In the following we describe a modification of the estimate in Corollary 12.1 which is weakly and strongly consistent (cf. Problem 12.4). Set                              Pn =  (M,K) ∈ N0 × N : 0 ≤ M...

In the following we describe a modification of the estimate in Corollary 12.1 which is weakly and strongly consistent (cf. Problem 12.4). Set

                             Pn =  (M,K) ∈ N0 × N : 0 ≤ M ≤ log(n), log(n) 2 ≤ K ≤ n1−δ ,

where 0 <>∈ Pn let Fn,(M,K) be the set of all piecewise polynomials of degree M (or less) w.r.t. an equidistant partition of [− log(n), log(n)] into K cells, and set βn = 5log(n) and penn((M,K)) = log(n) 4 K(M−1) n . Define the estimate mn by (12.5)–(12.10). Show that for n sufficiently large

                            E  |mn(x) − m(x)| 2 µ(dx)

      ≤ min (M,K)∈Pn  2 log(n) 4 K(M + 1) n + 2 inf f∈Fn,(M,K)  |f(x) − m(x)| 2 µ(dx)  +5 · 2568 log(n) 2 n

for every distribution of (X, Y ) such that |Y | is bounded a.s.