Lasso (statistics)
Elastic net

Then β ^ = β ^ ∗ 1 + λ 2 {\displaystyle {\hat {\beta }}={\frac {{\hat {\beta }}^{*}}{\sqrt {1+\lambda _{2}}}}}, which, when the covariates are orthogonal to each other, gives β ^ j = β ^ j ∗, OLS 1 + λ 2 max (0, 1 − λ ∗ | β ^ j ∗, OLS |) = β ^ j OLS 1 + λ 2 max (0, 1 − λ 1 | β ^ j OLS |) = (1 + λ 2) − 1 β ^ j lasso. {\displaystyle {\hat {\beta }}_{j}={\frac {{\hat {\beta }}{}_{j}^{\!\;*,{\text{OLS}}}}{\sqrt {1+\lambda _{2}}}}\max {\Biggl (}0,1-{\frac {\lambda ^{*}}{{\bigl |}{\hat {\beta }}{}_{j}^{\!\;*,{\text{OLS}}}{\bigr |}}}{\Biggr)}={\frac {{\hat {\beta }}{}_{j}^{\!\;{\text{OLS}}}}{1+\lambda _{2}}}\max {\Biggl (}0,1-{\frac {\lambda _{1}}{{\bigl |}{\hat {\beta }}{}_{j}^{\!\;{\text{OLS}}}{\bigr |}}}{\Biggr)}=(1+\lambda _{2})^{-1}{\hat {\beta }}{}_{j}^{\text{lasso}}.}
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