Additional File 2 of ”Goodness-of-fit tests and nonparamet- ric adaptive estimation for spike train analysis”: proof of Theorem 2 of [1]
Patricia Reynaud-Bouret
∗1and Vincent Rivoirard
2and Franck Grammont
1and Christine Tuleau- Malot
11Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France
2CEREMADE UMR CNRS 7534, Universit´e Paris Dauphine, Place du Mar´echal De Lattre De Tassigny, 75775 PARIS Cedex 16, France
Email: Patricia Reynaud-Bouret∗- [email protected]; Vincent Rivoirard - [email protected]; Franck Grammont - [email protected]; Christine Tuleau-Malot - [email protected];
∗Corresponding author
In this additional file, the intensityλ(.) should be understood as a function on the whole real line which is null outside [0, Tmax]. In particular, the Poisson processNa,n, with intensitynλ(.), exists now onR, but there is no points outside [0, Tmax] andNa,n(R) =Na,n([0, Tmax]). The proof is inspired by Proposition 2 of [2]. We set:
χ= (1 +η)(1 +kKk1)kKk2. Therefore, we have:
A(h) = sup
h′∈H
(
kλˆh,hn ′−λˆKnh′k2−χp
Na,n(R) n√
h′ )
+
and
ˆh= arg min
h∈H
(
A(h) +χp
Na,n(R) n√
h )
. For anyh∈ H,
kλˆGLn −λk2≤A1+A2+A3, with
A1:=kˆλGLn −ˆλˆh,hn k2≤A(h) +χp
Na,n(R) np
ˆh , A2:=kˆλˆh,hn −λˆKnhk2≤A(ˆh) +χp
Na,n(R) n√
h
and
A3:=kˆλKnh−λk2. By definition of ˆh, we have:
A1+A2≤2A(h) +2χp
Na,n(R) n√
h . Therefore, by setting
ζn(h) := sup
h′∈H
(
k(ˆλh,hn ′−E[ˆλh,hn ′])−(ˆλKnh′ −E[ˆλKnh′])k2−χp
Na,n(R) n√
h′ )
+
we have:
A1+A2 ≤ 2ζn(h) + 2 sup
h′∈HkE[ˆλh,hn ′]−E[ˆλKnh′]k2+2χp
Na,n(R) n√
h
≤ 2ζn(h) + 2 sup
h′∈HkKh⋆ Kh′⋆ λ−Kh′⋆ λk2+2χp
Na,n(R) n√
h
≤ 2ζn(h) + 2kKk1kKh⋆ λ−λk2+2χp
Na,n(R) n√
h . Finally, since (a+b+c)2≤3a2+ 3b2+ 3c2,
E[(A1+A2)2] ≤ 12E[ζn2(h)] + 12kKk21kKh⋆ λ−λk22+12χ2E[Na,n(R)]
n2h
≤ 12E[ζn2(h)] + 12kKk21kKh⋆ λ−λk22+12χ2kλk1
nh . For the last term, we obtain:
E[A23] = Z
Var(ˆλKnh(x))dx+kKh⋆ λ−λk22
= 1
n2 Z Z
Kh2(x−u)nλ(u)dudx+kKh⋆ λ−λk22
= kλk1
nh kKk22+kKh⋆ λ−λk22. Finally, replacingχwith its definition, we obtain: for anyh∈ H,
EkλˆGLn −λk22 ≤ 2E[(A1+A2)2] + 2E[A23]
≤ 2(1 + 12kKk21)kKh⋆ λ−λk22+ 2(1 + 12(1 +η)2(1 +kKk1)2)kλk1
nh kKk22+ 24E[ζn2(h)].
The result follows by using the next lemma.
Lemma 1. If H ⊂ {D−1: D= 1, ..., Dmax}with Dmax=δnfor some δ >0, and ifkλk∞<∞, then there exists a constantC depending onδ, η,kKk2,kKk1,kλk1 andkλk∞ such that for anyh∈ H
E[ζn2(h)]≤Cn−1.
Proof. First, we notice that for anyh∈ H ζn(h) ≤ sup
h′∈H
(
kλˆh,hn ′ −E[ˆλh,hn ′]k2+kˆλKnh′−E[ˆλKnh′]k2−χp
Na,n(R) n√
h′ )
+
≤ sup
h′∈H
(
(kKk1+ 1)kλˆKnh′ −E[ˆλKnh′]k2−χp
Na,n(R) n√
h′ )
+
≤ (kKk1+ 1)Sn, with
Sn:= sup
h∈H
(
kλˆKnh−E[ˆλKnh]k2−kKk2(1 +η)p
Na,n(R) n√
h
)
+
. We then have:
E[ζn2(h)]≤(kKk1+ 1)2(A+B) with
A:=E[S2n1{Na,n(R)≤(1−α)2nkλk1}], B:=E[Sn21{Na,n(R)>(1−α)2nkλk1}] for allα∈(0,1). We have:
Sn2 ≤ 2 sup
h∈HkλˆKnhk22+ 2 sup
h∈HkE[ˆλKnh]k22
≤ 2n−2sup
h∈H
Z dx
Z
Kh(x−u)dNa,n(u) 2
+ 2 sup
h∈HkKh⋆ λk22
≤ 2n−2sup
h∈H
Z Z
Kh2(x−u)dNa,n(u))dx×Na,n(R) + 2 sup
h∈HkKh⋆ λk22
≤ 2n−2sup
h∈H
kKk22
h ×Na,n(R)2+ 2 sup
h∈H
kKk22
h kλk21
≤ 2δnkKk22
Na,n(R)2
n2 +kλk21
. Therefore,
A≤4δnkKk22kλk21×P(Na,n(R)≤(1−α)2nkλk1).
