Show that if one replaces                                 1 0 |f(k) (x)| 2 dx by                                  R L |f(k) (x)| 2 dx in the definition of the penalized least...

Show that if one replaces

by

in the definition of the penalized least squares estimate for some

                    −∞ ≤ L ≤ min{X1,...,Xn} ≤ max{X1,...,Xn} ≤ R ≤ ∞,

then the values of the estimate on [min{X1,...,Xn}, max{X1,...,Xn}] do not change. Hint: Apply Lemma 20.2 with

                                 a = min{X1,...,Xn} − and b = max{X1,...,Xn} +

and show that a function f ∈ Ck(R), which minimizes

                              1 n n i=1 |f(Xi) − Yi| 2 + λ

satisfies f(k) (x) = 0 for L<>