Equivalence between adaptive Lasso and generalized ridge estimators in linear regression with orthogonal explanatory variables after optimizing regularization parameters

Mineaki Ohishi (), Hirokazu Yanagihara and Shuichi Kawano
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Mineaki Ohishi: Hiroshima University
Hirokazu Yanagihara: Hiroshima University
Shuichi Kawano: The University of Electro-Communications

Annals of the Institute of Statistical Mathematics, 2020, vol. 72, issue 6, No 8, 1516 pages

Abstract: Abstract In this paper, we deal with a penalized least-squares (PLS) method for a linear regression model with orthogonal explanatory variables. The used penalties are an adaptive Lasso (AL)-type $$\ell _1$$ ℓ 1 penalty (AL penalty) and a generalized ridge (GR)-type $$\ell _2$$ ℓ 2 penalty (GR penalty). Since the estimators obtained by minimizing the PLS methods strongly depend on the regularization parameters, we optimize them by a model selection criterion (MSC) minimization method. The estimators based on the AL penalty and the GR penalty have different properties, and it is universally recognized that these are completely different estimators. However, in this paper, we show an interesting result that the two estimators are exactly equal when the explanatory variables are orthogonal after optimizing the regularization parameters by the MSC minimization method.

Keywords: Adaptive Lasso; $$C_p$$ C p criterion; GCV criterion; Generalized ridge regression; GIC; Linear regression; Model selection criterion; Optimization problem; Regularization parameters; Sparsity (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10463-019-00734-2

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