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含噪声与高维尺度下的低秩及近低秩矩阵估计AI 解读

We study an instance of high-dimensional inference in which the goal is to estimate a matrix Θ∗∈ℝm1×m2 on the basis of N noisy observations. The unknown matrix Θ∗ is assumed to be either exactly low rank, or “near” low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider a standard M-estimator based on regularization by the nuclear or trace norm over matrices, and analyze its performance under high-dimensional scaling. We define the notion of…

局部化球面反卷积AI 解读

We provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behavior of our procedure for any $\mathbb {L}_{p}$ loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We…

复合假设检验、Hermite 多项式与非光滑泛函的最优估计AI 解读

A general lower bound is developed for the minimax risk when estimating an arbitrary functional. The bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional (1/n)∑|θi| from an observation Y∼N(θ, In). This problem exhibits some features that are significantly different from those that occur in estimating conventional smooth functionals. This is a setting where standard techniques fail to yield sharp results. A…

自适应马尔可夫链蒙特卡洛的渐近方差核估计AI 解读

We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in Lp and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the GARCH(1, 1) Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.

估计逆回归函数的两阶段混合程序AI 解读

We consider a two-stage procedure (TSP) for estimating an inverse regression function at a given point, where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two. We establish that the convergence rate of the second-stage estimate can attain the parametric n1/2 rate. Furthermore, a bootstrapped variant of TSP (BTSP) is introduced and its consistency properties studied.…

高维低秩矩阵的估计AI 解读

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T-matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the number mT of unknown entries can be much larger than the sample size N. Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink…

线性结构方程模型的全局可识别性AI 解读

Structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables. We consider the classical case of linear relationships and additive Gaussian noise terms. We give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms. Global identifiability is understood…

变阶可逆马尔可夫链的贝叶斯分析AI 解读

We define a conjugate prior for the reversible Markov chain of order r. The prior arises from a partially exchangeable reinforced random walk, in the same way that the Beta distribution arises from the exchangeable Polyá urn. An extension to variable-order Markov chains is also derived. We show the utility of this prior in testing the order and estimating the parameters of a reversible Markov model.

长时间区间内基于高频数据的 Lévy 过程估计AI 解读

In this paper, we study nonparametric estimation of the Lévy density for Lévy processes, with and without Brownian component. For this, we consider n discrete time observations with step Δ. The asymptotic framework is: n tends to infinity, Δ=Δn tends to zero while nΔn tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator of the Lévy density and to provide a bound for the global ${\mathbb{L}}^{2}$-risk. Estimators of the drift and of…

含噪观测下波动率推断的渐近等价性AI 解读

We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam’s sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function σ and a nonstandard noise level. As an application, new rate-optimal…

指数筛选与稀疏估计的最优速率AI 解读

In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in this setup is to assume that the linear combination is sparse in some sense, for example, that it involves only few covariates. We consider a general, nonnecessarily linear, regression with Gaussian noise and study a related question,…

对数凹分布逼近及其回归应用AI 解读

We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback–Leibler-type functional. We show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows distance D1(⋅, ⋅). This result implies consistency of the maximum…

连续状态空间中马尔可夫链拟蒙特卡洛的一致性AI 解读

The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0, 1) random variables. Tribble [Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences (2007) Stanford Univ.] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The previous theoretical justification for using anything other than…

面向大规模成像数据的 FDRL 多重检验AI 解读

The multiple testing procedure plays an important role in detecting the presence of spatial signals for large-scale imaging data. Typically, the spatial signals are sparse but clustered. This paper provides empirical evidence that for a range of commonly used control levels, the conventional FDR procedure can lack the ability to detect statistical significance, even if the p-values under the true null hypotheses are independent and uniformly distributed; more generally,…

错误发现比例的精确计算及其最不利配置应用AI 解读

In a context of multiple hypothesis testing, we provide several new exact calculations related to the false discovery proportion (FDP) of step-up and step-down procedures. For step-up procedures, we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable, which leads to explicit computations of the c.d.f., the sth moment and the mean of the FDP, the latter corresponding to the false discovery rate (FDR). For step-down…

利用检验功效增强多重决策并控制族错误率与错误发现率AI 解读

Improved procedures, in terms of smaller missed discovery rates (MDR), for performing multiple hypotheses testing with weak and strong control of the family-wise error rate (FWER) or the false discovery rate (FDR) are developed and studied. The improvement over existing procedures such as the Šidák procedure for FWER control and the Benjamini–Hochberg (BH) procedure for FDR control is achieved by exploiting possible differences in the powers of the individual tests. Results…

可分解协方差图模型的 Wishart 分布AI 解读

Gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph G. These models are distinctly different from the traditional concentration graph models (often also referred to as Gaussian graphical models or covariance selection models) since the zeros in the parameter are now reflected in the covariance matrix Σ, as compared to the concentration matrix Ω = Σ−1. The parameter space of interest for…

一般隐马尔可夫模型最大似然估计量的一致性AI 解读

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and…

单调谱密度估计AI 解读

We propose two estimators of a monotone spectral density, that are based on the periodogram. These are the isotonic regression of the periodogram and the isotonic regression of the log-periodogram. We derive pointwise limit distribution results for the proposed estimators for short memory linear processes and long memory Gaussian processes and also that the estimators are rate optimal.

协变量数量发散时聚类二元数据的 GEE 分析AI 解读

Clustered binary data with a large number of covariates have become increasingly common in many scientific disciplines. This paper develops an asymptotic theory for generalized estimating equations (GEE) analysis of clustered binary data when the number of covariates grows to infinity with the number of clusters. In this “large n, diverging p” framework, we provide appropriate regularity conditions and establish the existence, consistency and asymptotic normality of the GEE…

纵向数据的函数型单指标模型AI 解读

A new single-index model that reflects the time-dynamic effects of the single index is proposed for longitudinal and functional response data, possibly measured with errors, for both longitudinal and time-invariant covariates. With appropriate initial estimates of the parametric index, the proposed estimator is shown to be $\sqrt{n}$-consistent and asymptotically normally distributed. We also address the nonparametric estimation of regression functions and provide estimates…

网络异常簇的检测AI 解读

We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an “unusual behavior.” Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with…

Metropolis–Hastings 算法的简易 Rao–Blackwell 化AI 解读

Casella and Robert [Biometrika 83 (1996) 81–94] presented a general Rao–Blackwellization principle for accept-reject and Metropolis–Hastings schemes that leads to significant decreases in the variance of the resulting estimators, but at a high cost in computation and storage. Adopting a completely different perspective, we introduce instead a universal scheme that guarantees variance reductions in all Metropolis–Hastings-based estimators while keeping the computation cost…

曲面积分的非参数估计AI 解读

The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. 35 (2007) 1031–1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3-dimensional objects) via the consideration of shots at a box…