sufficiently large fixed budgets, looped worst-case risk vanishes while optimal worst-case untied risk remains bounded away from zero. The loop-to-untied risk ratio also tends to zero under specified growing-budget conditions. Regression, binary response, and energy-based generative models illustrate the theory.

Quantum state tomography provides a foundation for characterizing state preparation and predicting measurement outcomes. We characterize the minimax expected trace norm risk of quantum state tomography over classes with polynomial spectral decay under randomized nonadaptive single copy measurements. Under an explicit sufficient block size condition for the two layer periodic Clifford ensemble, we establish statewise expected trace norm oracle inequalities for operator norm minimum distance and measurement weighted projected least squares, with bounds that depend on each state's actual spectral tail, while the estimators require no prior knowledge of the state's structural characteristics, including its rank, eigenbasis, or spectral structure. Over these polynomial spectral decay classes, both estimators attain the minimax expected trace norm rates up to constants. As a direct consequence, the same estimators attain the minimax rate over every fixed rank constrained class, while remaining adaptive to the unknown structural characteristics of the state. For admissible logarithmic block sizes, the periodic Clifford measurements attain these guarantees with logarithmic elementary gate depth. The main technical ingredient is a uniform bound on the fourth moment of projectors for the periodic Clifford ensemble. The bound is sharp in dimension and holds for arbitrary test states, including those entangled across blocks. Covariance concentration and positivity yield adaptive oracle bounds, while a shared rotation information argument on spectral tail packings establishes matching lower bounds. We also provide polynomial time reconstruction algorithms, numerical comparisons, and Lean formalizations of all labeled theoretical results.

A classical question in statistics is which observable quantities to condition on when drawing inferences about unobservable targets. For conformal prediction in nonparametric uncertainty quantification, standard marginal validity offers limited resolution at the prediction values on which decisions are based, and fully conditional guarantees with respect to the covariates are provably unattainable. We address this gap by introducing a prediction-based conditioning framework that we refer to as Prediction-Interval-Conditional Prediction Intervals (PICPIs). Formally, a PICPI is an interval $I$ satisfying a self-consistency condition: $$\mathbb{E} [Y \mid p(X) \in I] \in I,$$ for predictive model $p$, contextual covariate $X$, and outcome $Y$. Thus, an interval simultaneously defines a stratum of prediction values and certifies that the mean outcome in that stratum lies in the same interval. This self-consistency condition yields data-adaptive strata without altering the original prediction. Such intervals can be constructed using practical algorithms. Under regularity of the prediction distribution, the constructed intervals cover all but an arbitrarily small fraction of prediction values and have widths that decrease at rate $n^{-1/3}$, up to logarithmic factors and the prediction error. Moreover, identifying these locally calibrated intervals can, in turn, inform downstream decision-making. We derive inference procedures for PICPIs in probabilistic prediction and multi-class classification, accompanied by theoretical guarantees. Empirical results are provided that compare PICPIs with existing interval-based baselines.

The exact distribution of the sample variance for $n=3$ is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with $S_3$ symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at $Y=1/4$, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about $10^{-3}$, an accuracy Royen's universal series reaches at about a hundred terms.

In this paper, we study nonasymptotic $L^p$ error bounds for interval length and conditional coverage in split conformalized quantile regression (CQR). Our bounds rely on local regularity conditions and accuracy guarantees for the estimated quantiles. We further instantiate our bounds for quantile regression with sparse ReLU neural networks. We also consider covariate shift, where the calibration and test covariates have different distributions, and derive nonasymptotic bounds for this setting. We obtain matching minimax upper and lower bounds in expectation for two constructed fixed-score calibration benchmarks under known covariate shift. The bounds match for every $p\in[1,\infty]$ in the scalar problem and for finite $p$ in the $K$-threshold problem; for the latter, a high-probability minimax lower bound holds for every $p\in[1,\infty]$.

