Estimation of Singularity Location for Poisson Process

Transcription

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Estimation of Singularity Location
for Poisson Process
S. Dachian
Laboratoire de Mathématiques Appliquées
Université Blaise Pascal
Clermont-Ferrand, FRANCE
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Statistique Asymptotique des Processus Stochastiques IV
Le Mans, December 19-20, 2002
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The model
– The process. — Poisson process of intensity function Sθ (·):
X = {X(t), 0 6 t 6 T }.
– The observations. — n independent realizations (trajectories) of X:
(X1 , . . . , Xn ) = X n .
– The hypotheses on Sθ (·) . — The intensity function Sθ (·) is regular everywhere on [0,T ] except at the point θ, where it has a singularity.
– The unknown parameter. — The location (the point) of the singularity:
θ ∈ Θ = (α,β) ⊆ (0,T ).
– The types of singularities. — Three types: “cusp”, singularity of “0”-type
and singularity of “∞”-type.
– The asymptotics. — n −→ ∞.
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S. Dachian, Clermont-Ferrand (France)
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(
We consider
Sθ (t) of the form Sθ (t) =
a |t − θ|p + Ψ(θ, t),
p
b |t − θ| + Ψ(θ, t),
if t 6 θ
if t > θ
.
We suppose that a2 +b2 > 0, Sθ (t) > 0 for all t 6= θ, the function Ψ(θ, t) is continuous
and uniformly in t Hölder continuous of order µ with respect to θ.
“cusp”
singularity of “0”-type
singularity of “∞”-type
0 < p < 1/2
0<p<1
−1 < p < 0
µ > p + 1/2
µ > (p + 1)/2
µ > (p + 1)/2
Ψ(θ, θ) > 0
Ψ(θ, θ) = 0
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The history of the problem
– Prakasa Rao, B.L.S., “Estimation of the location of the cusp of a continuous
density”, Annals of Mathematical Statistics, vol. 20, no. 1, pp. 76–87, 1968.
– Ibragimov, I.A. et Khasminskii, R.Z., “Statistical Estimation. Asymptotic
Theory”, Springer-Verlag, New York, 1981.
– Dachian, S., “Estimation of Cusp Location by Poisson Observations”, Statistical Inference for Stochastic Processes, to appear, 2001.
– Dachian, S., “Estimation of Singularity Location by Poisson Observations”, in
preparation, 2002.
– Dachian, S. et Kutoyants, Yu.A., “On Cusp Estimation of Ergodic Diffusion
Process”, Journal of Statistical Planning and Inference, to appear, 2001.
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The likelihood ratio is:
(
L(θ, X n ) = exp
n Z
X
ZT
T
ln Sθ (t) dXi (t) − n
i=1 0
)
£
¤
Sθ (t) − 1 dt
.
0
The maximum likelihood estimator (MLE) θbn is defined as one of the solutions of
the maximum likelihood equation
L(θbn , X n ) = sup L(θ, X n ).
θ∈Θ
The Bayesian estimator (BE) for prior density q(·) and quadratic loss function is
Zβ
¡
¢
¡ n¢
n
e
defined by θn = θ q θ|X dθ, where the posterior density q ·|X is given by:
α

¡
¢
n
q θ|X = L(θ, X n ) q(θ) 
Zβ
−1
L(θ, X n ) q(θ) dθ
.
α
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The “cusp” case. — We introduce the stochastic process (on R)
½
¾
1
2p+1
Z1 (u) = exp Γθ W p+1/2 (u) − Γ2θ |u|
2
¡
¢·
¸
2
2
p
+
1
,
p
+
1
B
a
+
b
− 2ab , 0 < Γ2θ < +∞,
where
Γ2θ =
Ψ(0, 0)
cos(πp)
and W H (·) is a fractional Brownian motion (fBm) of Hurst parameter H, that is, a
centered Gaussian process with covariance
h
i 1h
i
2H
2H
2H
H
H
E W (u1 ) W (u2 ) = |u1 | + |u2 | − |u1 − u2 |
.
