LASSO and adaptive LASSO are two different things, right? (To me, the penalties look different, but I’m just checking whether I miss something.)

When you generally speak about elastic net, is the special case LASSO or adaptive LASSO?

Which one does the glmnet package do, provided you choose alpha=1?

Adaptive LASSO works on milder conditions, right? Both have the oracle property in suitable data, right?

**Answer**

Brief answers to your questions:

- Lasso and adaptive lasso are different. (Check Zou (2006) to see how adaptive lasso differs from standard lasso.)
- Lasso is a special case of elastic net. (See Zou & Hastie (2005).)

Adaptive lasso is not a special case of elastic net.

Elastic net is not a special case of lasso or adaptive lasso. - Function
`glmnet`

in “glmnet” package in R performs lasso (not adaptive lasso) for`alpha=1`

. - Does lasso work under milder conditions than adaptive lasso? I cannot answer this one (should check Zou (2006) for insights).
- Only the adaptive lasso (but not lasso or elastic net) has the oracle property. (See Zou (2006).)

**References:**

- Zou, Hui. “The adaptive lasso and its oracle properties.”
*Journal of the American Statistical Association*101.476 (2006): 1418-1429. - Zou, Hui, and Trevor Hastie. “Regularization and variable selection via the elastic net.”
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*67.2 (2005): 301-320.

**Attribution***Source : Link , Question Author : Mr Validation , Answer Author : Richard Hardy*