Define the estimate mn by                                                         m˜ n(·) = arg min f∈Ck(R)  1 n n i=1 |f(Xi) − Yi| 2 I{Xi∈[−...

Define the estimate mn by

                                                        m˜ n(·) = arg min f∈Ck(R)  1 n n i=1 |f(Xi) − Yi| 2 I{Xi∈[− log(n),log(n)]}                      +λn

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∞ −∞ |f(k) (x)| 2 dx

and

                                   mn(x) = Tlog(n)m˜ n(x) · I{x∈[− log(n),log(n)]}.

Show that mn is strongly universally consistent provided

                                     λn → 0 (n → ∞) and nλn → ∞ (n → ∞).

Hint: Use the error decomposition

|mn(x) − m(x)| 2 µ(dx) =

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R\[− log(n),log(n)] |mn(x) − m(x)| 2 µ(dx) +E{|mn(X) − Y | 2 · I{X∈[− log(n),log(n)]} | Dn} −E{|m(X) − Y | 2 · I{X∈[− log(n),log(n)]}}.