1,820 research outputs found
TIGER: A Tuning-Insensitive Approach for Optimally Estimating Gaussian Graphical Models
We propose a new procedure for estimating high dimensional Gaussian graphical
models. Our approach is asymptotically tuning-free and non-asymptotically
tuning-insensitive: it requires very few efforts to choose the tuning parameter
in finite sample settings. Computationally, our procedure is significantly
faster than existing methods due to its tuning-insensitive property.
Theoretically, the obtained estimator is simultaneously minimax optimal for
precision matrix estimation under different norms. Empirically, we illustrate
the advantages of our method using thorough simulated and real examples. The R
package bigmatrix implementing the proposed methods is available on the
Comprehensive R Archive Network: http://cran.r-project.org/
Statistical Inferences Using Large Estimated Covariances for Panel Data and Factor Models
While most of the convergence results in the literature on high dimensional
covariance matrix are concerned about the accuracy of estimating the covariance
matrix (and precision matrix), relatively less is known about the effect of
estimating large covariances on statistical inferences. We study two important
models: factor analysis and panel data model with interactive effects, and
focus on the statistical inference and estimation efficiency of structural
parameters based on large covariance estimators. For efficient estimation, both
models call for a weighted principle components (WPC), which relies on a high
dimensional weight matrix. This paper derives an efficient and feasible WPC
using the covariance matrix estimator of Fan et al. (2013). However, we
demonstrate that existing results on large covariance estimation based on
absolute convergence are not suitable for statistical inferences of the
structural parameters. What is needed is some weighted consistency and the
associated rate of convergence, which are obtained in this paper. Finally, the
proposed method is applied to the US divorce rate data. We find that the
efficient WPC identifies the significant effects of divorce-law reforms on the
divorce rate, and it provides more accurate estimation and tighter confidence
intervals than existing methods
Fast and Adaptive Sparse Precision Matrix Estimation in High Dimensions
This paper proposes a new method for estimating sparse precision matrices in
the high dimensional setting. It has been popular to study fast computation and
adaptive procedures for this problem. We propose a novel approach, called
Sparse Column-wise Inverse Operator, to address these two issues. We analyze an
adaptive procedure based on cross validation, and establish its convergence
rate under the Frobenius norm. The convergence rates under other matrix norms
are also established. This method also enjoys the advantage of fast computation
for large-scale problems, via a coordinate descent algorithm. Numerical merits
are illustrated using both simulated and real datasets. In particular, it
performs favorably on an HIV brain tissue dataset and an ADHD resting-state
fMRI dataset.Comment: Maintext: 24 pages. Supplement: 13 pages. R package scio implementing
the proposed method is available on CRAN at
https://cran.r-project.org/package=scio . Published in J of Multivariate
Analysis at
http://www.sciencedirect.com/science/article/pii/S0047259X1400260
Adaptive estimation of covariance matrices via Cholesky decomposition
This paper studies the estimation of a large covariance matrix. We introduce
a novel procedure called ChoSelect based on the Cholesky factor of the inverse
covariance. This method uses a dimension reduction strategy by selecting the
pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be
interpreted as a graph estimation procedure for directed Gaussian graphical
models. Our approach is particularly relevant when the variables under study
have a natural ordering (e.g. time series) or more generally when the Cholesky
factor is approximately sparse. ChoSelect achieves non-asymptotic oracle
inequalities with respect to the Kullback-Leibler entropy. Moreover, it
satisfies various adaptive properties from a minimax point of view. We also
introduce and study a two-stage procedure that combines ChoSelect with the
Lasso. This last method enables the practitioner to choose his own trade-off
between statistical efficiency and computational complexity. Moreover, it is
consistent under weaker assumptions than the Lasso. The practical performances
of the different procedures are assessed on numerical examples
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