2015/09/30 by Lee H. Dicker · 43 citations
Mathematics · #Advanced Statistical Methods and Models #Asymptotic analysis #Asymptotic distribution #Dimension (graph theory) #Distribution (mathematics) #Estimator #Linear regression #Minimax #Point processes and geometric inequalities #Ridge #Statistical Methods and Inference #math.ST #stat.TH
paper · pdf · doi:10.3150/14-bej609
published in Bernoulli 22(1) (Chapman and Hall London) · Published at http://dx.doi.org/10.3150/14-BEJ609 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2015/09/30 · arxiv created 2016/01/15 · arxiv updated 2016/01/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study asymptotic minimax problems for estimating a d-dimensional regression parameter over spheres of growing dimension (d→∞). 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→ρ∈(0,∞).