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ScreeNOT：相关噪声下精确 MSE 最优的奇异值阈值化AI 解读

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell’s ever-popular but vague scree plot heuristic from 1966. ScreeNOT has a surprising…

Fréchet 均值的 Stein 效应AI 解读

The Fréchet mean is a useful description of location for a probability distribution on a metric space that is not necessarily a vector space. This article considers simultaneous estimation of multiple Fréchet means from a decision-theoretic perspective, and in particular, the extent to which the unbiased estimator of a Fréchet mean can be dominated by a generalization of the James–Stein shrinkage estimator. It is shown that if the metric space satisfies a nonpositive…

高斯从属长程依赖过程的最优分块重抽样AI 解读

Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine…

积分 Copula 谱AI 解读

Frequency domain methods form a ubiquitous part of the statistical toolbox for time-series analysis. In recent years, considerable interest has been given to the development of new spectral methodology and tools capturing dynamics in the entire joint distributions, and thus avoiding the limitations of classical, L2-based spectral methods. Most of the spectral concepts proposed in that literature suffer from one major drawback, though: their estimation requires the choice of…

一比特量化下的协方差估计AI 解读

We consider the classical problem of estimating the covariance matrix of a sub-Gaussian distribution from i.i.d. samples in the novel context of coarse quantization, that is, instead of having full knowledge of the samples, they are quantized to one or two bits per entry. This problem occurs naturally in signal processing applications. We introduce new estimators in two different quantization scenarios and derive nonasymptotic estimation error bounds in terms of the operator…

协方差算子秩的检验AI 解读

How can we discern whether the covariance operator of a stochastic process is of reduced rank, and if so, what its precise rank is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for functional data, which require low-dimensional representations whether by functional PCA or other methods. The difficulty is that the determination is to be made on the basis of i.i.d. replications of the process observed discretely and…

多元非可逆扩散的上确界范数自适应漂移估计AI 解读

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key…

在持久性波动与平稳波动之间进行选择AI 解读

This paper suggests a multiplicative volatility model where volatility is decomposed into a stationary and a nonstationary persistent part. We provide a testing procedure to determine which type of volatility is prevalent in the data. The persistent part of volatility is associated with a nonstationary persistent process satisfying some smoothness and moment conditions. The stationary part is related to stationary conditional heteroskedasticity. We outline theory and…

高维随机森林的渐近性质AI 解读

As a flexible nonparametric learning tool, the random forests algorithm has been widely applied to various real applications with appealing empirical performance, even in the presence of high-dimensional feature space. Unveiling the underlying mechanisms has led to some important recent theoretical results on the consistency of the random forests algorithm and its variants. However, to our knowledge, almost all existing works concerning random forests consistency in a…

条件独立性的局部置换检验AI 解读

In this paper, we investigate local permutation tests for testing conditional independence between two random vectors X and Y given Z. The local permutation test determines the significance of a test statistic by locally shuffling samples, which share similar values of the conditioning variables Z, and it forms a natural extension of the usual permutation approach for unconditional independence testing. Despite its simplicity and empirical support, the theoretical…

平均奖励马尔可夫决策过程中的批量策略学习AI 解读

We consider the batch (off-line) policy learning problem in the infinite horizon Markov Decision Process. Motivated by mobile health applications, we focus on learning a policy that maximizes the long-term average reward. We propose a doubly robust estimator for the average reward and show that it achieves semiparametric efficiency. Further we develop an optimization algorithm to compute the optimal policy in a parameterized stochastic policy class. The performance of the…

高斯过程协方差参数的贝叶斯固定域渐近理论AI 解读

Gaussian process models typically contain finite-dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matérn covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter…

主题模型中稀疏主题分布的似然估计及 Wasserstein 文档距离应用AI 解读

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in the context of topic models. The data consists of observed multinomial counts of p words across n independent documents. In topic models, the p×n expected word frequency matrix is assumed to be factorized as a p×K word-topic matrix A and a K×n topic-document matrix T. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns…

