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Wednesday, 4 November 2026: 8 new papers, 9 cross-lists and 10 replacements in Statistics Theory (math.ST), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study the multi-modal Laplace approximation, given by a weighted Gaussian mixture centred at the local minima of the negative log-density, for sampling from probability measures that concentrate around finitely many modes as an inverse-temperature parameter $n$ grows. Such distributions occur in Bayesian inference in the small noise or large data regime. We derive explicit, non-asymptotic bounds on the Hellinger distance between the multi-modal Laplace approximation and the concentrating posterior. The bound decays like $n^{-1/2}$ with $n$ and the constants depend only polynomially on the dimension $d$. Moreover, we analyse the Laplace mixture as a proposal distribution for self-normalised importance sampling and independent Metropolis--Hastings (IMH) targeting concentration multimodal distributions. We derive asymptotic results, as $n\to\infty$, on the effective sample size in case of importance sampling as well as on the average acceptance probability, lag-one autocorrelation, and spectral gap in case of IMH. Numerical experiments illustrate and verify our theoretical findings.
We introduce a one-parameter family of continuous distributions on the unit interval whose cumulative distribution functions are given by the odd-height power towers \(x\uparrow\uparrow n\) (\(n=1,3,5,\dots\)). The associated densities admit a simple recursive formula obtained by logarithmic differentiation. We prove that every such density is strictly positive on \((0,1)\) and integrates to one, thereby establishing that the construction yields a genuine probability distribution. Numerical investigation reveals a characteristic peak--dip--rise shape: a sharp mode lying just below \(e^{-e}\) for all examined heights, followed by a trough and a gradual recovery toward density value~1 at the right endpoint. We then identify the limiting distribution as the height tends to infinity through odd integers.
We establish minimax separation rates across sparsity levels for regression signal detection under $(\epsilon, \delta)$-central differential privacy, considering linear regression with an unknown error distribution and generalised linear models (GLMs) with an unknown intercept. Without sparsity restrictions, the separation rates for linear regression and GLMs with binary responses match the private Gaussian mean testing rate up to polylogarithmic factors. For GLMs with more than two possible response outcomes, additional information in the marginal response distribution permits faster detection of sufficiently small signals when the dimension exceeds the sample size $n$. Our upper bounds are based on a reduction to private mean testing, tests of dependence between the response and sparse covariate projections, and a response-only test that is adjusted for the unknown intercept. The resulting procedure adapts to unknown sparsity and matches our lower bounds up to polylogarithmic factors. We also construct polynomial-time tests that attain these rates when $\epsilon\gtrsim n^{-1/2}$, and compare their numerical performance with existing private procedures.
The classical model-based optimal design criteria (D-, T-, L-, C-, and A-optimality) are usually motivated via the Cram\'er--Rao bound as optimizing scalar functions of the Fisher information matrix (FIM). This paper offers a complementary reading in which each criterion appears asymptotically as $N \to \infty$ as the leading-order term of an expected dissimilarity between prior and posterior, with the FIM entering through the Bernstein--von Mises limit. The Kullback--Leibler (KL) divergence of the posterior relative to the prior recovers D-optimality (classically known); a reverse KL of the prior relative to the posterior recovers T-optimality (here defined as maximization of the trace of the FIM, following one common convention) and its weighted-trace generalization; and a first-order approximation of the squared 2-Wasserstein distance recovers L-, C- and A-optimality. The five criteria emerge as instances of a common Bayesian-geometric template, indexed by three choices: the dissimilarity, the prior parametrization, and the approximation scheme used to evaluate it (e.g.\ Taylor expansions of the matrix square root). For the weighted criteria, the Gaussian prior covariance encodes the criterion's weight matrix, so any choice of weights corresponds to a specific prior in the family. The T-, L-, C-, and A-optimality derivations appear novel to the best of my knowledge. A sixth criterion, E-optimality, is recovered not as an independent expected-dissimilarity limit but as a min-max robustification of C-optimality against uncertainty in the modeler's prior direction. The framework is extended to distinguish between a designer's prior over unknown parameters at design time and a modeler's prior used for posterior inference, recovering the classical expected information gain objective as a special case.
A system contains an unknown number $N$ of units, each failing at an unknown common rate, each unit is removed once it fails. The failure-rate declines as units are removed, but after only a few failures the decline is too slight to distinguish from a constant rate, and the maximum-likelihood estimate of $N$ is infinite with substantial probability. This is a well-known feature of the model, but its probability has not been quantified. We show that about $N^{2/3}$ failures are needed before a finite population can be distinguished from an inexhaustible one, and that at this scale the profile log-likelihood ratio converges to that of a Gaussian shift experiment, in which an infinite estimate occurs with a calculable probability. The exponent is not a property of the model alone. Profiling annihilates the Taylor terms that the nuisance-parameters can reproduce, and detectability is governed by the first surviving term. For example, unknown drift in testing-effort reproduces the linear term, leaving a quadratic survivor and raising the requirement to $N^{4/5}$; in general the critical scale is $N^{2q/(2q+1)}$ for a survivor of order $q$. We obtain the limiting-law of the estimator, the null likelihood ratio distribution, and confidence sets that may be unbounded.