A well-known phenomenon in statistical analyses of populations of phylogenetic trees in the Billera-Holmes-Vogtmann space is that the topology of the Fréchet mean tree can contain multifurcations (i.e., internal nodes with more than two children), which raises the practical question of whether this reflects a population-level branching structure (hard polytomy) or merely sampling variability in the data (soft polytomy). This is an instance of the more general phenomenon of "stickiness" in non-Euclidean statistics, whereby the sample Fréchet mean in certain non-positively curved stratified spaces becomes permanently trapped in a lower-dimensional stratum. In this work, we identify a particular multidimensional random walk embedded within the Fréchet mean process, and we show that the time at which stickiness occurs is determined by the largest last-passage time above zero of the coordinates of this random walk. Using this representation, we develop a fully nonparametric procedure for estimating the probability that trifurcations in a sample Fréchet mean tree will bifurcate at some future time if more observations are collected. Lastly, we apply our methodology to a problem in phylogenetics where we consider whether an observed trifurcation in the species tree of primates, glires, and tree shrews is genuinely trifurcated at the population level.

In the planted clique problem, one observes either an Erdős–Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = \Theta(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = \lambda\sqrt{n}$ planted in a random location with $\lambda > 1 / \sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande–Montanari (2015) that succeeds when $\lambda > 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $\lambda$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{\omega + o(1)}$ for $\omega$ the constant of square matrix multiplication and succeeds when $\lambda > 0.3320$; under the folklore conjecture that $\omega = 2$, this runs in the nearly-linear time of the algorithm of Deshpande–Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon–Krivelevich–Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $\lambda$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $\lambda$.

We propose a kernel-based estimation framework for Multivariate Generalized Gamma convolutions (MGGC), a class of probability distributions on R d + . MGGC are useful for risk modeling e.g. Their densities are in general not explicit. The construction relies on a new Stein operator for the MGGC class, derived from a partial differential equation satisfied by the characteristic function and expressed as an integral against the underlying Thorin measure. This leads to construct a new kernel and an estimation procedure in the derived reproducing kernel Hilbert space (RKHS); with consistency results. Numerical experiments illustrate the practical behaviour of the proposed estimator.

Two-point correlation functions are a standard summary statistic in cosmic shear and large-scale-structure analyses. Their values are, however, constrained: non-negativity of the underlying power spectrum restricts any admissible sequence of correlation coefficients $(r_1,\ldots,r_N)$ to a bounded convex region, described in the one-dimensional case by the recursive interval geometry of Schneider & Hartlap (2009). Their formalism introduces an affine variable $x_n$ that maps the admissible interval for $r_n$, given $r_1,\ldots,r_{n-1}$, to $[-1,+1]$. We identify $x_n$ with the partial autocorrelation coefficient $\alpha_n$ of the associated positive Toeplitz correlation matrix, a classical quantity in time-series analysis. The partial autocorrelations thus provide natural coordinates on the admissible region, each $\alpha_n$ varying independently in $[-1,+1]$. This identification makes the standard partial-autocorrelation toolbox directly applicable. The admissible intervals for $r_n$, cumbersome to obtain at higher order in the original determinant-based formalism, now follow from an $\mathcal{O}(N^2)$ recursion at arbitrary order. The inverse hyperbolic tangent used in the quasi-Gaussian likelihood construction of Wilking & Schneider (2013) becomes Fisher's $z$-transformation of partial autocorrelations, providing a classical statistical interpretation of its empirical Gaussianising effect. Numerical experiments illustrate the practical use of these natural coordinates in the tested one-dimensional settings. Higher-dimensional isotropic constraints require additional geometric input and lie outside the direct scope of this identification.

We study the problem of nonparametric estimation of the interaction function in particle systems whose interaction enters through the diffusion coefficient. The particle system is observed discretely over a fixed time interval, and the interaction function is assumed to belong to a suitable nonparametric function class. We propose an estimation procedure based on empirical risk minimization over finite-dimensional sieve spaces, introducing a computable contrast function that exploits the local volatility of the observed trajectories. We derive a general non-asymptotic error bound under the intrinsic norm associated with the statistical problem and, focusing the analysis on multiresolution wavelet approximation spaces, we obtain explicit polynomial convergence rates in two distinct regimes, depending on whether the particle or discretization error dominates. Finally, we establish convergence rates for the classical $L^2$-norm.

