Prove XXXXXXXXXXHint: Show that, for (k, λ) ∈K× Λn,                                  penn(k, λ) ≥ c1 log n n · λ1/(2k) + c2 log n n and apply (21.6). Formulate and prove a multivariate version of...

Prove (21.10). Hint: Show that, for (k, λ) ∈K× Λn,

                                 penn(k, λ) ≥ c1 log n n · λ1/(2k) + c2 log n n

and apply (21.6).

Formulate and prove a multivariate version of Lemma 21.1. Use it to define adaptive penalized least squares estimates for multivariate data and formulate and prove a multivariate version of Theorem 21.2.