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On the asymptotic risk of ridge regression with many predictors

Author

Listed:
  • Krishnakumar Balasubramanian

    (University of California Davis)

  • Prabir Burman

    (University of California Davis)

  • Debashis Paul

    (University of California Davis
    Indian Statistical Institute)

Abstract

This work is concerned with the properties of the ridge regression where the number of predictors p is proportional to the sample size n. Asymptotic properties of the means square error (MSE) of the estimated mean vector using ridge regression is investigated when the design matrix X may be non-random or random. Approximate asymptotic expression of the MSE is derived under fairly general conditions on the decay rate of the eigenvalues of $$X^{T}X$$ X T X when the design matrix is nonrandom. The value of the optimal MSE provides conditions under which the ridge regression is a suitable method for estimating the mean vector. In the random design case, similar results are obtained when the eigenvalues of $$E[X^{T}X]$$ E [ X T X ] satisfy a similar decay condition as in the non-random case.

Suggested Citation

  • Krishnakumar Balasubramanian & Prabir Burman & Debashis Paul, 2024. "On the asymptotic risk of ridge regression with many predictors," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(3), pages 1040-1054, September.
    • Handle: RePEc:spr:indpam:v:55:y:2024:i:3:d:10.1007_s13226-024-00646-9
    DOI: 10.1007/s13226-024-00646-9
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