To bound the last term, we use, for instance, Inequality (5.2) of [3] (withξ= (2α−α2)nkλk1 and with the functionf ≡ −1), which shows that there existsα′ >0 only depending onαsuch that
P(Na,n(R)≤(1−α)2nkλk1)≤exp(−α′kλk1×n).
This shows that
A≤CAn−1,
whereCA depends onα,δ,kKk2 andkλk1. Now, we deal with the term B by fixing the previous valueα:
we setα= min(η/2,1/4). This implies
(1 +η)(1−α)≥1 + η 4 and
B = E[Sn21{Na,n(R)>(1−α)2nkλk1}]
≤ E
sup
h∈H
(
kˆλKnh−E[ˆλKnh]k2−(1 + η4)kKk2
pkλk1
√nh
)2
+
=
Z +∞ 0
P
sup
h∈H
(
kλˆKnh−E[ˆλKnh]k2−(1 +η4)kKk2
pkλk1
√nh
)2
+
≥x
dx
≤ X
h∈H
Z +∞ 0
P
(
kλˆKnh−E[ˆλKnh]k2−(1 +η4)kKk2
pkλk1
√nh
)2
+
≥x
dx.
To conclude, it remains to control for anyx≥0, the probability inside the integral. For this purpose, we apply Corollary 2 of [3] and we set
U(x) = ˆλKnh(x)−E[ˆλKnh(x)] = 1 n
Z
Kh(x−u)dNa,n(u)−(Kh⋆ λ)(x).
IfAis a countable dense subset of the unit ball ofL2(R), we have:
kUk2 = sup
a∈A
Z
a(t)U(t)dt
= sup
a∈A
Z a(t)
1 n
Z
Kh(t−u)dNa,n(u)− Z
Kh(t−u)λ(u)du
dt
= sup
a∈A
Z
ψa(u)(dNa,n(u)−nλ(u))du, with for anya∈ Aand anyu∈R,
ψa(u) = 1 n
Z
a(t)Kh(t−u)dt.
We have for anyu,
ψ2a(u)≤ 1 n2
Z
a2(t)dt Z
Kh2(t−u)dt=kKk22
n2h . So, ifb= kKk2
n√
h, kψak∞≤b.We have E[kUk22] =
Z
E[U2(t)]dt= Z
var(ˆλKnh(t))dt= kλk1
nh kKk22,
which implies
E[kUk2]≤ pkλk1
√nh kKk2. Finally,
v := sup
a∈A
Z
ψ2a(x)nλ(x)dx
= sup
a∈A
Z Z a(t)
n Kh(t−x)dt 2
nλ(x)dx
≤ 1 n
Z
λ(x)dx Z
a2(t)|Kh(t−x)|dt Z
|Kh(t−x)|dt
≤ kKk21kλk∞
n .
Corollary 2 of [3] gives: for anyǫ >0, for anyu >0, P kλˆKnh−E[ˆλKnh]k2≥(1 +ǫ)
pkλk1
√nh kKk2+ r12
nkKk21kλk∞u+ 5
4+32 ǫ
kKk2 n√
hu
!
≤exp(−u).
Then, we conclude by using exactly the same computation as in the proof of Lemma 1 of [4, p 32-34] that gives:
B≤CBn−1, whereCB is a constant depending onδ, η,kKk2,kKk1andkλk∞.
The lemma leads to the result.
References
1. Reynaud-Bouret P, Rivoirard V, Grammont F, Tuleau-Malot C:Goodness-of-fit tests and nonparametric adaptive estimation for spike train analysis. Tech. rep., http://hal.archives-ouvertes.fr/hal-00789127 2013.
2. Doumic M, Hoffmann M, Reynaud-Bouret P, Rivoirard V:Nonparametric estimation of the division rate of a size-structured population.SIAM J. Numer. Anal.2012,50(2):925–950.
3. Reynaud-Bouret P: Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities.Probab. Theory Related Fields 2003,126:103–153.
4. Doumic M, Hoffmann M, Reynaud-Bouret P, Rivoirard V:Nonparametric estimation of the division rate of a size-structured population. Tech. rep., http://hal.archives-ouvertes.fr/hal-00578694 2012.