We study the problem of adversarially robust PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies every perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the optimal $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/\epsilon + \log(1/\delta)/\epsilon \big)$ in the realizable setting and $\mathcal{O} \big( d/\epsilon^2 + \log(1/\delta)/\epsilon^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs no additional distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call binomial-bagging. We believe that binomial-bagging and its analysis may be of independent interest.

Shah and Peters showed that a conditional independence test with finite-sample or uniformly-controlled level has only trivial power against any alternative. Most practical tests, however, only claim pointwise asymptotic level. There have been incorrect claims in the literature of conditional independence tests with pointwise asymptotic level and consistency against any alternative; whether such a test actually exists has remained open. We resolve this question. Even restricting to hypotheses with a bounded density on compact subsets of Euclidean spaces, for any sequence of (possibly randomized) tests with pointwise asymptotic level $\alpha$, for every $\varepsilon > 0$ there is a conditionally dependent distribution where the test's limsup power is at most $\alpha + \varepsilon$. This remains true when the level control is required only over conditionally independent distributions with a continuous density and a uniformly continuous conditional law, and the conditionally dependent distributions have smooth densities. On the other hand, the extreme limitation of trivial power against any alternative for uniform-level tests does not apply to tests with pointwise level. We exhibit a test that, without regularity assumptions, has pointwise asymptotic level $\alpha$, strictly higher power for all alternatives, and power tending to one for all distributions whose dependence exceeds a chosen scalar threshold.

Object-valued time series, including distributions, covariance matrices, networks, and compositions, lack the linear structure required by conventional autocovariance-based definitions of long memory. We develop an intrinsic framework for defining and estimating long memory in metric spaces of negative type. An isometric embedding yields a centered Hilbert-valued process, and the trace of its lag-covariance operator provides a signed, additive measure of temporal dependence. Crucially, this trace equals the difference between the marginal mean pairwise distance and expected lagged distance, so the framework and its estimators use only distances between the original objects. We define the memory parameter through the nonsummable decay of this trace and show that it coincides with the usual parameter in Hilbert-valued settings. Estimation uses Bartlett aggregates of distance-based lag measures. Estimating the common marginal distance from the same dependent sample induces a common-centering bias in these aggregates. We derive its finite-sample form and propose iterated block-difference corrections, log-ratio and multi-bandwidth log-slope estimators, and a localized self-consistency refinement. We establish consistency, document substantial bias reduction in simulations, and find long-memory evidence in foreign-exchange return distributions and U.S. electricity-generation compositions.

Large-scale multipurpose cohort studies and biobanks often omit covariates needed for specific downstream analyses. We study target-population inference for generalized linear regression when key covariates are completely absent from the target data but observed in a related source population. Standard missing covariate methods are not directly applicable because they require at least partial observation of the covariates in the target population. We develop a doubly robust transfer learning framework under a sub-population shift assumption, which allows the distribution of the observed outcome and covariates to differ between populations while requiring the conditional distribution of the missing covariates given the observed variables to be shared. Unlike linear regression, nonlinear target estimating equations require parameter-indexed conditional functionals of the missing covariates rather than a fixed collection of low-order conditional moments. Our estimator combines importance-weighted source estimating equations with imputation of these conditional functionals. Under the identifying assumption, the estimator remains consistent when either the density-ratio model or the conditional-covariate model is correctly specified. Under regularity conditions, it is root-$n$ consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

Let $\Pi$ denote the independence copula and $M,W$ the upper and lower Fréchet–Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $\Pi$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2(C,\Pi)\leq 1/10$, with equality if and only if $C\in\{M,W\}$. We also determine explicitly the optimal Monge map from $\Pi$ to $M$. The proof is constructive and combines the optimal transport from independence to the diagonal, a sharp convex-order inequality for 1-Lipschitz functions, the conditional convex order, and a coupling construction based on conditional comonotonicity and the supermodular order. As a consequence, we obtain a normalized Wasserstein-based dependence measure that characterizes independence and attains its maximal value exactly for comonotone and countermonotone dependence.