We propose a new estimator of low-frequency skewness (monthly) that exploits high-frequency data (5 min) through a direct functional mapping consisting of layers of Convolutional Neural Networks followed by Multi Layer Perceptrons. Monte Carlo experiments involving stochastic volatility models, with and without jumps, show that the estimator reduces finite-sample bias relative to existing realized-skewness estimators and remains accurate under model misspecification. Empirically, our estimator exhibits temporal stability and delivers cross-sectional performance in actual skewness-sorted portfolios. We characterize the finite-sample bias of conventional plug-in skewness estimators and establish risk consistency properties of the proposed NN estimator under standard regularity conditions.
Complex data are increasingly represented as hypergraphs. We establish bounds for approximating joint vertex-degree distributions by independent multivariate Poisson and correlated normal distributions in hypergraph models with independent hyperedges. We discuss the approximation of the distribution under different density regimes which are further supported by simulation studies.
Conformal prediction gives finite-sample coverage when calibration and test points are exchangeable, but exchangeability can fail when deployment changes the data distribution. We study conformal prediction in performative settings, where the deployed predictor affects the data distribution, and seek prediction sets valid at performative equilibrium: coverage must hold under $D(f^\star)$ for a performatively stable predictor $f^\star$. Under total-variation $L$-sensitivity of the distribution map, we prove a phase transition at $L=1$. If $L \ge 1$, no finite-sample procedure can uniformly guarantee both equilibrium-valid coverage and nontrivial prediction sets. If $L<1$, the minimax excess miscoverage over informative procedures is $\widetilde{\Omega}(1/((1-L)\sqrt{n}))$, implying a sample-complexity inflation of order $1/(1-L)^2$ for fixed excess miscoverage. Our main technical tool is a transfer theorem: any procedure returning a predictor within $\varepsilon$ of equilibrium yields equilibrium-valid coverage up to an additive $L\varepsilon$ term after split-conformal calibration. We instantiate the theorem with two retraining algorithms: a model-free method for discretizable classes and a parametric tail-averaged method that matches the $n$- and $L$-dependence of this rate up to logarithmic factors. Experiments, including semi-synthetic credit scoring, demonstrate the predicted threshold behavior and scaling laws.
We propose logit-based IV and augmented logit-based IV estimators that serve as alternatives to the traditionally used 2SLS estimator in the model where both the endogenous treatment variable and the corresponding instrument are binary. Our novel estimators are as easy to compute as the 2SLS estimator but have an advantage over the 2SLS estimator in terms of causal interpretability. In particular, in certain cases where the probability limits of both our estimators and the 2SLS estimator take the form of weighted-average treatment effects, our estimators are guaranteed to yield non-negative weights whereas the 2SLS estimator is not.
We extend the categorical treatment of Bayesian filters to models with feedback, whose transitions and observations may depend on the past observations. We work in the stochastic category, whose objects are measurable spaces and whose morphisms are Markov kernels, which we refer to as stochastic morphisms. We systematically parameterize stochastic morphisms by a measurable space, and develop the calculus of conditioning in a notation intuitively compatible with classical probability theory, requiring neither string diagrams nor densities. A stochastic morphism into a product space is regarded as a parameterized family of joint laws, and its conditional law is again a stochastic morphism, so that the rules of conditioning become equalities of stochastic morphisms. We define the Bayesian inverse with respect to a parameterized prior and prove Bayesian coherence. A model with feedback is then a model whose parameter space grows with the observations, and we derive the Bayesian filter, prediction and smoothing for such models, with standard Borel hidden states and countably generated observation spaces. The affine Gaussian stochastic morphisms parameterized by a measurable space form a family strictly larger than the Markov category $\mathbf{Gauss}$, closed under composition and Bayesian inversion, in which we obtain the Kalman filter and the Rauch-Tung-Striebel smoother with feedback.
Learn-then-Test (LTT) provides a general framework for certifying deployment policies against a prescribed risk target, but its standard formulation targets a fixed population risk estimated from calibration losses. In sequential deployment, losses may instead be dependent and nonstationary, so the risk of a fixed policy can evolve with the observed history. We introduce StreamLTT, which replaces this fixed population risk with average predictable risk, defined by conditioning each decision's loss on the information available before its outcome is revealed. Conditional centering turns empirical deviations from this target into a martingale sum, enabling both a test that adapts to predictable variance and a finite-sample Hoeffding alternative. Combined with family-wise error rate control, StreamLTT yields a simultaneously certified policy region that supports subsequent deployment selection. Across seven forecasting datasets and seven deployment risks, StreamLTT places the certified frontier substantially closer to the prescribed risk target than IID, robust-variance, block-bootstrap, and concentration-based alternatives, while empirical family-wise error closely follows the prescribed level. A finite-sample ablation further shows that the predictable-risk formulation drives the tighter risk frontier, while adaptation to predictable variance makes more effective use of the available family-wise error budget. StreamLTT therefore extends risk-certified policy selection to dependent deployment streams by addressing both what risk should be certified and how it should be tested.
