Papers

Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(\phi_n\), and let \(N_n=\phi_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur–Rao–Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental–boundary–persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.

We consider sparsely observed functional data generated by a latent Gauss-Markov process modeled through a linear stochastic differential equation (SDE). Motivated by an application to tumor growth data, we allow for outcome-dependent dropout, where sampling terminates once the most recent observation exceeds a prescribed threshold. Such observation schemes arise naturally in longitudinal studies and violate the fundamental missing completely at random assumption commonly imposed in the analysis of partially observed functional data. We develop a likelihood-based estimation framework that, in contrast to existing moment-based methods, avoids the bias induced by outcome-dependent dropout. We establish convergence rates for the proposed estimators and show that they are mini-max optimal up to logarithmic factors, with the diffusion coefficient admitting a faster rate than the drift coefficients under a random initial condition for the SDE. An application to tumor growth data further demonstrates the advantages of a fully probabilistic functional data model for tasks beyond second-order inference, including prediction bands, first-passage times, and tumor-age estimation.

The classical semiparametric convolution theorem characterizes the limiting distributions of regular estimators of pathwise differentiable parameters. We extend this analysis to directionally pathwise differentiable functionals. Regularity along a subspace or convex cone forces the directional derivative to agree there with a bounded linear functional, and every such estimator has a Gaussian convolution factor determined by that functional. We characterize the relevant subspaces, compare their efficiency bounds, and identify the influence functions of efficient estimators. These comparisons show when a stronger regularity requirement increases the variance bound. Under an additivity condition, the Gaussian limit experiment separates into a linear estimation problem and a nonlinear one. Their minimax risks add under squared error; for more general losses, the Gaussian component enters through convolution of the loss. Applications to treatment values, instrumental-variable bounds, and calibration give explicit efficient influence functions, compare existing procedures, and establish feasible attainment in finite-stratum models.

We study exact sign recovery for the PCA connectivity construction of Ding and He (2004), based on the first K-1 principal components. For any fixed number of components with affinely independent means, we derive the oracle signal projection explicitly. Under anisotropic sub-Gaussian noise and random component sizes, a rowwise eigenspace bound yields a sufficient condition, governed by the weakest signal direction, for recovering all connectivity signs without requiring a bounded condition number of the nonzero signal spectrum. Under isotropic Gaussian noise with a fixed mean shape, we obtain an asymptotically sharp cutoff, with an explicit formula when the mixing proportions are equal and the means form a regular simplex.

Conformal prediction intervals are used to construct marginally well-calibrated regions around point predictions. Recent work has focused on extending these ideas to conformal predictive distributions, whose marginal probability integral transform (PIT) follows a Unif[0, 1] distribution. In this work, we investigate how to recover predictive densities from conformal predictive distributions. We extend the theory of conformal predictive distributions by proving asymptotic marginal validity for a tail-corrected version of conformal predictive distributions. We propose a method called quantile matching, which preserves an upper bound on the deviation of the marginal PIT from uniformity. Furthermore, we show that the distribution induced by quantile matching is equivalent to the crisp version of conformal predictive distributions when the number of quantiles equals the size of the calibration set. For the recovery of conformal densities, we construct a fidelity-constrained bandwidth optimization for kernel smoothing that preserves asymptotic marginal validity and has a closed-form solution. We apply and compare our proposed methodology on a large simulated real estate transactions dataset based on the Hierarchical Trend Model.

\citet{barczy2014asymptotic} showed that the ordinary least-squares (OLS) estimator of the innovation mean is inconsistent for a unit-root INAR(2) process. We construct a consistent intercept estimator using inverse-time weighted least squares (WLS) and derive its mixed-rate asymptotics: the intercept estimator is asymptotically normal at rate $\sqrt{\log n}$, with a Gaussian limit independent of the autoregressive limits, while the autoregressive estimators retain their OLS rates. We develop an OLS unit-root test calibrated with WLS nuisance estimates. Under the maintained unit root, residual-score Gaussian intervals for the innovation mean and long-run drift remain asymptotically valid after test nonrejection. Simulations show that WLS reduces root mean squared error for both quantities relative to OLS, with further gains from imposing the unit root. An application to two Canadian flood inventories shows that conclusions about persistence depend on the inventory and weighting offset.

