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Consistency of the objective general index in high-dimensional settings

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  • Bando, Takuma
  • Sei, Tomonari
  • Yata, Kazuyoshi

Abstract

The objective general index is a scale-invariant weighting method for ranking of multivariate data. We show that the sample objective general index is a consistent estimator of the population counterpart in high-dimensional settings under p/n→0 together with a set of conditions, where p and n denote the dimension and the sample size. The proof is based on a recent result on random matrix theory. We also evaluate the tail probability of the estimator for normal samples based on the large deviation theory. Numerical experiments are conducted to support the theoretical result. An example of real data analysis suggests an application of the weight to variable selection.

Suggested Citation

  • Bando, Takuma & Sei, Tomonari & Yata, Kazuyoshi, 2022. "Consistency of the objective general index in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:jmvana:v:189:y:2022:i:c:s0047259x21002037
    DOI: 10.1016/j.jmva.2021.104938
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    References listed on IDEAS

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    1. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "PCA consistency for the power spiked model in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 334-354.
    2. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    3. Sei, Tomonari, 2016. "An objective general index for multivariate ordered data," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 247-264.
    4. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
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