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收缩先验下线性模型的后验一致性AI 解读

We investigate the asymptotic behaviour of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution concentrates in neighbourhoods of the true parameter under simple sufficient conditions. These conditions hold under popular shrinkage priors given some sparsity assumptions.

连接集合卡尔曼滤波与粒子滤波AI 解读

In many applications of Monte Carlo nonlinear filtering, the propagation step is computationally expensive, and hence the sample size is limited. With small sample sizes, the update step becomes crucial. Particle filtering suffers from the well-known problem of sample degeneracy. Ensemble Kalman filtering avoids this, at the expense of treating non-Gaussian features of the forecast distribution incorrectly. Here we introduce a procedure that makes a continuous transition…

稀疏主成分分析：最优收敛率与自适应估计AI 解读

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty…

控制错误发现率的高斯图模型估计AI 解读

This paper studies the estimation of a high-dimensional Gaussian graphical model (GGM). Typically, the existing methods depend on regularization techniques. As a result, it is necessary to choose the regularized parameter. However, the precise relationship between the regularized parameter and the number of false edges in GGM estimation is unclear. In this paper we propose an alternative method by a multiple testing procedure. Based on our new test statistics for conditional…

高维稀疏主子空间估计的极小极大理论AI 解读

We study sparse principal components analysis in high dimensions, where $p$ (the number of variables) can be much larger than $n$ (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary notions of $\ell_{q}$ subspace sparsity: row sparsity and column sparsity. We prove nonasymptotic lower and upper bounds on the minimax subspace estimation error…

高维时间序列的协方差与精度矩阵估计AI 解读

We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance matrices evolve smoothly in time, thus forming a covariance matrix function. Using the functional dependence measure of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150–14154 (electronic)], we obtain the rate of convergence for the thresholded estimate and illustrate how the…

稀疏回归中的置信集AI 解读

The problem of constructing confidence sets in the high-dimensional linear model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that…

稀疏与列间依赖下高维均值检验的 Higher Criticism 替代方法AI 解读

We consider two alternative tests to the Higher Criticism test of Donoho and Jin [Ann. Statist. 32 (2004) 962–994] for high-dimensional means under the sparsity of the nonzero means for sub-Gaussian distributed data with unknown column-wise dependence. The two alternative test statistics are constructed by first thresholding $L_{1}$ and $L_{2}$ statistics based on the sample means, respectively, followed by maximizing over a range of thresholding levels to make the tests…

高维随机向量和最大值的高斯近似与乘子自助法AI 解读

We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors ($p$) is large compared to the sample size ($n$); in fact, $p$ can be much larger than $n$, without…

完全子集回归AI 解读

This paper proposes a new method for combining forecasts based on complete subset regressions. For a given set of potential predictor variables we combine forecasts from all possible linear regression models that keep the number of predictors fixed. We explore how the choice of model complexity, as measured by the number of included predictor variables, can be used to trade off the bias and variance of the forecast errors, generating a setup akin to the efficient frontier…

降秩惩罚样条平滑的边际方法及其多层函数数据应用AI 解读

Multilevel functional data is collected in many biomedical studies. For example, in a study of the effect of Nimodipine on patients with subarachnoid hemorrhage (SAH), patients underwent multiple 4-hour treatment cycles. Within each treatment cycle, subjects' vital signs were reported every 10 minutes. This data has a natural multilevel structure with treatment cycles nested within subjects and measurements nested within cycles. Most literature on nonparametric analysis of…

通过嵌套高斯过程实现局部自适应贝叶斯非参数回归AI 解读

We propose a nested Gaussian process (nGP) as a locally adaptive prior for Bayesian nonparametric regression. Specified through a set of stochastic differential equations (SDEs), the nGP imposes a Gaussian process prior for the function’s mth-order derivative. The nesting comes in through including a local instantaneous mean function, which is drawn from another Gaussian process inducing adaptivity to locally varying smoothness. We discuss the support of the nGP prior in…

