Formulate and prove a version of Theorem 7.2 which uses crossvalidation instead of splitting of the data. Show that, under the conditions of part (a) of Theorem 8.1,                                   ...

Formulate and prove a version of Theorem 7.2 which uses crossvalidation instead of splitting of the data.

Show that, under the conditions of part (a) of Theorem 8.1,

                                   E{  |m(Hn) n (x) − m(x)| 2 µ(dx)} ≤ ∆(h¯n) n + c · |Qn| √n

for some constant c depending only on L and ρ.

Hint: Use Theorem A.3 for the treatment of

                                     |m(h) n (x) − m(x)| 2 µ(dx) − ∆(h) n (h ∈ Qn),

 further Problem 8.2 and Theorem 8.1 (a).