This paper establishes time-uniform accuracy guarantees for deterministic square-root ensemble Kalman filters in linear–Gaussian state-space models under perfect-model dynamics. For hyperbolic, detectable systems, we prove that the ensemble means and covariances approximate their Kalman filter counterparts uniformly in time, with high probability and without inflation or resampling. The required ensemble size depends on the effective rank of the initial covariance and the dimension of the unstable subspace, rather than on the ambient state dimension. We also analyze a localized square-root ensemble Kalman filter for weakly coupled spatial systems. We show that the stabilizing Kalman covariance inherits spatial decay from the dynamics and derive accuracy bounds that separate sampling error from localization bias. For the localized filter, the required ensemble size depends on the local intrinsic dimension and only logarithmically on the number of spatial blocks, while the localization bias scales with the interaction strength. Our analysis combines stability and forgetting for possibly singular Riccati recursions, nonasymptotic covariance concentration, and perturbation estimates for localized covariance dynamics.

For Gaussian location mixtures with identity covariance, we prove that the nonparametric maximum likelihood estimator of the mixing distribution is unique for Lebesgue-almost every dataset, for every sample size and dimension. In particular, uniqueness holds almost surely whenever the joint distribution of the observations is absolutely continuous. The proof uses the finite-support theorem for Gaussian location NPMLEs and almost-everywhere differentiability of the optimal log-likelihood. Wherever this optimal value is differentiable, all NPMLEs have the same density derivatives at the observations. A minimal linear dependence among Gaussian evaluation vectors then rules out distinct solutions. An appendix extends the result to arbitrary fixed, known, observation-specific positive-definite covariances.

The high-frequency spectral roll-off of continuous-time ARMA (CARMA) models can magnify the effect of narrow-band disturbances when aliasing is weak, making standard maximum-likelihood Whittle estimation sensitive to affected ordinates. We show that a logarithmic transformation $r_k = \log \rho_k$ converts the Whittle scale problem into a Gumbel location problem, connecting robust spectral estimation to the classical minimax theory of Huber and Rieder. A piecewise centering correction—closed-form for small $b$, implicit closed-form for the practitioner range—preserves Fisher consistency for any clipping level without numerical optimisation. Clipping the Gumbel score symmetrically and transforming back yields a two-sided clipped Gumbel score (the standard Rieder–Hampel bounded-influence form) whose normalised influence curve is proved locally asymptotically minimax under shrinking gross-error contamination. The efficiency loss is quantified by a single scalar $K(b)$: at $b = 1.5$, only $16\%$ nominal asymptotic variance overhead. In the stated AR(2) Monte Carlo design, the sample-size trends are compatible with the asymptotic rate, and at $b=1.5$ the bias reductions are about $40\%$, $90\%$, and $96\%$ for $a_1$, $a_2$, and $\lambda$, respectively.

Modern conditional generative models face significant challenges when learning complex covariate dependencies. While sufficient dimension reduction (SDR) provides a principled approach to compress these dependencies, traditional SDR frameworks were not formulated for conditional generation. To bridge this gap, we propose Belted Engression, a unified and architecturally parameter-efficient framework for generative distributional regression. Our approach establishes an end-to-end compress-then-generate paradigm driven by sufficient representation learning, embedding a structural bottleneck into the generative architecture. Theoretically, we prove that the standard SDR condition is equivalent to a law-preserving generative factorization, which is achieved at the global optimum of the population Belted Engression objective. Furthermore, by uncovering a localized Bernstein-type control for the energy-score loss, we establish finite-sample convergence rates that are sharper than those of existing results. We also prove that this belted architecture is strictly smaller, operating with an asymptotically vanishing parameter count relative to the unstructured baseline. Extensive simulations and real-world applications demonstrate that Belted Engression achieves superior distributional prediction and SDR recovery with fewer trainable parameters.