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High dimensional linear discriminant analysis: optimality, adaptive algorithm and missing data

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  • T. Tony Cai
  • Linjun Zhang

Abstract

The paper develops optimality theory for linear discriminant analysis in the high dimensional setting. A data‐driven and tuning‐free classification rule, which is based on an adaptive constrained l1‐minimization approach, is proposed and analysed. Minimax lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces. In addition, we consider classification with incomplete data under the missingness completely at random model. An adaptive classifier with theoretical guarantees is introduced and the optimal rate of convergence for high dimensional linear discriminant analysis under the missingness completely at random model is established. The technical analysis for the case of missing data is much more challenging than that for complete data. We establish a large deviation result for the generalized sample covariance matrix, which serves as a key technical tool and can be of independent interest. An application to lung cancer and leukaemia studies is also discussed.

Suggested Citation

  • T. Tony Cai & Linjun Zhang, 2019. "High dimensional linear discriminant analysis: optimality, adaptive algorithm and missing data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(4), pages 675-705, September.
    • Handle: RePEc:bla:jorssb:v:81:y:2019:i:4:p:675-705
    DOI: 10.1111/rssb.12326
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    Cited by:

    1. Rasoul Lotfi & Davood Shahsavani & Mohammad Arashi, 2022. "Classification in High Dimension Using the Ledoit–Wolf Shrinkage Method," Mathematics, MDPI, vol. 10(21), pages 1-13, November.
    2. Pun, Chi Seng & Hadimaja, Matthew Zakharia, 2021. "A self-calibrated direct approach to precision matrix estimation and linear discriminant analysis in high dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).

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