Statistical Details
Statistical Details for Estimation Methods
Ridge RegressionAn l2 penalty is applied to the regression coefficients during ridge regression. Ridge regression coefficient estimates are given by the following:
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is the l2 penalty, λ is the tuning parameter, N is the number of rows, and p is the number of variables.
Lasso RegressionAn l1 penalty is applied to the regression coefficients during Lasso. Coefficient estimates for the Lasso are given by the following:
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is the l1 penalty, λ is the tuning parameter, N is the number of rows, and p is the number of variables
Elastic NetThe Elastic Net combines both l1 and l2 penalties. Coefficient estimates for the Elastic Net are given by the following:
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is the |
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is the |
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λ is the tuning parameter
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N is the number of rows
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p is the number of variables
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Adaptive MethodsThe adaptive Lasso method uses weighted penalties to provide consistent estimates of coefficients. The weighted form of the l1 penalty is
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is the MLE when the MLE exists. If the MLE does not exist and the response distribution is normal, estimation is done using least squares and 
is the solution obtained using a generalized inverse. If the response distribution is not normal, 
is the ridge solution.For the adaptive Lasso, this weighted form of the l1 penalty is used in determining the 
coefficients.
coefficients.The adaptive Elastic Net uses this weighted form of the l1 penalty and also imposes a weighted form of the l2 penalty. The weighted form of the l2 penalty for the adaptive Elastic Net is
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is the MLE when the MLE exists. If the MLE does not exist and the response distribution is normal, estimation is done using least squares and 
is the solution obtained using a generalized inverse. If the response distribution is not normal, 
is the ridge solution.