Spectral calibration of exponential Levy models ??
21
7.1. Lower bound for μ in the case σ = 0
Fix a positive integer j. Let ψ(j) ∈ C∞ (R) be some function with
support in [0, 1] satisfying kψ(j) kL2 = 1, R ψ (j) (x)e-2-j x dx = 0 and
R ∖F ψ (j ) ( u ) u-212 du < ∞. Certainly, there are infinitely many functions ψ(j )
fulfilling these requirements; the last property follows for instance if ψ is the
second derivative of an L2-function. Introduce the wavelet-like notation
ψjk ( x ) := 2j/2 ψ (j )(2j x - k ), j > 0, k = 0,..., 2j - 1.
Consider for any r = (rk ) ∈ {- 1, +1}2j and some β > 0 the perturbed Levy
triplets Tr = (0,γ0, μr ) with
2j
μr (x) = μo(x) + β2-j(s+1 /2) X rkψjk(x), x ∈ R .
k=1
We note that due to Fψjk (0) = 0 and e-xψjk (x) dx = 0 the triplet
Tr satisfies the martingale condition such that Tr ∈ Gs (R, 0) holds for a
sufficiently small choice of the constant β > 0.
The Gaussian likelihood ratio of the observations under the probabilities
corresponding to Tr0 and Tr under the law of Tr for some r, r0 with rk = rk0
for all k except one k0 is given by
Λ(r0,r) = expf [ (Oro-Or)(x)ε-1 dW(x)-- I ∣Oro-Or)(x)|2ε-2 dx´ .
-∞ 2 -∞
Hence, the Kullback-Leibler divergence (relative entropy) between the two
observation models equals
KL(Tro ∖Tr) = 1 ∞ |(Oro
2 -∞
- Or)(x)|2 ε-2 dx.
The standard Assouad Lemma (Korostelev and Tsybakov 1993, Thm.
2.6.4) now yields the lower bound for the risk of any estimator μ of μ
inf sup Eτ h ∕∣^(x) - μ(x) |2 dxi & 2j ∖∖μr - μro ∣∣2 2 ~ 2-2js,
μ T =(0,γ,μ)∈Gs(R,0) '-J j
provided the Kullback-Leibler divergence K L(Tro |Tr) stays uniformly
bounded by a small constant. It remains to determine a minimal rate for
2j → ∞ such that this holds when the noise level tends to zero.
Arguing in the spectral domain and using the general estimate |ez - 1| 6
2∖z∖, for ∖z∖ 6 δ and some small δ > 0, together with ∣∣φτ,r0∣ψτ,r∣∣∞ → 1 for
2j → ∞, we obtain for all sufficiently large j
KL(Tro |Tr)
1∞
= ∏2< J∞F( Or
- Or)(u)|2 du
6ε
-2
Γ∣
-∞
φτ,r (u - i) - Ψτ,rθ (u - i )
u(u - i)
∣2
∣∣ du
More intriguing information
1. The name is absent2. The name is absent
3. PERFORMANCE PREMISES FOR HUMAN RESOURCES FROM PUBLIC HEALTH ORGANIZATIONS IN ROMANIA
4. CAPACITAÇÃO GERENCIAL DE AGRICULTORES FAMILIARES: UMA PROPOSTA METODOLÓGICA DE EXTENSÃO RURAL
5. Innovation Policy and the Economy, Volume 11
6. Visual Perception of Humanoid Movement
7. Tissue Tracking Imaging for Identifying the Origin of Idiopathic Ventricular Arrhythmias: A New Role of Cardiac Ultrasound in Electrophysiology
8. Gender and aquaculture: sharing the benefits equitably
9. Problems of operationalizing the concept of a cost-of-living index
10. The name is absent