Given observed data from a normal model, $Y\sim N(X\beta, \sigma^2 I)$, where $X$ is a $n\times p$ matrix of known predictors, I want to compute a confidence interval for $||\beta||_2$, the L2 norm. I initially thought that I could invert the standard F-statistic -- $\hat{\beta}^\top(X^\top X)\hat{\beta}/(p\hat{\sigma}^2)$ has a non-central F distribution -- but the mapping from L2-norms to non-centrality parameters isn't one-to-one, so it seems this approach is bound to fail. Are there any standard methods for this?
My fallback would be to compute Bayesian credible intervals by sampling from a posterior distribution.
I should mention that in the intended application I expect the columns of $X$ to be weakly correlated, but I would be interested in a method that assumes orthogonality.