2
We introduce equally the random variables ξ1 and ζ1 by
Z1 (ξ1 ) = sup Z1 (u)
u∈R
and
−1
 +∞
+∞
Z
Z
Z1 (u) du .
u Z1 (u) du 
ζ1 =
−∞
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−∞
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Theorem. — In the case of “cusp”, we have the following lower bound on the
risks of all the estimators of θ:
³
lim lim inf
δ→0 n→∞ θ n
sup
1/(2p+1)
Eθ n
|θ−θ0 |<δ
¡
¢ ´2
θn − θ
> Eζ12
for all θ0 ∈ Θ, where inf is taken on the set of all the estimators θn of θ.
Definition. — We say that an estimator θn is asymptotically efficient if
³
lim lim
sup
δ→0 n→∞ |θ−θ0 |<δ
Eθ n
1/(2p+1)
¡
θn − θ
¢ ´2
= Eζ12
for all θ0 ∈ Θ.
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Theorem. — In the case of “cusp”, the BE θen and the MLE θbn have uniformly
in θ ∈ K (for any compact K ⊂ Θ) the following properties:
• θen and θbn are consistent, that is,
Pθ
θ
θen −→
and
Pθ
θ,
θbn −→
• the limit distributions of θen and θbn are given by
¡
¡
¢
¢
1/(2p+1) e
1/(2p+1) b
n
θn − θ =⇒ ζ1 and n
θn − θ =⇒ ξ1 ,
• for any k > 0 the convergence of moments equally holds:
¯
¢¯¯k
¯ 1/(2p+1) ¡ e
k
lim Eθ ¯n
θn − θ ¯ = E |ζ1 | ,
n→∞
¯
¢¯¯k
¯ 1/(2p+1) ¡ b
k
lim Eθ ¯n
θn − θ ¯ = E |ξ1 | .
n→∞
Moreover, the BE θen are asymptotically efficient.
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The “0”-type and “∞”-type singularity cases. — We introduce
the stochastic process (on R)
(
Z2 (u) = exp
+∞
Z
¯
¢ a − b p+1
u ¯¯ ¡
¯
p
ln ¯1 − ¯ ν(dz) − Eν(dz) −
|u|
sign(u) +
z
p+1
−∞
+ ln
½
where d(z) =
a,
b,
a
b
Zu
0
)
+∞·
¸
Z
¯
¯
¯
¯
u ¯p
u¯
¯
¯
p
ν(dz) −
¯1 − ¯ − 1 − p ln ¯1 − ¯ d(z) |z| dz ,
z
z
−∞
if z 6 0
if z > 0
p
and ν is a Poisson process of intensity d(z) |z| .
We introduce equally the random variables ξ2 (in the case of “0”-type singularity
only) and ζ2 (in both cases) by
and
Z2 (ξ2 ) = sup Z2 (u)
u∈R
 +∞
−1
+∞
Z
Z
ζ2 =
u Z2 (u) du 
Z2 (u) du .
−∞
&
−∞
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Theorem. — In the case of “0”-type or “∞”-type singularity, we have the
following lower bound on the risks of all the estimators of θ:
³
¡
¢´2
1/(p+1)
lim lim inf sup Eθ n
θn − θ
> Eζ22
δ→0 n→∞ θ n
|θ−θ0 |<δ
for all θ0 ∈ Θ, where inf is taken on the set of all the estimators θn of θ.
Definition. — We say that an estimator θn is asymptotically efficient if
³
lim lim
sup
δ→0 n→∞ |θ−θ0 |<δ
Eθ n
1/(p+1)
¡
¢´2
θn − θ
= Eζ22
for all θ0 ∈ Θ.
