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不可忽略无响应数据的半参数最优估计AI 解读

When the response mechanism is believed to be not missing at random (NMAR), a valid analysis requires stronger assumptions on the response mechanism than standard statistical methods would otherwise require. Semiparametric estimators have been developed under the parametric model assumptions on the response mechanism. In this paper, a new statistical test is proposed to guarantee model identifiability without using instrumental variable assumption. Furthermore, we develop…

连接稳健主成分分析中的凸与非凸优化：噪声、异常值及缺失数据AI 解读

This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis in the presence of random noise)…

隐私的代价：差分隐私参数估计的最优收敛率AI 解读

Privacy-preserving data analysis is a rising challenge in contemporary statistics, as the privacy guarantees of statistical methods are often achieved at the expense of accuracy. In this paper, we investigate the tradeoff between statistical accuracy and privacy in mean estimation and linear regression, under both the classical low-dimensional and modern high-dimensional settings. A primary focus is to establish minimax optimality for statistical estimation with the…

Kronecker 协方差极大似然估计的存在性与唯一性AI 解读

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the…

具有集合结构的全局经验风险最小化估计在一般维度下达到最优速率AI 解读

Entropy integrals are widely used as a powerful empirical process tool to obtain upper bounds for the rates of convergence of global empirical risk minimizers (ERMs), in standard settings such as density estimation and regression. The upper bound for the convergence rates thus obtained typically matches the minimax lower bound when the entropy integral converges, but admits a strict gap compared to the lower bound when it diverges. Birgé and Massart (Probab. Theory Related…

高斯混合模型中谱聚类的最优性AI 解读

Spectral clustering is one of the most popular algorithms to group high- dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been fully understood. In this paper, we show that spectral clustering is minimax optimal in the Gaussian mixture model with isotropic covariance matrix, when the number of clusters is fixed and the signal-to-noise ratio is large enough.…

Stein 正态近似方法：回忆与思考AI 解读

This paper is a short exposition of Stein’s method of normal approximation from my personal perspective. It focuses mainly on the characterization of the normal distribution and the construction of Stein identities. Through examples, it provides glimpses into the many approaches to constructing Stein identities and the diverse applications of Stein’s method to mathematical problems. It also includes anecdotes of historical interest, including how Stein discovered his method…

查尔斯·斯坦与不变性：从 Hunt–Stein 定理谈起AI 解读

When statistical decision theory was emerging as a promising new paradigm, Charles Stein was to play a major role in the development of minimax theory for invariant statistical problems. In some of his earliest work with Gil Hunt, he set out to prove that, in problems where invariant procedures have constant risk, any best invariant test would be minimax among all tests. Although finding it not quite true in general, this led to the legendary Hunt–Stein theorem, which…

We revisit a paper by Charles Stein, and discuss its follow-up.

连接存在依赖时的社区检测AI 解读

In network analysis, within-community members are more likely to be connected than between-community members, which is reflected in that the edges within a community are intercorrelated. However, existing probabilistic models for community detection such as the stochastic block model (SBM) are not designed to capture the dependence among edges. In this paper, we propose a new community detection approach to incorporate intra-community dependence of connectivities through the…

高斯过程建模中参考后验分布的正常性AI 解读

In a seminal article, Berger, De Oliveira and Sansó [J. Amer. Statist. Assoc. 96 (2001) 1361–1374] compare several objective prior distributions for the parameters of Gaussian process models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels: Spherical, Exponential with power ρ<2, Matérn with smoothness ν<1. This paper provides a proof…

时间序列中变点的最优估计AI 解读

This paper establishes asymptotic theory for optimal estimation of change points in general time series models under α-mixing conditions. We show that the Bayes-type estimator is asymptotically minimax for change-point estimation under squared error loss. Two bootstrap procedures are developed to construct confidence intervals for the change points. An approximate limiting distribution of the change-point estimator under small change is also derived. Simulations and real…

