本文精选了统计学领域国际顶刊《The Annals of Statistics》近期发表的论文，提供统计学领域最新的学术动态。

Statistical inference for rough volatility: Minimax theory

原刊和作者：

The Annals of Statistics Volume 52, Issue 4

Carsten H. Chong （Hong Kong University of Science and Technology）

Marc Hoffmann （University Paris Dauphine-PSL）

Yanghui Liu （Baruch College CUNY）

Mathieu Rosenbaum （CMAP）

Grégoire Szymansky （CMAP）

Abstract

In recent years, rough volatility models have gained considerable attention in quantitative finance. In this paradigm, the stochastic volatility of the price of an asset has quantitative properties similar to that of a fractional Brownian motion with small Hurst index H<1/2. In this work, we provide the first rigorous statistical analysis of the problem of estimating H from historical observations of the underlying asset. We establish minimax lower bounds and design optimal procedures based on adaptive estimation of quadratic functionals based on wavelets. We prove in particular that the optimal rate of convergence for estimating H based on price observations at n time points is of order n−1/(4H 2) as n grows to infinity, extending results that were known only for H>1/2. Our study positively answers the question whether H can be inferred, although it is the regularity of a latent process (the volatility); in rough models, when H is close to 0, we even obtain an accuracy comparable to usual -consistent regular statistical models.

Link: https://doi.org/10.1214/23-AOS2343

Forecasting the market value of power battery industry chain: A novel RRMIDAS-SVR model

原刊和作者：

The Annals of Statistics Volume 52, Issue 4

Randolf Altmeyer （University of Cambridge）

Anton Tiepner （Aarhus University）

Martin Wahl （Universität Bielefeld）

Abstract

The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is nondecreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic evolution equation, a Gaussian lower bound scheme is introduced. As a result, minimax optimality of the rates as well as sufficient and necessary conditions for consistent estimation are established.

Link: https://doi.org/10.1214/24-AOS2364

Optimal estimation of Schatten norms of a rectangular matrix

原刊和作者：

The Annals of Statistics Volume 52, Issue 4

Solène Thépaut （Safran）

Nicolas Verzelen （Université de Montpellier）

Abstract

We consider the twin problems of estimating the effective rank and the Schatten norms As of a rectangular p×q matrix A from noisy observations. When s is an even integer, we introduce a polynomial-time estimator of As that achieves the minimax rate (pq)1/4. Interestingly, this optimal rate does not depend on the underlying rank of the matrix A. When s is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate (q∧p)(pq)1/4, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.

Link: https://doi.org/10.1214/24-AOS2374

Higher-order coverage errors of batching methods via Edgeworth expansions on t-statistics

原刊和作者：

The Annals of Statistics Volume 52, Issue 4

Shengyi He （Columbia University）

Henry Lam （Columbia University）

Abstract

While batching methods have been widely used in simulation and statistics, their higher-order coverage behaviors and relative advantages in this regard remain open. We develop techniques to obtain higher-order coverage errors for batching methods by building Edgeworth-type expansions on t-statistics. The coefficients in these expansions are intricate analytically, but we provide algorithms to estimate the coefficients of the n−1 error terms via Monte Carlo simulation. We provide insights on the effect of the number of batches on the coverage error, where we demonstrate generally nonmonotonic relations. We also compare different batching methods both theoretically and numerically, and argue that none of the methods is uniformly better than others in terms of coverage. However, when the number of batches is large, sectioned jackknife has the best coverage among all.

Link: https://doi.org/10.1214/24-AOS2377

On the approximation accuracy of Gaussian variational inference

The Annals of Statistics Volume 52, Issue 4

Anya Katsevich （Massachusetts Institute of Technology）

Philippe Rigollet （Massachusetts Institute of Technology）

Abstract

The main computational challenge in Bayesian inference is to compute integrals against a high-dimensional posterior distribution. In the past decades, variational inference (VI) has emerged as a tractable approximation to these integrals, and a viable alternative to the more established paradigm of Markov chain Monte Carlo. However, little is known about the approximation accuracy of VI. In this work, we bound the TV error and the mean and covariance approximation error of Gaussian VI in terms of dimension and sample size. Our error analysis relies on a Hermite series expansion of the log posterior whose first terms are precisely cancelled out by the first order optimality conditions associated to the Gaussian VI optimization problem.

Link: https://doi.org/10.1214/24-AOS2393