We study principal component analysis (PCA) under memory constraints, a setting that is increasingly important in large-scale data analysis. Our focus is on Oja's algorithm, which is a one-pass, memory-efficient algorithm requiring only $O(p)$ storage in the rank-one case and $O(pk)$ storage for $k$- PCA. The main goal is to develop a procedure based on Oja's algorithm that is optimal without prior knowledge of the eigengap, which is otherwise needed to tune the learning rates effectively. We do this by first introducing a geometric perspective on the convergence of $k$-PCA: by embedding the Oja iterates into the exterior space, we establish an exact equivalence between $k$-PCA in the ambient space and $1$-PCA in the exterior space. This geometric viewpoint enables us to establish several convergence guarantees, including one based only on a single learning-rate schedule. Building on this geometric perspective, we develop an averaging theory for $k$-PCA and show that the resulting averaged estimator is adaptive: it achieves the nearly optimal convergence rate without prior knowledge of the eigengap. As an application, we construct memory-efficient estimators for elliptical component analysis (ECA) \cite{han2014scale,han2018eca}. Simulation studies and real data analysis are conducted to demonstrate the benefits of our proposed algorithms.

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of $\mathbb R^d$ but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani–Fourier family $\mathrm T_{s,p}$, the weighted $L^p$ norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on $\mathbb R^d$. For $1\le p<\infty$ we show that $d/p<s<1+d/p$ is exactly the window in which $\mathrm T_{s,p}$ is finite on $\mathcal P_p(\mathbb R^d)$, both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies $\mathrm T_{s,p}$ as an integral probability metric over a homogeneous Fourier–Lebesgue ball, with an explicit extremizer when $1<p<\infty$. We prove the global bound $\mathrm T_{s,p}\lesssim W_p^{\,s-d/p}$, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with $W_p$ on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at $p=2$. There, $\mathrm T_{s,2}$ is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

In a spiked covariance model, let $\mathcal{H}$ and $\mathcal{B}$ be selected sample and population eigenvector matrices. Their alignment matrix $\Xi=\mathcal{H}^\top\mathcal{B}$ is a latent oracle diagnostic of direction-specific correspondence beyond subspace overlap. We study its uniform pathwise recovery in a nested high-dimensional, low-sample-size experiment with fixed sample size and growing dimension. At every fixed model, an observable spectral contrast consistently estimates the row Gram matrix $\Xi\Xi^\top$. For two or more spikes, however, no measurable estimator sequence achieves uniformly reliable almost-sure path recovery of $\Xi$ over a family having common spike strengths and a common signal subspace but different ordered eigenframes. For every $0<\epsilon<1/\sqrt{2}$, a finite orbit forces each estimator to have some orbit point at which the approximate path-recovery probability is at most a variational bound $u(\epsilon)$, where $u(\epsilon)$ tends to zero as $\epsilon$ tends to zero. The worst-model exact-recovery probability over a countable orbit is zero. Thus the Gram summary is recoverable model by model, while the full alignment matrix is not uniformly recoverable along the nested path.

We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at $N$ equidistant times through a superposition of $m$ independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every $N\ge2$, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér–Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.

We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.

We study clustering of categorical sequences using the Optimal Matching (OM) distance under finite mixtures of finite-state Markov chains. We show that the normalized OM distance between two independent chains converges almost surely to a deterministic population quantity, concentrates exponentially around its finite-horizon mean, and admits an $O(\sqrt{\log n/n})$ convergence rate when the two chains have the same transition kernel. These population quantities yield a natural separation condition: the largest within-component limit must be smaller than the smallest between-component limit. Under this condition, hierarchical clustering with any bracketed linkage and Partitioning Around Medoids consistently recover the latent mixture partition. We also propose a consistent estimator of the number of components based on empirical OM distance profiles. The results extend to finite-state hidden Markov models and multichannel categorical observations. Overall, they provide a statistical justification for standard OM-based clustering methods for categorical time series.