流形上的局部线性回归及其几何解释AI 解读

High-dimensional data analysis has been an active area, and the main focus areas have been variable selection and dimension reduction. In practice, it occurs often that the variables are located on an unknown, lower-dimensional nonlinear manifold. Under this manifold assumption, one purpose of this article is regression and gradient estimation on the manifold, and another is developing a new tool for manifold learning. As regards the first aim, we suggest directly reducing…

含高维协变量的广义可加部分线性模型AI 解读

This paper studies generalized additive partial linear models with high-dimensional covariates. We are interested in which components (including parametric and nonparametric components) are nonzero. The additive nonparametric functions are approximated by polynomial splines. We propose a doubly penalized procedure to obtain an initial estimate and then use the adaptive least absolute shrinkage and selection operator to identify nonzero components and to obtain the final…

交互固定效应面板模型中的同质性检验AI 解读

This paper proposes a residual-based Lagrange Multiplier (LM) test for slope homogeneity in large-dimensional panel data models with interactive fixed effects. We first run the panel regression under the null to obtain the restricted residuals and then use them to construct our LM test statistic. We show that after being appropriately centered and scaled, our test statistic is asymptotically normally distributed under the null and a sequence of Pitman local alternatives. The…

高维影响度量AI 解读

Influence diagnosis is important since presence of influential observations could lead to distorted analysis and misleading interpretations. For high-dimensional data, it is particularly so, as the increased dimensionality and complexity may amplify both the chance of an observation being influential, and its potential impact on the analysis. In this article, we propose a novel high-dimensional influence measure for regressions with the number of predictors far exceeding the…

高维样本协方差矩阵特征向量经验谱分布的收敛速率AI 解读

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Marčenko–Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite…

超高维非凸惩罚回归的校准AI 解读

We investigate high-dimensional non-convex penalized regression, where the number of covariates may grow at an exponential rate. Although recent asymptotic theory established that there exists a local minimum possessing the oracle property under general conditions, it is still largely an open problem how to identify the oracle estimator among potentially multiple local minima. There are two main obstacles: (1) due to the presence of multiple minima, the solution path is…

假设检验中距离统计量与 RKHS 统计量的等价性AI 解读

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a…

向量自回归与多元非线性时间序列模型的自动设定检验AI 解读

This article introduces an automatic test for the correct specification of a vector autoregression (VAR) model. The proposed test statistic is a Portmanteau statistic with an automatic selection of the order of the residual serial correlation tested. The test presents several attractive characteristics: simplicity, robustness, and high power in finite samples. The test is simple to implement since the researcher does not need to specify the order of the autocorrelation…

高维半参数双图模型AI 解读

In multivariate analysis, a Gaussian bigraphical model is commonly used for modelling matrix-valued data. In this paper, we propose a semiparametric extension of the Gaussian bigraphical model, called the nonparanormal bigraphical model. A projected nonparametric rank-based regularization approach is employed to estimate sparse precision matrices and produce graphs under a penalized likelihood framework. Theoretically, our semiparametric procedure achieves the parametric…

半参数模型中的降维与预测变量选择AI 解读

Dimension reduction in semiparametric regressions includes construction of informative linear combinations and selection of contributing predictors. To reduce the predictor dimension in semiparametric regressions, we propose an ℓ1-minimization of sliced inverse regression with the Dantzig selector, and establish a non-asymptotic error bound for the resulting estimator. We also generalize the regularization concept to sliced inverse regression with an adaptive Dantzig…

主正交补阈值化方法与高维协方差估计AI 解读

Summary The paper deals with the estimation of a high dimensional covariance with a conditional sparsity structure and fast diverging eigenvalues. By assuming a sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the principal orthogonal complement thresholding method ‘POET’ to explore such an approximate factor structure with sparsity.…

惩罚样条两种框架下的平滑参数选择AI 解读

Summary There are two popular smoothing parameter selection methods for spline smoothing. First, smoothing parameters can be estimated by minimizing criteria that approximate the average mean-squared error of the regression function estimator. Second, the maximum likelihood paradigm can be employed, under the assumption that the regression function is a realization of some stochastic process. The asymptotic properties of both smoothing parameter estimators for penalized…