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

Author

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  • Mineaki Ohishi

    (Hiroshima University)

  • Hirokazu Yanagihara

    (Hiroshima University)

  • Shuichi Kawano

    (The University of Electro-Communications)

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.

Suggested Citation

  • Mineaki Ohishi & Hirokazu Yanagihara & Shuichi Kawano, 2020. "Equivalence between adaptive Lasso and generalized ridge estimators in linear regression with orthogonal explanatory variables after optimizing regularization parameters," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1501-1516, December.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:6:d:10.1007_s10463-019-00734-2
    DOI: 10.1007/s10463-019-00734-2
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
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    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    5. Francis K. C. Hui & David I. Warton & Scott D. Foster, 2015. "Tuning Parameter Selection for the Adaptive Lasso Using ERIC," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 262-269, March.
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