何谓分辨率：超分辨显微成像的统计极小极大检验视角AI 解读

As a general rule of thumb the resolution of a light microscope (i.e., the ability to discern objects) is predominantly described by the full width at half maximum (FWHM) of its point spread function (psf)—the diameter of the blurring density at half of its maximum. Classical wave optics suggests a linear relationship between FWHM and resolution also manifested in the well-known Abbe and Rayleigh criteria, dating back to the end of the 19th century. However, during the last…

分布序列变点的在线监测AI 解读

We propose a method for the detection of a change point in a sequence {Fi} of distributions, which are available through a large number of observations at each i≥1. Under the null hypothesis, the distributions Fi are equal. Under the alternative hypothesis, there is a change point i∗>1, such that Fi=G for i≥i∗ and some unknown distribution G, which is not equal to F1. The change point, if it exists, is unknown, and the distributions before and after the potential change…

用于α散度最小化的无限维梯度下降AI 解读

This paper introduces the (α,Γ)-descent, an iterative algorithm which operates on measures and performs α-divergence minimisation in a Bayesian framework. This gradient-based procedure extends the commonly-used variational approximation by adding a prior on the variational parameters in the form of a measure. We prove that for a rich family of functions Γ, this algorithm leads at each step to a systematic decrease in the α-divergence and derive convergence results. Our…

非参数多元有限混合模型的成分数量估计AI 解读

We propose a novel estimator for the number of mixture components (denoted by M) in a nonparametric finite mixture model. The setting that we consider is one where the analyst has repeated observations of K≥2 variables that are conditionally independent given a finitely supported latent variable with M support points. Under a mild assumption on the joint distribution of the observed and latent variables, we show that an integral operator T that is identified from the data…

极小极大最优的条件独立性检验AI 解读

We consider the problem of conditional independence testing of X and Y given Z where X,Y and Z are three real random variables and Z is continuous. We focus on two main cases—when X and Y are both discrete, and when X and Y are both continuous. In view of recent results on conditional independence testing [Ann. Statist. 48 (2020) 1514–1538], one cannot hope to design nontrivial tests, which control the type I error for all absolutely continuous conditionally independent…

含时变协变量的非参数风险函数提升估计AI 解读

Given functional data from a survival process with time-dependent covariates, we derive a smooth convex representation for its nonparametric log-likelihood functional and obtain its functional gradient. From this we devise a generic gradient boosting procedure for estimating the hazard function nonparametrically. An illustrative implementation of the procedure using regression trees is described to show how to recover the unknown hazard. The generic estimator is consistent…

扩展可容许决策程序及其非标准贝叶斯风险AI 解读

For finite parameter spaces, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it is Bayes. Various relaxations of this classical equivalence have been established for infinite parameter spaces, but these extensions are each subject to technical conditions that limit their applicability, especially to modern (semi and nonparametric) statistical problems. Using results in mathematical logic and nonstandard…

多维保序回归及其他单调模型的置信区间AI 解读

We consider the problem of constructing pointwise confidence intervals in the multiple isotonic regression model. Recently, Han and Zhang (2020) obtained a pointwise limit distribution theory for the so-called block max–min and min–max estimators (Fokianos, Leucht and Neumann (2020); Deng and Zhang (2020)) in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the…

波动率估计的耦合理论AI 解读

This paper provides a strong approximation, or coupling, theory for spot volatility estimators formed using high-frequency data. We show that the t-statistic process associated with the nonparametric spot volatility estimator can be strongly approximated by a growing-dimensional vector of independent variables defined as functions of Brownian increments. We use this coupling theory to study the uniform inference for the volatility process in an infill asymptotic setting.…

马尔可夫蒙特卡洛的 Peskun–Tierney 排序：超越可逆情形AI 解读

Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key role in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterise and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are…

二阶 Stein 公式：SURE 的风险估计及高维推断应用AI 解读

Stein’s formula states that a random variable of the form z⊤f(z)−divf(z) is mean-zero for all functions f with integrable gradient. Here, divf is the divergence of the function f and z is a standard normal vector. This paper aims to propose a second-order Stein formula to characterize the variance of such random variables for all functions f(z) with square integrable gradient, and to demonstrate the usefulness of this second-order Stein formula in various applications. In…

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