ISSN 0253-2778

CN 34-1054/N

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Open AccessOpen Access JUSTC Original Paper

Scalable confidence intervals of precision matrices in high dimensions

  • International Institure of Finance, School of Management, University of Science and Technology of China,Hefei 230601, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.06.006
  • Received Date: 26 December 2019
  • Accepted Date: 03 June 2020
  • Rev Recd Date: 03 June 2020
  • Publish Date: 30 June 2020
    Abstract

  • Abstract

    In order to solve the problem of the computational inefficiency in confidence intervals of high-dimensional precision matrices, the De-SCIO was proposed. Compared with other methods, the computational efficiency of the confidence intervals based on De-SCIO statistic are greatly improved, and their average coverage is closer to the true level. The construction of the De-SCIO statistic is simple and avoids complicated theoretical derivation. Under reasonable assumptions, the asymptotic normality of the De-SCIO statistic was proved. The advantages of this method in average coverage and computational efficiency were demonstrated by the numerical studies and real data example.

    Abstract

    In order to solve the problem of the computational inefficiency in confidence intervals of high-dimensional precision matrices, the De-SCIO was proposed. Compared with other methods, the computational efficiency of the confidence intervals based on De-SCIO statistic are greatly improved, and their average coverage is closer to the true level. The construction of the De-SCIO statistic is simple and avoids complicated theoretical derivation. Under reasonable assumptions, the asymptotic normality of the De-SCIO statistic was proved. The advantages of this method in average coverage and computational efficiency were demonstrated by the numerical studies and real data example.
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    • References(26)
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    BICKEL P J, LEVINA E. Covariance regularization by thresholding[J]. The Annals of Statistics, 2008, 36(6): 2577-2604.
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    CAI T, LIU W, LUO X. A constrained 1 minimization approach to sparse precision matrix estimation[J]. Journal of the American Statistical Association, 2011, 106: 594-607.
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    NICKL R,VAN DE GEER S. Confidence sets in sparse regression[J]. The Annals of Statistics, 2013, 41(6): 2852-2876.
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    VAN DE GEER S, BUHLMANN P, RITOV Y, et al. On asymptotically optimal confidence regions and tests for high dimensional models[J]. The Annals of Statistics, 2014, 42(3): 1166-1202.
    [16]
    MEINSHAUSEN N. Assumption-free confidence intervals for groups of variables in sparse high-dimensional regression [DB/OL]. [2019-12-01]. https://arxiv.org/abs/1309.3489.
    [17]
    ZHANG C H, ZHANG S S. Confidence intervals for low-dimensional parameters in high dimensional liner models[J]. Journal of the Royal Statistical Society: Series B, 2014, 76: 217-242.
    [18]
    JANKOVA J, VAN DE GEER S. Confidence intervals for high-dimensional inverse covariance estimation [J]. Electronic Journal of Statistics, 2015, 9(1): 1205-1229.
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    JANKOVA J,VAN DE GEER S. Honest confidence regions and optimality in high-dimensional precision matrix estimation[J]. TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2017, 26(1): 143-162.
    [20]
    HUANG X, LI M. Confidence intervals for sparse precision matrix estimation via Lasso penalized D-trace loss[J]. Communications in Statistics: Theory and Methods, 2017, 46(24): 12299-12316.
    [21]
    JANKOVA J,VAN DE GEER S. Inference in high dimensional graphical models [DB/OL]. [2019-12-01]. https://arxiv.org/abs/1801.08512.
    [22]
    YUAN M, LIN Y. Model selection and estimation in the Gaussian graphical model[J]. Biometrika, 2007, 94(1): 19-35.
    [23]
    STRANGER B E, NICA A C, FORREST M S, et al. Population genomics of human gene expression[J]. Nature Genetics, 2007, 39(10): 1217-1224.
    [24]
    BHADRA A, MALLICK B K. Joint high-dimensional Bayesian variable and covariance selection with an application to eQTL analysis[J]. Biometrics, 2013, 69(2): 447-457.
    [25]
    DURRETT R. Probability:Theory and Examples [M]. Cambridge: Cambridge University Press, 2010.
    [26]
