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February 2016 Ridge regression and asymptotic minimax estimation over spheres of growing dimension
Lee H. Dicker
Bernoulli 22(1): 1-37 (February 2016). DOI: 10.3150/14-BEJ609


We study asymptotic minimax problems for estimating a $d$-dimensional regression parameter over spheres of growing dimension ($d\to\infty$). Assuming that the data follows a linear model with Gaussian predictors and errors, we show that ridge regression is asymptotically minimax and derive new closed form expressions for its asymptotic risk under squared-error loss. The asymptotic risk of ridge regression is closely related to the Stieltjes transform of the Marčenko–Pastur distribution and the spectral distribution of the predictors from the linear model. Adaptive ridge estimators are also proposed (which adapt to the unknown radius of the sphere) and connections with equivariant estimation are highlighted. Our results are mostly relevant for asymptotic settings where the number of observations, $n$, is proportional to the number of predictors, that is, $d/n\to\rho\in(0,\infty)$.


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Lee H. Dicker. "Ridge regression and asymptotic minimax estimation over spheres of growing dimension." Bernoulli 22 (1) 1 - 37, February 2016.


Received: 1 January 2013; Revised: 1 September 2013; Published: February 2016
First available in Project Euclid: 30 September 2015

zbMATH: 06543262
MathSciNet: MR3449775
Digital Object Identifier: 10.3150/14-BEJ609

Keywords: adaptive estimation , Equivariance , Marčenko–Pastur distribution , Random matrix theory

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 1 • February 2016
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