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共同函数主成分AI 解读

Functional principal component analysis (FPCA) based on the Karhunen–Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation, but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the log-returns of…

一种通用的估计量聚合程序AI 解读

In this paper we study the aggregation problem that can be formulated as follows. Assume that we have a family of estimators $\mathcal{F}$ built on the basis of available observations. The goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family. We propose a general aggregation scheme that is universal in the following sense: it applies for families of arbitrary estimators and a wide variety of models and global…

多变量数据中的异常值传播AI 解读

We investigate the performance of robust estimates of multivariate location under nonstandard data contamination models such as componentwise outliers (i.e., contamination in each variable is independent from the other variables). This model brings up a possible new source of statistical error that we call “propagation of outliers.” This source of error is unusual in the sense that it is generated by the data processing itself and takes place after the data has been…

离散观测过程中的跳跃检验AI 解读

We propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes. As the sampling interval tends to 0, our test statistic converges to 1 if there are jumps, and to another deterministic and known value (such as 2) if there are no jumps. The test is valid for all Itô semimartingales, depends neither on the law of the process nor on the coefficients of the equation which it solves, does not require a preliminary estimation of…

离散观测扩散过程的蒙特卡洛最大似然估计AI 解读

This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity…

随机矩阵理论：SAMSI 研究计划介绍AI 解读

Craig Tracy and I became coorganizers and the program's scope was enlarged.It now included regularization issues, inference in graphical models, Bayesian multivariate analysis in such models, applications of random matrix theory particularly in electrical engineering, as well as applications of regularization and high dimensional multivariate analysis in geophysical models.The program was realized as a six-month session in 2006-2007 and featured not only the usual opening…

高斯对称矩阵特征值与特征向量的推断AI 解读

This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic background radiation data, where the observations are, respectively, 3×3 and 2×2 symmetric…

高斯图模型中的灵活协方差估计AI 解读

In this paper, we propose a class of Bayes estimators for the covariance matrix of graphical Gaussian models Markov with respect to a decomposable graph G. Working with the WPG family defined by Letac and Massam [Ann. Statist. 35 (2007) 1278–1323] we derive closed-form expressions for Bayes estimators under the entropy and squared-error losses. The WPG family includes the classical inverse of the hyper inverse Wishart but has many more shape parameters, thus allowing for…

主成分分析的有限样本近似：矩阵扰动方法AI 解读

Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, and those of the limiting population PCA as n→∞. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness…

多元分析与 Jacobi 系综：最大特征值、Tracy–Widom 极限及收敛速率AI 解读

Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A + B)(-1)B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and, scaling, the distribution is approximated to second-order, O(p(-2/3)), by the Tracy-Widom law. The…

大型Wishart矩阵的统计特征值推断AI 解读

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., “eigen-inference”). Results found in the literature establish the asymptotic normality of the…

利用随机矩阵理论估计高维协方差矩阵的谱AI 解读

Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely used techniques, in particular in principal component analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample size, n, is of the same order of magnitude as the number of variables p. Random…

高维稀疏协方差矩阵的算子范数一致估计AI 解读

Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n×p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly. In this “large n, large p” setting, it is sometimes the case that practitioners are willing to assume…

基于特征筛减独立规则的高维分类AI 解读

Classification using high-dimensional features arises frequently in many contemporary statistical studies such as tumor classification using microarray or other high-throughput data. The impact of dimensionality on classifications is largely poorly understood. In a seminal paper, Bickel and Levina (2004) show that the Fisher discriminant performs poorly due to diverging spectra and they propose to use the independence rule to overcome the problem. We first demonstrate that…

正则化样本协方差矩阵的中心极限定理AI 解读

We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our…

通过阈值化实现协方差正则化AI 解读

This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and (log p)/n→0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an…

中心极限定理型随机场的倾斜欧拉示性数密度及其“bubbles”数据应用AI 解读

Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level u chosen to control the tail probability or p-value of its maximum. This p-value is approximated by the expected Euler characteristic (EC) of the excursion set of the test statistic field above u, denoted $\mathbb{E}\varphi(A_{u})$. Under isotropy, one can use the expansion $\mathbb{E}\varphi(A_{u})=\sum_{k}\mathcal{V}_{k}\rho_{k}(u)$, where $\mathcal{V}_{k}$…

通过镜像平均学习AI 解读

Given a finite collection of estimators or classifiers, we study the problem of model selection type aggregation, that is, we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original nonlinear one and constitutes a special case of the mirror…

平稳系统中的多参数局部 Whittle 估计AI 解读

Moving from univariate to bivariate jointly dependent long-memory time series introduces a phase parameter (γ), at the frequency of principal interest, zero; for short-memory series γ=0 automatically. The latter case has also been stressed under long memory, along with the “fractional differencing” case γ=(δ2−δ1)π/2, where δ1, δ2 are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the…

参数维数增长时的剖面核似然推断AI 解读

The generalized varying coefficient partially linear model with growing number of predictors arises in many contemporary scientific endeavor. In this paper we set foot on both theoretical and practical sides of profile likelihood estimation and inference. When the number of parameters grows with sample size, the existence and asymptotic normality of the profile likelihood estimator are established under some regularity conditions. Profile likelihood ratio inference for the…

长记忆与短记忆时间序列的残差经验过程AI 解读

This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov–Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992–1024] does not hold when the stochastic regression…

通过马尔可夫链论证评价形式后验分布AI 解读

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function φ of a parameter when the loss is quadratic. If the posterior mean of φ is admissible for all bounded φ, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient…

修剪与似然：椭圆模型中的稳健位置及离散程度估计AI 解读

Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or k)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to…

利用 Malliavin 演算的高斯过程漂移 Stein 估计AI 解读

We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James–Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate…