اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHigh dimensional precision matrix estimation under weak sparsity
191
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper, we estimate the high dimensional precision matrix under the weak sparsity condition where many entries are nearly zero. We study a Lasso-type method for high dimensional precision matrix estimation and derive general error bounds under the weak sparsity condition. The common irrepresentable condition is relaxed and the results are applicable to the weak sparse matrix. As applications, we study the precision matrix estimation for the heavy-tailed data, the non-paranormal data, and the matrix data with the Lasso-type method.
قيم البحث
اقرأ أيضاً
The purpose of this thesis is to develop new theories on high-dimensional structured signal recovery under a rather weak assumption on the measurements that only a finite number of moments exists. High-dimensional recovery has been one of the emergin
g topics in the last decade partly due to the celebrated work of Candes, Romberg and Tao (e.g. [CRT06, CRT04]). The original analysis there (and the works thereafter) necessitates a strong concentration argument (namely, the restricted isometry property), which only holds for a rather restricted class of measurements with light-tailed distributions. It had long been conjectured that high-dimensional recovery is possible even if restricted isometry type conditions do not hold, but the general theory was beyond the grasp until very recently, when the works [Men14a, KM15] propose a new small-ball method. In these two papers, the authors initiated a new analysis framework for general empirical risk minimization (ERM) problems with respect to the square loss, which is robust and can potentially allow heavy-tailed loss functions. The materials in this thesis are partly inspired by [Men14a], but are of a different mindset: rather than directly analyzing the existing ERMs for signal recovery for which it is difficult to avoid strong moment assumptions, we show that, in many circumstances, by carefully re-designing the ERMs to start with, one can still achieve the minimax optimal statistical rate of signal recovery with very high probability under much weaker assumptions than existing works.
We consider high-dimensional measurement errors with high-frequency data. Our focus is on recovering the covariance matrix of the random errors with optimality. In this problem, not all components of the random vector are observed at the same time an
d the measurement errors are latent variables, leading to major challenges besides high data dimensionality. We propose a new covariance matrix estimator in this context with appropriate localization and thresholding. By developing a new technical device integrating the high-frequency data feature with the conventional notion of $alpha$-mixing, our analysis successfully accommodates the challenging serial dependence in the measurement errors. Our theoretical analysis establishes the minimax optimal convergence rates associated with two commonly used loss functions. We then establish cases when the proposed localized estimator with thresholding achieves the minimax optimal convergence rates. Considering that the variances and covariances can be small in reality, we conduct a second-order theoretical analysis that further disentangles the dominating bias in the estimator. A bias-corrected estimator is then proposed to ensure its practical finite sample performance. We illustrate the promising empirical performance of the proposed estimator with extensive simulation studies and a real data analysis.
Motivated by establishing theoretical foundations for various manifold learning algorithms, we study the problem of Mahalanobis distance (MD), and the associated precision matrix, estimation from high-dimensional noisy data. By relying on recent tran
sformative results in covariance matrix estimation, we demonstrate the sensitivity of MD~and the associated precision matrix to measurement noise, determining the exact asymptotic signal-to-noise ratio at which MD fails, and quantifying its performance otherwise. In addition, for an appropriate loss function, we propose an asymptotically optimal shrinker, which is shown to be beneficial over the classical implementation of the MD, both analytically and in simulations. The result is extended to the manifold setup, where the nonlinear interaction between curvature and high-dimensional noise is taken care of. The developed solution is applied to study a multiscale reduction problem in the dynamical system analysis.
Several novel statistical methods have been developed to estimate large integrated volatility matrices based on high-frequency financial data. To investigate their asymptotic behaviors, they require a sub-Gaussian or finite high-order moment assumpti
on for observed log-returns, which cannot account for the heavy tail phenomenon of stock returns. Recently, a robust estimator was developed to handle heavy-tailed distributions with some bounded fourth-moment assumption. However, we often observe that log-returns have heavier tail distribution than the finite fourth-moment and that the degrees of heaviness of tails are heterogeneous over the asset and time period. In this paper, to deal with the heterogeneous heavy-tailed distributions, we develop an adaptive robust integrated volatility estimator that employs pre-averaging and truncation schemes based on jump-diffusion processes. We call this an adaptive robust pre-averaging realized volatility (ARP) estimator. We show that the ARP estimator has a sub-Weibull tail concentration with only finite 2$alpha$-th moments for any $alpha>1$. In addition, we establish matching upper and lower bounds to show that the ARP estimation procedure is optimal. To estimate large integrated volatility matrices using the approximate factor model, the ARP estimator is further regularized using the principal orthogonal complement thresholding (POET) method. The numerical study is conducted to check the finite sample performance of the ARP estimator.
This paper deals with the dimension reduction for high-dimensional time series based on common factors. In particular we allow the dimension of time series $p$ to be as large as, or even larger than, the sample size $n$. The estimation for the factor
loading matrix and the factor process itself is carried out via an eigenanalysis for a $ptimes p$ non-negative definite matrix. We show that when all the factors are strong in the sense that the norm of each column in the factor loading matrix is of the order $p^{1/2}$, the estimator for the factor loading matrix, as well as the resulting estimator for the precision matrix of the original $p$-variant time series, are weakly consistent in $L_2$-norm with the convergence rates independent of $p$. This result exhibits clearly that the `curse is canceled out by the `blessings in dimensionality. We also establish the asymptotic properties of the estimation when not all factors are strong. For the latter case, a two-step estimation procedure is preferred accordingly to the asymptotic theory. The proposed methods together with their asymptotic properties are further illustrated in a simulation study. An application to a real data set is also reported.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
