M-estimators of scatter with eigenvalue shrinkage

金融应用中非正态模型的非线性收缩估计

Mens: Nonlinear shrinkage estimation in nonparanormal models for financial applications

研究针对非正态模型开发非线性收缩协方差估计理论，提出无边际非线性收缩估计器MENS，给出其经验谱分布收敛律及渐近最优性，建立尖峰相变，通过模拟和回测验证其优势，兼具稳健性与效率，对高维配置决策有实用价值。

Fixed support positive-definite modification of covariance matrix estimators via linear shrinkage

In this work, we study the positive definiteness (PDness) problem in covariance matrix estimation. For high dimensional data, many regularized estimators are proposed under structural assumptions on the true covariance matrix including sparsity. They are shown to be asymptotically consistent and rate-optimal in estimating the true covariance matrix and its structure. However, many of them do not take into account the PDness of the estimator and produce a non-PD estimate. To achieve the PDness, researchers consider additional regularizations (or constraints) on eigenvalues, which make both the …

Improved Covariance Matrix Estimator using Shrinkage Transformation and Random Matrix Theory

One of the major challenges in multivariate analysis is the estimation of population covariance matrix from sample covariance matrix (SCM). Most recent covariance matrix estimators use either shrinkage transformations or asymptotic results from Random Matrix Theory (RMT). Shrinkage techniques help in pulling extreme correlation values towards certain target values whereas tools from RMT help in removing noisy eigenvalues of SCM. Both of these techniques use different approaches to achieve a similar goal which is to remove noisy correlations and add structure to SCM to overcome the bias-varianc…

收缩协方差估计器谱泛函中的误差传播：扰动界与校准推断

Error Propagation in Spectral Functionals of Shrinkage Covariance Estimators: Perturbation Bounds and Calibrated Inference

研究收缩协方差估计器谱泛函中的误差传播，通过推导特征子空间的零律、利用无分布方法校准测试等，证明尺度不变标量泛函的免疫性，给出高维无偏估计器，经模拟验证，用于总体未知时的诊断。

多元回归中高维残差协方差矩阵的受限非线性收缩

Restricted nonlinear shrinkage of high-dimensional residual covariance matrices in multivariate regressions

研究非正态模型的非线性收缩协方差估计，提出无边际非线性收缩估计器MENS，给出其经验谱分布收敛性及渐近最优性，建立相变，通过模拟和回测证实理论，展示其在高维配置和决策中的实用价值。

Asymptotics of Empirical Eigen-structure for Ultra-high Dimensional Spiked Covariance Model

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spike magnitude of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights into the roles the leading eigenvalues, sample size, and dimensionality play in principal component analysis. The results are proven by a technical device, which swaps the role of rows and columns and converts the high-dimensional problems into low-dimensional one…

特征间隙稀疏性用于协方差简洁性

Eigengap Sparsity for Covariance Parsimony

本文提出基于特征间隙稀疏性的协方差简洁性方法，通过投影梯度下降在单调锥上求解，将协方差简洁性与收缩方法联系起来。

Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data

We study ridge estimation of the precision matrix in the high-dimensional setting where the number of variables is large relative to the sample size. We first review two archetypal ridge estimators and note that their utilized penalties do not coincide with common ridge penalties. Subsequently, starting from a common ridge penalty, analytic expressions are derived for two alternative ridge estimators of the precision matrix. The alternative estimators are compared to the archetypes with regard to eigenvalue shrinkage and risk. The alternatives are also compared to the graphical lasso within th…

Regularized LRT for Large Scale Covariance Matrices: One Sample Problem

The main theme of this paper is a modification of the likelihood ratio test (LRT) for testing high dimensional covariance matrix. Recently, the correct asymptotic distribution of the LRT for a large-dimensional case (the case $p/n$ approaches to a constant $γ\in (0,1]$) is specified by researchers. The correct procedure is named as corrected LRT. Despite of its correction, the corrected LRT is a function of sample eigenvalues that are suffered from redundant variability from high dimensionality and, subsequently, still does not have full power in differentiating hypotheses on the covariance ma…

A large covariance matrix estimator under intermediate spikiness regimes

The present paper concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. In this approach, the low rank plus sparse decomposition of the covariance matrix is recovered by least squares minimization under nuclear norm plus $l_1$ norm penalization. This paper proposes a new estimator of that family based on an additional least-squares re-optimization step aimed at un-shrinking the eigenvalues of the low rank component estimated at the first step. We prove that such un-shrinkage causes the final estimate to approach the targe…