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Theorem. — The BE θen (in both cases) and the MLE θbn (in the case of
“0”-type singularity only) have uniformly in θ ∈ K (for any compact K ⊂ Θ) the
following properties:
• θen and θbn are consistent, that is,
Pθ
θen −→
θ
and
Pθ
θbn −→
θ,
• the limit distributions of θen and θbn are given by
¡
¢
¡
¢
1/(p+1) e
1/(p+1) b
n
θn − θ =⇒ ζ2 and n
θn − θ =⇒ ξ2 ,
• for any k > 0 the convergence of moments equally holds:
¯
¢¯¯k
¯ 1/(p+1) ¡ e
k
lim Eθ ¯n
θn − θ ¯ = E |ζ2 | ,
n→∞
¯
¢¯¯k
¯ 1/(p+1) ¡ b
k
lim Eθ ¯n
θn − θ ¯ = E |ξ2 | .
n→∞
Moreover, the BE θen are asymptotically efficient.
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Ideas of the proof
We use the Ibragimov and Khasminskii method which consist in studying the normalized likelihood ratio process
L (θu , X n )
Zn (u) =
,
L (θ, X n )
u ∈ Un ,
where we denote θu = θ + u n−1/ν (with ν = 2p + 1 or ν = p + 1) and the set
¡ 1/ν
¢
1/ν
Un = n (α − θ) , n (β − θ) , and establishing the three following properties:
– The finite-dimensional distributions of Zn (u) converge to those of Z(u) (with
Z = Z1 or Z = Z2 ) uniformly in θ ∈ K.
¯
¯2
¯ 1/2
¯
ν
1/2
– Eθ ¯Zn (u1 ) − Zn (u2 )¯ 6 C |u1 − u2 | uniformly in θ ∈ K.
©
νª
1/2
– Eθ Zn (u) 6 exp −c |u| uniformly in θ ∈ K.
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The simulations
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– The model. — Sθ (t) = 2 − |t − θ|
– The true value of the parameter. — θ = 1,5
– The rate of convergence of the BE and the MLE. — n1/(2p+1) = n5/6
– The limit distributions of the BE and the MLE. —
¡
¡
¢
¢
1
1
5/6 e
5/6 b
n
θn − θ =⇒ ζ1 = ζ ≈ 14ζ and n
θn − θ =⇒ ξ1 = ξ ≈ 14ξ,
γ
γ
2/(2p+1)
where γ = Γθ
≈ 0,07 and the random variables ξ and ζ are defined by
 +∞
−1
+∞
Z
Z
Z(ξ) = sup Z(u) and ζ =
u Z(u) du 
Z(u) du ,
u∈R
½
where Z(u) = exp W p+1/2 (u) −
&
−∞
1
2p+1
|u|
2
−∞
¾
½
= exp W 3/5 (u) −
¾
1
6/5
|u|
.
2
%
12
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
1/10
The intensity function Sθ (t) = 2 − |t − θ|
2
for θ = 1,5
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20
15
10
5
0
-5
-10
-15
-20
-25
-30
0
0.5
1
1.5
2
A realization of the log-likelihood for n = 1000
14
17.5
17
16.5
16
15.5
15
1.47
1.48
1.49
1.5
1.51
1.52
1.53
A “zoom” of the log-likelihood in the vicinity of the true value θ = 1,5
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0.2
0.1
0
-0.1
-0.2
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
The error θbn − 1.5 of the MLE for n = 50, 100, . . . , 10000
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100
0
-100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
The normalized error n5/6 (θbn − 1.5) of the MLE for n = 50, 100, . . . , 10000
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0.2
0.1
0
-0.1
-0.2
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
The error θen − 1.5 of the BE for n = 100, 200, . . . , 10000
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100
0
-100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
The normalized error n5/6 (θen − 1.5) of the BE for n = 100, 200, . . . , 10000
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1000
900
800
700
600
500
400
300
200
100
0
0
100
200
300
400
500
600
700
800
900
1000
1100
The empirical variances of n5/6 (θen − 1.5) for BE and n5/6 (θbn − 1.5) for MLE
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Estimation of Singularity Location for Poisson Process

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