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MINIMAX ESTIMATION OF LARGE PRECISION MATRICES WITH BANDABLE CHOLESKY FACTORS

Liu, Yu (2018) MINIMAX ESTIMATION OF LARGE PRECISION MATRICES WITH BANDABLE CHOLESKY FACTORS. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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Status: Unpublished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:
CreatorsEmailPitt UsernameORCID
Liu, Yuyul125@pitt.eduyul1250000-0001-7891-5030
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairZhao, Rzren@pitt.edu
Committee MemberJung, Ssungkyu@pitt.edu
Committee MemberIyengar, Sssi@pitt.edu
Committee MemberJing, Ljinglei@andrew.cmu.edu
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 2018 18:59
URI: http://d-scholarship.pitt.edu/id/eprint/35144

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