The adaptive lasso penalises each coefficient differently by employing a weighted vector. Becuase of this modification from the original lasso, if combine with the additional conditions such as, $\lambda_{n}$$/$$\sqrt{n}$ $\rightarrow$ $0$ and $\lambda_{n}$$n^{(\gamma-1)/2}$ $\rightarrow$ $\infty$, then we say that the adaptive lasso has achieved the oracle properties $.$

1.Could you simply explain why that is the case that the adaptive lasso is oracle. More specifically, why do we need those two conditions and, isn't the speed of $\lambda_{n}$ is slower than root n in this case?

2.More importantly, since the adaptive lasso has achieved the oracle property, which means that it could identify the true model asymptotically. Can I assume that there is a shift in power (trade-off) from being good at prediction, like the lasso, to more robust in producing the right explanation?

Let's focus on the low dimensional cases, where the number of variables is fixed.

Thanks a lot for your help, and let me know if anything can be made more clearly.



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