= Residual-score decomposition from square-root Lasso
For a fixed residual direction $R=Z-X\widetilde\beta_\gamma\ne0$ obtained by <Square-root Lasso>, its optimality condition gives $\|X^TR\|_\infty\leq\sqrt n\,\gamma\|R\|_2$. In a fixed-design <normal linear model>, the score $R^T(Y-X\widehat\beta)/\|R\|_2$ equals a $N(0,\sigma^2)$ component plus a remainder bounded by $\sqrt n\,\gamma\|\beta^0-\widehat\beta\|_1$. The direction must be fixed or independent of the response noise for that normality claim.
Back to article page