    WANG C, JIANG B. An efficient ADMM algorithm for high dimensional precision matrix estimation via penalized quadratic loss[J]. Computational Statistics & Data Analysis, 2020, 142: Article 106812.
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    [1]
    SCHFER J, STRIMMER K. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics[J]. Statistical Applications in Genetics and Molecular Biology, 2005, 4(1): Article 32.
    [2]
    CHEN X, LIU Y, LIU H, et al. Learning spatial-temporal varying graphs with applications to climate data analysis[C]// Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence. Palo Alto, CA: Association for the Advancement of Artificial Intelligence, 2010: 425-430.
    [3]
    FAN J, LIAO Y, LIU H. An overview of the estimation of large covariance and precision matrices [J]. Econometrics Journal, 2016, 19(1): C1-C32.
    [4]
    WAINWRIGHT M J, JORDAN M I. Graphical Models, Exponential Families, and Variational Inference [M]. Hanover,MA: Now, 2008.
    [5]
    LIU H, LAFFERTY J, WASSERMAN L. The nonparanormal: Semiparametric estimation of high dimensional undirected graphs [J]. Journal of Machine Learning Research, 2009, 10(3): 2295-2328.
    [6]
    LAURITZEN S. Graphical Models [M]. New York: Oxford Univ Press, 1996.
    [7]
    MEINSHAUSEN N, BHLMANN P. High-dimensional graphs and variable selection with the Lasso [J]. The Annals of Statistics, 2006, 34(3): 1436-1462.
    [8]
    BICKEL P J, LEVINA E. Covariance regularization by thresholding[J]. The Annals of Statistics, 2008, 36(6): 2577-2604.
    [9]
    RAVIKUMAR P, WAINWRIGHT M J, RASKUTTI G, et al. High-dimensional covariance estimation by minimizing 1-penalized log-determinant divergence[J]. Electronic Journal of Statistics, 2011, 5: 935-980.
    [10]
    LIU W, LUO X. Fast and adaptive sparse precision matrix estimation in high dimensions[J]. Journal of Multivariate Analysis, 2015, 135:153-162.
    [11]
    REN Z, SUN T, ZHANG C, et al. Asymptotic normality and optimalities in estimation of large Gaussian graphical models[J]. The Annals of Statistics, 2015, 43(3): 991-1026.
    [12]
    FAN Y, LV J. Innovated scalable efficient estimation in ultra-large Gaussian graphical models[J]. The Annals of Statistics, 2016, 44(5): 2098-2126.
    [13]
    CAI T, LIU W, LUO X. A constrained 1 minimization approach to sparse precision matrix estimation[J]. Journal of the American Statistical Association, 2011, 106: 594-607.
    [14]
    NICKL R,VAN DE GEER S. Confidence sets in sparse regression[J]. The Annals of Statistics, 2013, 41(6): 2852-2876.
    [15]
    VAN DE GEER S, BUHLMANN P, RITOV Y, et al. On asymptotically optimal confidence regions and tests for high dimensional models[J]. The Annals of Statistics, 2014, 42(3): 1166-1202.
    [16]
    MEINSHAUSEN N. Assumption-free confidence intervals for groups of variables in sparse high-dimensional regression [DB/OL]. [2019-12-01]. https://arxiv.org/abs/1309.3489.
    [17]
    ZHANG C H, ZHANG S S. Confidence intervals for low-dimensional parameters in high dimensional liner models[J]. Journal of the Royal Statistical Society: Series B, 2014, 76: 217-242.
    [18]
    JANKOVA J, VAN DE GEER S. Confidence intervals for high-dimensional inverse covariance estimation [J]. Electronic Journal of Statistics, 2015, 9(1): 1205-1229.
    [19]
    JANKOVA J,VAN DE GEER S. Honest confidence regions and optimality in high-dimensional precision matrix estimation[J]. TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2017, 26(1): 143-162.
    [20]
    HUANG X, LI M. Confidence intervals for sparse precision matrix estimation via Lasso penalized D-trace loss[J]. Communications in Statistics: Theory and Methods, 2017, 46(24): 12299-12316.
    [21]
    JANKOVA J,VAN DE GEER S. Inference in high dimensional graphical models [DB/OL]. [2019-12-01]. https://arxiv.org/abs/1801.08512.
    [22]
    YUAN M, LIN Y. Model selection and estimation in the Gaussian graphical model[J]. Biometrika, 2007, 94(1): 19-35.
    [23]
    STRANGER B E, NICA A C, FORREST M S, et al. Population genomics of human gene expression[J]. Nature Genetics, 2007, 39(10): 1217-1224.
    [24]
    BHADRA A, MALLICK B K. Joint high-dimensional Bayesian variable and covariance selection with an application to eQTL analysis[J]. Biometrics, 2013, 69(2): 447-457.
    [25]
    DURRETT R. Probability:Theory and Examples [M]. Cambridge: Cambridge University Press, 2010.
    [26]
    WANG C, JIANG B. An efficient ADMM algorithm for high dimensional precision matrix estimation via penalized quadratic loss[J]. Computational Statistics & Data Analysis, 2020, 142: Article 106812.
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