We study nonparametric estimation and forecasting for GARCH-type processes and propose an iterated neural estimator that alternates between estimating the recursion map and updating the unobserved conditional variances. Under mild conditions, we establish finite-sample bounds for the time-averaged in-sample variance-path error and show that the estimator is minimax rate-optimal up to logarithmic factors. Under stationarity and mixing conditions, we establish the same rate for recursion-map estimation and derive finite-sample bounds for out-of-sample variance forecasts and expected excess Gaussian quasi-likelihood (QLIKE) loss. The convergence rates depend on the true lag orders and remain unchanged when the fitted orders are overspecified. Simulation studies illustrate the effectiveness of our estimators under nonlinear and time-varying volatility dynamics. Real-data applications suggest that our neural methods perform competitively and can incorporate option-price and high-frequency information.
Model-based Small Area Estimation (SAE) has seen rapid growth over the past 25 years or so due to the rapidly increasing demand for reliable small area estimators of totals, means and more complex area parameters. Direct area-specific estimators are either unreliable (due to small area-specific sample sizes) or not feasible (due to zero area-specific sample sizes). It is therefore necessary to borrow strength from related areas through implicit or explicit models linking related areas. We focus on explicit area-level and unit-level models. After a brief historical sketch of the evolution of SAE, we focus on selective recent developments (mostly after 2015). Developments before 2015 are mostly covered in the comprehensive Wiley book by Rao and Molina (2015). Under a basic area-level model, we consider semi-parametric and robust estimation of small area means under model misspecifications. Under a basic unit level model, methods for estimating complex parameters, such as poverty indicators, will be presented, including robust estimation in the presence of outliers. A comparison of area-level and unit-level models when unit level data are available will also be presented. Estimation methods studied include empirical best linear unbiased prediction (EBLUP), empirical best (EB) and hierarchical Bayes (HB).
Distributed statistical learning has recently attracted significant attention owing to its computational efficiency and scalability in large-scale systems. However, the non-smooth quantile loss poses substantial challenges to the development of efficient decentralized algorithms for quantile regression. Existing methods may either suffer from slow convergence or rely on smoothing-based approximations that require additional stringent assumptions, particularly in high-dimensional settings. To address these issues, we propose a decentralized framework that integrates gradient tracking with iterative hard thresholding for sparse quantile estimation. The proposed method avoids smoothing approximations and relaxes the initialization requirement. Theoretically, we show that the proposed algorithm achieves linear convergence under stated regularity conditions and that the resulting estimator attains the same order of statistical accuracy as its centralized counterpart. We further develop a decentralized information criterion for selecting the working sparsity level and establish support screening guarantees. Numerical experiments and a real-data study illustrate the finite-sample performance of the proposed method.
The Onsager--Machlup functional of a probability measure is a generalisation of the negative logarithm of the probability density function, defined precisely using the ratios of masses of balls in the limit of vanishing radius. It plays an important role in several areas of applied mathematics because its minimisers can be understood as modes of the measure. We show that the Onsager--Machlup functional of the angular central Gaussian measure on the unit sphere of an infinite-dimensional Hilbert space---i.e., the radial projection of a centred Gaussian measure with covariance operator $C$---is constant on the finite-dimensional sphere spanned by the eigenvectors of leading eigenvalue under $C$, and takes the value $+\infty$ elsewhere. We discuss the consequences of this extreme concentration of mass for maximum a posteriori estimation in Bayesian inverse problems with an angular central Gaussian prior.
We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $\epsilon$ using $d^{O(k)}2^{O_k(\log^3(1/\epsilon))}$ time and samples, improving over the $d^{O(k^2/\epsilon^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{\Omega(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\mathrm{poly}_k(1/\epsilon)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/\epsilon$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.
Distribution shifts, including covariate shifts, are ubiquitous in real-world applications of machine learning. Empirical evidence suggests that, across a variety of models with and without downstream adaptation, ensembles tend to be more robust to covariate shift than single models. We extend the theory of covariate shift severity from single models to ensembles, providing theoretical justification for the empirical trends. We introduce new discrepancy measures of covariate shift severity to control the risk of partition-based kernel estimators. We then show that with sufficient ensemble size, these models can achieve minimax-optimal risk convergence rates under covariate shift. Our results suggest that modelers can expect to need larger ensemble sizes for more severe covariate shifts.