 Solution

Choose $({\lambda }_1,{\lambda }_2)$ on a grid by K-fold cross-validation, comparing the same held-out prediction loss and optionally applying the one-standard-error rule for a simpler model. A genuinely untouched test set can then estimate final prediction error.

Ordinary model-based intervals after selecting nonzero coefficients ignore selection and are generally invalid. Valid approaches include Debiased Lasso or a selective-inference procedure under its assumptions, sample splitting followed by an unpenalized refit and inference on the independent half, or a bootstrap that repeats both tuning and fitting and is interpreted with care near the nonsmooth zero threshold.