16
Denis Belomestny and Markus Reiβ
An application of Proposition 1(d) therefore shows for σ > 0
∞ I E[ O)( x ) -O ( x )] I d x 6 e-A
-∞
+ 3∆2 + 2C0∆2 + 4C2∆e-(A-∆)
∆2.
In the case σ = 0 we consider the index jt with Xj-- 1 6 γT < xj- and
face an additional error estimated by
χxi*
xxi--1
IE[O)(x) - O(x)]I dx 6
xi-
k(O
xi--1
β0),∣∣L∞ I
2(x - xj--1)(xj- - x)
xj- - xj--1
dx
6 ∣(O - β0)0∣L∞ (xj- - xj--1)2
We infer that this error term is also of order ∆2 and thus does not enlarge
the convergence rate.
6.3. Upper bound for σ2
The rate for σ2 follows once the general risk estimate
E[∣σ2 - σ212] . U-2(s+3) + E(Tσ2U2)U-1 ε2 + E(Tσ2maxU2)2U4ε4 (32)
has been shown for U . ∆-1 uniformly over Gs (R, σmax ), since the explicit
choice of U renders the second and third term asymptotically negligible.
Consider in the definition (12) of ψ separately the linearisation L, ne-
glecting the stabilisation by κ, and the remainder term R:
L(u) := T-1 φτ(u - i)-1(u - i)uF(O) - O)(u), (33)
R(u) := ψ)(u) - ψ(u) - L(u). (34)
When neglecting the remainder term, we may view ψ)(u) as observation
of ψ(u) in additive noise, whose intensity grows like ∣φT(u-i) ∣-11(u-i)u∣ ~
u2 eTσ2 u2 for IuI → ∞. This heteroskedasticity reflects the degree of ill-
posedness of the estimation problem.
Lemma 1. For all u ∈ R the remainder term satisfies
∣R(u)∣ 6 T-1κ(u)-2(u4 +u2)∣F(O) - O)(u)∣2.
Proof. Let us set ⅛)T(u - i) := 1 - u(u - i)FO(u) which equals eTψ(u) if
-⅛
∣⅛)T(u - i) I > κ(u). Using ∣eTψ(u) ∣ > κ(u), u ∈ R, we obtain by a second-
order expansion of the logarithm
∣Tψ(u) - log(φT(u - i))) - φT(u - i)-1(eTψ(u) - φT(u - i))∣
6 1 κ(u)-2 ∣eTψ(u) - φT(u - i) ∣2.
This gives the result whenever ∣⅛)T(u - i)∣ > κ(u). For the other values u
the inequalities ∣⅛)T(u - i)∣ < κ(u) 6 ∣φT(u - i)∣/2 imply 1 6 ∣⅛)T(u - i) -
φT(u - i) ∣κ(u)-1 and hence
∣φT(u - i)-1(eTψ(u) - ⅛)T(u - i))∣ 6 2κ(u)-21⅛)T(u - i) - φT(u - i)∣2