Liu, Yu
(2018)
MINIMAX ESTIMATION OF LARGE PRECISION MATRICES WITH BANDABLE CHOLESKY FACTORS.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
This is the latest version of this item.
Abstract
Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted
through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this thesis, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the
equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based block-thresholding rule. Lepski's method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model, by applying our local cropping procedure to the rank-based estimators. Numerical studies are carried out to confirm our theoretical findings.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
27 September 2018 |
| Date Type: |
Publication |
| Defense Date: |
31 July 2018 |
| Approval Date: |
27 September 2018 |
| Submission Date: |
31 July 2018 |
| Access Restriction: |
1 year -- Restrict access to University of Pittsburgh for a period of 1 year. |
| Number of Pages: |
107 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Statistics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
optimal rate of convergence
precision matrix
Cholesky factor
operator norm
Frobenius norm
adaptive estimation |
| Date Deposited: |
27 Sep 2018 18:59 |
| Last Modified: |
27 Sep 2019 05:16 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/35144 |
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MINIMAX ESTIMATION OF LARGE PRECISION MATRICES WITH BANDABLE CHOLESKY FACTORS. (deposited 27 Sep 2018 18:59)
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