协方差核的补全AI 解读

We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I×I to a covariance kernel on the entire domain I×I. For a broad class of domains Ω called serrated domains, we are able to present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of…

重尾总体样本相关矩阵特征值的极限分布AI 解读

Consider a p-dimensional population x∈Rp with i.i.d. coordinates that are regularly varying with index α∈(0,2). Since the variance of x is infinite, the diagonal elements of the sample covariance matrix Sn=n−1 ∑ i=1nxixi′ based on a sample x1,…,xn from the population tend to infinity as n increases and it is of interest to use instead the sample correlation matrix Rn={diag(Sn)}−1/2Sn{diag(Sn)}−1/2. This paper finds the limiting distributions of the eigenvalues of Rn when…

稳健马尔可夫决策过程的理论理解：样本复杂度与渐近性质AI 解读

In this paper, we study the nonasymptotic and asymptotic performances of the optimal robust policy and value function of robust Markov Decision Processes (MDPs), where the optimal robust policy and value function are estimated from a generative model. While prior work focusing on nonasymptotic performances of robust MDPs is restricted in the setting of the KL uncertainty set and (s,a)-rectangular assumption, we improve their results and also consider other uncertainty sets,…

公平统计决策问题的优化层级AI 解读

Data-driven decision making has drawn scrutiny from policy makers due to fears of potential discrimination, and a growing literature has begun to develop fair statistical techniques. However, these techniques are often specialized to one model context and based on ad hoc arguments, which makes it difficult to perform theoretical analysis. This paper develops an optimization hierarchy, which is a sequence of optimization problems with an increasing number of constraints, for…

多任务学习的无免费午餐定理AI 解读

Multitask learning and related areas such as multisource domain adaptation address modern settings where data sets from N related distributions {Pt} are to be combined toward improving performance on any single such distribution D. A perplexing fact remains in the evolving theory on the subject: while we would hope for performance bounds that account for the contribution from multiple tasks, the vast majority of analyses result in bounds that improve at best in the number n…

通过弱庞加莱不等式比较马尔可夫链及其伪边际 MCMC 应用AI 解读

We investigate the use of a certain class of functional inequalities known as weak Poincaré inequalities to bound convergence of Markov chains to equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis–Hastings sampler and pseudo-marginal methods for intractable likelihoods, the latter being subgeometric in many practical settings. These results rely on novel…

协变量不平衡递减且协变量数量增长时的再随机化AI 解读

Completely randomized experiments have been the gold standard for drawing causal inference because they can balance all potential confounding on average. However, they may suffer from unbalanced covariates for realized treatment assignments. Rerandomization, a design that rerandomizes the treatment assignment until a prespecified covariate balance criterion is met, has recently got attention due to its easy implementation, improved covariate balance and more efficient…

潜变量模型可识别性的半途径准则AI 解读

We consider linear structural equation models with latent variables and develop a criterion to certify whether the direct causal effects between the observable variables are identifiable based on the observed covariance matrix. Linear structural equation models assume that both observed and latent variables solve a linear equation system featuring stochastic noise terms. Each model corresponds to a directed graph whose edges represent the direct effects that appear as…

依赖情形下控制错误发现率的条件校准AI 解读

We introduce a new class of methods for finite-sample false discovery rate (FDR) control in multiple testing problems with dependent test statistics where the dependence is known. Our approach separately calibrates a data-dependent p-value rejection threshold for each hypothesis, relaxing or tightening the threshold as appropriate to target exact FDR control. In addition to our general framework, we propose a concrete algorithm, the dependence-adjusted Benjamini–Hochberg…

空间干扰下速率最优的整群随机化设计AI 解读

We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares outcomes under the counterfactuals that all or no units are treated. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect…

利用残差依赖度量估计时间序列模型AI 解读

We propose new estimation methods for time series models, possibly noncausal and/or noninvertible, using serial dependence information from the characteristic function of model residuals. This allows to impose the i.i.d. or martingale difference assumptions on the model errors to identify the unknown location of the roots of the lag polynomials for ARMA models without resorting to higher order moments or distributional assumptions. We consider generalized spectral density…