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High-dimensional consistency of rank estimation criteria in multivariate linear model

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  • Fujikoshi, Yasunori
  • Sakurai, Tetsuro

Abstract

This paper is concerned with consistency properties of rank estimation criteria in a multivariate linear model, based on the model selection criteria AIC, BIC and Cp. The consistency properties of these criteria are studied under a high-dimensional framework with two different assumptions on the noncentrality matrix such that the number of response variables and the sample size tend to infinity. In general, it is known that under a large-sample asymptotic framework, the criteria based on AIC and Cp are not consistent, but the criterion based on BIC is consistent. However, we note that there are cases that the criteria based on AIC and Cp are consistent, but the criterion based on BIC is not consistent. Such consistency properties are also shown for the generalized criteria with a tuning parameter. Further, the modified criteria with a ridge-type estimator are also examined. Through a Monte Carlo simulation experiment, our results are checked numerically, and the estimation criteria are compared.

Suggested Citation

  • Fujikoshi, Yasunori & Sakurai, Tetsuro, 2016. "High-dimensional consistency of rank estimation criteria in multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 199-212.
    • Handle: RePEc:eee:jmvana:v:149:y:2016:i:c:p:199-212
    DOI: 10.1016/j.jmva.2016.04.005
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    References listed on IDEAS

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    1. Fujikoshi, Yasunori & Sakurai, Tetsuro & Yanagihara, Hirokazu, 2014. "Consistency of high-dimensional AIC-type and Cp-type criteria in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 184-200.
    2. Ming Yuan & Ali Ekici & Zhaosong Lu & Renato Monteiro, 2007. "Dimension reduction and coefficient estimation in multivariate linear regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 329-346, June.
    3. Izenman, Alan Julian, 1975. "Reduced-rank regression for the multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 5(2), pages 248-264, June.
    4. Gunderson, Brenda K. & Muirhead, Robb J., 1997. "On Estimating the Dimensionality in Canonical Correlation Analysis," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 121-136, July.
    5. Lisha Chen & Jianhua Z. Huang, 2012. "Sparse Reduced-Rank Regression for Simultaneous Dimension Reduction and Variable Selection," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1533-1545, December.
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    Cited by:

    1. Hideto Nakashima & Piotr Graczyk, 2022. "Wigner and Wishart ensembles for sparse Vinberg models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(3), pages 399-433, June.
    2. Oda, Ryoya & Suzuki, Yuya & Yanagihara, Hirokazu & Fujikoshi, Yasunori, 2020. "A consistent variable selection method in high-dimensional canonical discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    3. Yasunori Fujikoshi & Tetsuro Sakurai, 2023. "High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures," Mathematics, MDPI, vol. 11(3), pages 1-15, January.

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