Direct shrinkage estimation of large dimensional precision matrix

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• 作者：Taras Bodnar^a ; ^{taras.bodnar@math.su.se" class="auth_mail" title="E-mail the corresponding author} ; Arjun K. Gupta^b ; Nestor Parolya^c
• 关键词：60B20 ; 62H12 ; 62G20 ; 62G30
• 刊名：Journal of Multivariate Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：146
• 期：Complete
• 页码：223-236
• 全文大小：510 K

文摘

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0047259X15002249&_mathId=si10.gif&_user=111111111&_pii=S0047259X15002249&_rdoc=1&_issn=0047259X&md5=211bc87654522ffe9c2d5bd39e7d22ef" title="Click to view the MathML source">p→∞gif" overflow="scroll">p→∞ and the sample size g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0047259X15002249&_mathId=si11.gif&_user=111111111&_pii=S0047259X15002249&_rdoc=1&_issn=0047259X&md5=b3513106eef3959a991e0d85391e2f4d" title="Click to view the MathML source">n→∞gif" overflow="scroll">n→∞ so that g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0047259X15002249&_mathId=si12.gif&_user=111111111&_pii=S0047259X15002249&_rdoc=1&_issn=0047259X&md5=9800fe1285503eb2ed71b2de1267c211" title="Click to view the MathML source">p/n→c∈(0,+∞)gif" overflow="scroll">p/n→c∈(0,+∞). The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. Using this result, we construct a bona fide optimal linear shrinkage estimator for the precision matrix in case g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0047259X15002249&_mathId=si13.gif&_user=111111111&_pii=S0047259X15002249&_rdoc=1&_issn=0047259X&md5=24e5fb6654be079576fa56e2e74330b4" title="Click to view the MathML source">c<1gif" overflow="scroll">c<1. At the end, a simulation is provided where the suggested estimator is compared with the estimators proposed in the literature. The optimal shrinkage estimator shows significant improvement even for non-normally distributed data.