Ex. 8.4

Ex. 8.4

Consider the bagging method of Section 8.7. Let our estimate f^(x) be the B-spline smoother μ^(x) of Section 8.2.1. Consider the parametric bootstrap of equation (8.6), applied to this estimator. Show that if we bag f^(x), using the parametric bootstrap to generate the bootstrap samples, the bagging estimate f^bag(x) converges to the original estimate f^(x) as B.

Soln. 8.4

By definition of bagging we get

(1)f^bag(x)=1Bb=1Bf^b(x)

where

f^b(x)=Sy=S(f^(x)+ϵb)
S=N(NTN)1NT

and ϵbN(0,σ2) for B-spline smoother. Note that S2=S, we obtain

f^b(x)=S(f^(x)+ϵb)=S(Sy+ϵb)=Sy+Sϵb.

Therefore (1) reduces to

f^bag(x)=Sy+S(1Bb=1Bϵb).

From here it's easy to see that

limBf^bag(x)=Sy=f^(x).