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In some papers about $\mathcal{l}_1$-penalized regression, $$ \hat{\beta}=\underset{\beta\in\mathbb{R}^{p}}{\operatorname{\arg\min}}\|Y-X\beta\|_2^2+\lambda\|\beta\|_1, $$ the authors say that they obtained some theoretical results for the estimators. One is the consistency of the estimator $\hat{\beta}$.

In high-dimensional settings, what does consistency really mean? Does consistency have different meanings in different places?

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    $\begingroup$ Consistency of an estimator $\hat\theta_n$ means that $P(\lVert \theta - \hat\theta_n \rVert > \varepsilon) \to 0$ as the sample size $n$ grows to infinity, for any small $\varepsilon>0$. I think in high-dimensional settings you'll simply use the appropriate multidimensional Euclidean norm, $\lVert x \rVert = \sqrt{\sum_i x_i^2}$, but online people write $||$ or $\operatorname{plim}$ and don't discuss the norm involved. $\endgroup$ Commented Nov 11, 2024 at 10:31
  • $\begingroup$ What is a "high" dimension for you? $\endgroup$ Commented Nov 11, 2024 at 12:35
  • $\begingroup$ @whuber p$\gg$ n $\endgroup$ Commented Nov 11, 2024 at 12:50
  • $\begingroup$ And just how meaningful would that be for any kind of a mathematical definition of consistency? $\endgroup$ Commented Nov 11, 2024 at 14:39
  • $\begingroup$ I didn't express myself clearly. What I want to ask is what the term 'consistency' usually means in high-dimensional statistical papers, because it seems to have different definitions in different places, and it seems to be true. $\endgroup$ Commented Nov 12, 2024 at 2:32

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