Masked diffusions are popular generative models for discrete distributions. Unlike standard autoregressive sampling, they reveal several coordinates in parallel, approximating each block's joint conditional law by a product of one-coordinate conditionals. The resulting procedure, usually called tau-leaping, reduces computational cost but introduces a factorization error ($\varepsilon_\text{fact}$), even with perfectly learned predictors. We study the resulting tradeoff between generative accuracy and computational cost, focusing on how to choose a denoising schedule to minimize $\varepsilon_\text{fact}$ for a fixed sampling budget. To do so, we establish an exact integral representation of $\varepsilon_\text{fact}$ separating the schedule from the target's dependence structure, summarized by a dependence density $\rho$. This representation yields recursive stationarity equations for optimal schedules and allows us to quantify how estimation errors in $\rho$ affect schedule selection. As the dimension $N$ and sampling budget grow, we characterize the optimal schedule and quantify the cost of random block sizes relative to a deterministic planner. We highlight a fundamental dichotomy: if $\rho$ converges uniformly to a strictly positive continuous profile as $N\to\infty$, schedule optimization can only improve the leading constant of $\varepsilon_\text{fact}$, while if $\rho$ degenerates, schedule optimization can improve the asymptotic order. Examples based on stationary processes and exchangeable mixtures illustrate these regimes.

Covariate matching between treatment and control groups remains a foundational methodology in observational causal inference, designed to replicate the covariate balance of randomized controlled trials. Despite substantial methodological advancements, applied researchers continue to face an inherent trade-off between achieving rigorous balance and preserving the statistical power of the matched sample. We introduce a novel framework for a matching method named Optimal Study Interior Partitioning (OSIP). OSIP frames observational matching as a joint optimization problem governed simultaneously by a maximum cardinality constraint on the number of strata, and a strict fine balance threshold on the matching carried out in each strata. OSIP is based on two-stage optimization algorithm. In both stages it considers a projection of the (high-dimensional) data into one-dimensional space by evaluating an estimated propensity score for each unit. The method is based on enforcing a partition of the input to subsets of consecutive intervals of estimated propensity scores. The novelty arises because we consider this task as a global optimization problem. In the first stage, we employ an optimal partitioning algorithm with respect to an auxiliary (one-dimensional) goal function. In the second phase, we go back to the original distance function and apply heuristics approaches locally optimizing the solution from the first stage. We compare our framework with established methodologies from the literature across well-studied empirical benchmarks. We use the standard benchmark of Lindner cardiovascular dataset to demonstrate that OSIP provides a highly efficient sweet spot solution for the corresponding multi-criteria goal. This conclusion is verified by considering also other benchmark datasets.

In this paper, we address the joint parameter estimation of drift and diffusion coefficients for heterogeneously interacting particle systems. Unlike the homogeneous setting, the interactions are governed by a graphon-weighted mean-field framework, which introduces significant analytical complexity: while homogeneous systems yield i.i.d. limits, our setting results in particles that are independent but non-identically distributed in the limit. Based on discrete observations of the system over a fixed time interval $[0, T]$, we propose a contrast function based on a pseudo-likelihood approach. We prove the consistency of the resulting estimators as the discretization step $\Delta_n \to 0$ and the number of particles $N \to \infty$. Furthermore, we establish asymptotic normality under the additional constraint $N \Delta_n \to 0$, demonstrating that the underlying graphon structure can be rigorously handled despite the lack of identical distribution in the limit.

This paper proposes a new small area estimation approach that integrates Partial Least Squares within the Fay-Herriot model to address the challenges posed by high-dimensional and highly correlated auxiliary variables sets. The resulting Partial Fay-Herriot (PFH) estimator constructs supervised components that maximize their association with the target variable, enhancing model stability and predictive efficiency. Monte Carlo simulations demonstrate that PFH estimator achieves lower mean squared error than the standard Fay-Herriot estimator and outperforms principal components-based alternatives while relying on fewer latent dimensions. The methodology is applied to the estimation of district-level per capita consumption in Mozambique, where the survey data source is complemented by large set of correlated census variables. The resulting estimates highlight pronounced geographic heterogeneity and uncover spatial clusters of deprivation. Overall, the findings show that the proposed supervised dimension-reduction approach represents an effective and easily interpretable tool for producing reliable indicators in high-dimensional contexts.

We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.

We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric $\alpha$-stable jumps. A finite-dimensional Markov embedding provides Kramers–Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional $L^2$ error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian–Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.

Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances. Yet, their use in large-scale scientific problems is limited by computational cost. Equivariant neural networks are common but typically lack the uncertainty quantification offered by GPs, which is valuable in applications such as molecular research. High-dimensional inputs and large symmetry groups further demand scalability. We establish results pertaining to the interplay of GP equivariance and conditioning and leverage them to obtain equivariant sparse GPs through suitable mean functions and covariance kernels. We instantiate this framework with a flexible class of integration-free equivariant kernels, yielding scalable and data-efficient GP inference. In particular, we introduce triply scalable equivariant Gaussian processes. We employ equivariant sparse variational Gaussian processes for $\mathrm{SO}(2)$-equivariant vector fields and molecular property prediction. Alongside the SVGP, we develop a matrix-free equivariant full-GP implementation that combines an exact Kronecker reduction with preconditioned conjugate-gradient solves, enabling fast and scalable evaluation of the full joint predictive density. We further compare different approaches for selecting inducing points in the equivariant sparse GP models. Our test cases include synthetic $\mathrm{SO}(2)$-equivariant fields as well as the prediction of electric dipole moments of N-methylformamide based on quantum chemistry simulations, achieving accurate, uncertainty-aware predictions at a fraction of the computational cost of classical GP inference.

We characterize, up to universal constants, the total variation distance between finite products of arbitrary probability measures by a simple formula. A theorem of Latala reduces this expression to one scalar equation. As an application, we characterize the sample complexity of equal-prior binary hypothesis testing uniformly in the weak-detection regime, where squared Hellinger distance alone does not determine the answer, and also characterize the TV distance between multinomial distributions.

We study data-driven early stopping for spectral regularisation methods in the classical non-parametric regression setting. Building on the discrepancy principle, we propose a multi-scale stopping rule that applies to general kernel estimators and show that, unlike previous approaches, it achieves full adaptivity over all smoothness levels in the well-specified case. A key contribution of our work is an extension based on random feature approximations, which reduces computational cost on large datasets while preserving minimax-optimal statistical guarantees. Our procedure not only selects an optimal stopping time but also provides a fully data-driven choice of the number of random features needed to achieve optimal rates. Through the established connection between random features and neural networks in the neural tangent kernel regime, our method further yields a principled, data-driven recommendation for the network width. We prove that the resulting simultaneously chosen width and stopping time allow neural networks to attain minimax-optimal learning rates without prior knowledge of smoothness or capacity parameters.

Latent variable generative models are commonly fit using simple priors over latent variables, but draws from these priors often fail to produce realistic data. This failure is due to a mismatch between the prior and the aggregated posterior, the distribution of latent variables induced by the fitted model and the data. This mismatch is often viewed as evidence that the prior is misspecified and should be replaced. Alternatively, in modern generative models, a two-stage strategy is increasingly used where first, the model is fit, and second, the aggregated posterior is estimated (van den Oord et al.,2017; Rombach et al., 2022.). Synthetic data are then obtained by sampling from this aggregated posterior instead of the prior. To check such procedures, we introduce the aggregated posterior predictive check (APPC). Theoretically, we establish sufficient conditions under which the APPC is asymptotically calibrated. For probabilistic principal component analysis, we show that the APPC can remain calibrated under a misspecified latent prior when pervasive factors permit recovery of the signal space. Experiments with variational autoencoders show that aggregated posterior sampling improves generation for heavy-tailed and clustered data relative to Gaussian prior sampling while performing comparably to models with more flexible latent priors.

We consider the problem of testing Granger causality in time series, specifically, whether the future outcome $Y_{t+1}$ and the exposure history $\bar X_t$ are conditionally independent given the history of $\bar Y_t$ and a set of confounding variables $\bar Z_t$ up to time $t$. This testing procedure distinguishes true causal effects from associations driven by common external processes, supporting reliable decision-makings in applications such as finance, neuroscience, and climate science. While traditional approaches assume a linear vector autoregressive (VAR) model and are vulnerable to misspecification, we instead address a model-free version of the problem. We propose nonlinearly regressing both the outcome and exposure on the joint history of $Y$ and $Z$, and calculating a test statistic based on the sample covariance of residuals, we call the Generalised Temporal Covariance Measure (GTCM). To account for heteroscedasticity and improve power against local alternatives, we incorporate variance weights and employ a data-adaptive test based on polynomial lag expansions.The type I error control of the test relies on the relatively weak assumption that user-chosen regression procedures estimate conditional means at a sufficiently fast rate that is slow enough to accommodate nonparametric settings. By further assuming stability of the regression procedures and weak dependence in the time series, we can utilise the entire dataset to estimate the conditional means without splitting the time series into subsets.

Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\(n\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density \(f\), the optimal location estimation error with sample size \(n\) under failure probability \(\delta\) is known to be Le Cam's two-point rate: \[ \sup\left\{r>0:\mathsf{H}^2\left(f_0, f_{2r}\right)\lesssim \frac{\log(1/\delta)}{n}\right\}. \] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only \(O(\log(n))\) time on sorted samples.

Principal component analysis (PCA) is a fundamental tool for learning low-dimensional structure from high-dimensional data. When data are collected from multiple sources, the underlying task distributions may exhibit unknown degrees of similarity, with some tasks potentially arising from arbitrary distributions. We propose new multi-task PCA procedures that exploit similarity structure across tasks to improve eigenspace estimation while remaining robust to outlier tasks. We establish non-asymptotic convergence rates and show that the proposed procedures attain minimax optimal rates in a range of regimes. One of the procedures builds on the matrix-depth notion of Chen, Gao, and Ren (2018) and can achieve the optimal dependence of the estimation error on the proportion of outlier tasks, addressing a key challenge in robust multi-task learning. Extensive simulations and real-data analyses demonstrate the effectiveness of the proposed methods.

As a fundamental task across computational statistics, scientific computing and machine learning, sampling from high-dimensional probability distributions has received increasing attention in recent years. Numerous sampling algorithms have been proposed, among which underdamped Langevin Monte Carlo (ULMC) methods based on underdamped Langevin dynamics (ULD) have emerged as a class of efficient ones. In this work, we introduce a “universal" predictor-corrector formulation that bridges Euler-type, UBU-type and randomized schemes through different choices of method parameters. Notably, the “universal" integrator induces two novel classes of low-cost integrators, termed low-cost randomized integrators (LC-RIs) and low-cost UBU integrators (LC-UBUIs), as well as their exponential-free variants based on polynomial and rational approximations. The resulting new UBU-type and randomized schemes require only one gradient evaluation and two Gaussians per iteration, considerably reducing the number of gradient evaluations or Gaussians per iteration required by existing counterparts. Further, a general framework of long-time error analysis is developed for general discretization schemes in a probability metric. Under certain smoothness and non-log-concavity conditions, we rely on the unified framework to establish non-asymptotic $\mathcal{W}_1$-error bounds of both old and new schemes, revealing convergence rates of order $\mathcal{O}(d^{\frac{1}{2}}h)$ for Euler-type schemes, order $\mathcal{O}(d h^2)$ for UBU-type ones and order $\mathcal{O}(d^{\frac{1}{2}}h^{\frac{3}{2}})$ for randomized ones. In the strongly convex setting, the same non-asymptotic error bounds can be recovered in $\mathcal{W}_2$-distance. Numerical experiments corroborate the theoretical findings.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

In a statistical factor model, principal components (or eigenvectors) of a sample covariance matrix serve as estimates of principal directions, the true drivers of co-movement of a collection of observed variables. We write the often substantial error in these estimates as a sum of two interpretable terms, which we show have almost sure asymptotic limits as the number of variables grows with sample size bounded. This scenario is commonplace in financial economics, genomics, machine learning and signal processing. Out-of-subspace error measures the distance from an estimate to the subspace spanned by population factor exposures. It can be expressed in terms of data, providing an estimable floor for asymptotic error. In-subspace error is driven by a rotation error within that subspace, which arises from the fixed sample size of the latent factor returns and cannot be estimated from data alone. We illustrate our error analysis with a three-factor simulation of the US public equity market, showing the dependence of the magnitude of the error and its components on dimension and sample size. In that simulation, out-of-subspace error dominates. Researchers who rely on principal component analysis to estimate factor models can use our results to quantify errors in model-based predictions and attributions.