统计理论：2025–2026 年度综述

An annual survey of the math.ST arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 132 papers and organised into 37 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Robust Statistical Estimation and Uncertainty Quantification (36.8%) — Focuses on developing robust estimators, hypothesis testing frameworks, and uncertainty guarantees under model misspecification, missing data, or adversarial perturbations.
• Fundamental Limits and Asymptotic Statistical Inference (17.5%) — Investigates theoretical lower bounds, minimax rates, and asymptotic efficiency for statistical inference problems in classical and high-dimensional settings.
• High-Dimensional Spectral and Matrix Estimation Theory (15.8%) — Analyzes statistical properties, convergence rates, and algorithmic performance for covariance, spiked matrix, and spectral estimation in high-dimensional regimes.
• Nonparametric and Parametric Estimation Asymptotics (13.0%) — Examines convergence rates, bias correction, and optimality of estimators for densities, distributions, and parameters using kernel, likelihood, and U-statistic methods.
• Stochastic Processes and Functional Data Analysis (6.5%) — Develops theoretical tools for estimating parameters and analyzing dependence structures in stochastic processes, functional data, and time-series models.
• Gaussian Processes and Markov Chain Monte Carlo Methods (3.3%) — Studies sampling algorithms, mixing properties, and regret bounds for Gaussian processes, Markov chains, and mixture models in statistical learning.
• Causal Inference and Experimental Design Theory (2.7%) — Addresses identification, estimation, and design stability of causal effects in randomized and observational studies with interference or continuous treatments.
• Information Geometry and Optimal Transport Metrics (2.6%) — Explores geometric structures, Wasserstein distances, and information-theoretic bounds for statistical models and latent factor representations.
• Bayesian Inference and Variational Approximation (1.4%) — Analyzes posterior contraction rates, variational inference efficiency, and exact conditioning methods in parametric and nonparametric Bayesian models.
• Noise Models and Conformal Prediction Theory (0.2%) — Investigates statistical lower bounds and prediction intervals under additive noise, confounding, and inhomogeneous noise structures.
• Partially Observed Stochastic Process Estimation (0.1%) — Focuses on parameter estimation and likelihood construction for partially observed Ornstein-Uhlenbeck and epidemic processes on dynamic networks.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.