On the asymptotic risk of ridge regression with many predictors

Krishnakumar Balasubramanian (), Prabir Burman () and Debashis Paul ()
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Krishnakumar Balasubramanian: University of California Davis
Prabir Burman: University of California Davis
Debashis Paul: University of California Davis

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 3, 1040-1054

Abstract: 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.

Keywords: Double asymptotics; Eigenvalue decay; Multicollinearity; Ridge regression; Random matrix theory (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-024-00646-9

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