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Estimation of Singularity Location for Poisson Process S. Dachian Laboratoire de Math ematiques Appliqu ees Universit e Blaise Pascal Clermont-Ferrand, FRANCE Statistique Asymptotique des Processus

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Estimation of Singularity Location for Poisson Process

  • S. Dachian

Laboratoire de Math´ ematiques Appliqu´ ees Universit´ e 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 t T}. – The observations. — n independent realizations (trajectories) of X: (X1, . . . , Xn) = Xn. – The hypotheses on Sθ(·) . — The intensity function Sθ(·) is regular every- where 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 − → ∞.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪ We consider Sθ(t) of the form Sθ(t) = a |t − θ|p + Ψ(θ, t), if t θ b |t − θ|p + Ψ(θ, t), if t θ . We suppose that a2+b2 > 0, Sθ(t) > 0 for all t = θ, the function Ψ(θ, t) is continuous and uniformly in t H¨

  • lder 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

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

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”, Statis- tical 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.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪ The likelihood ratio is: L(θ, Xn) = exp n

  • i=1

T

  • ln Sθ(t) dXi(t) − n

T

  • Sθ(t) − 1
  • dt
  • .

The maximum likelihood estimator (MLE) θn is defined as one of the solutions of the maximum likelihood equation L( θn, Xn) = sup

θ∈Θ

L(θ, Xn). The Bayesian estimator (BE) for prior density q(·) and quadratic loss function is defined by θn =

β

  • α

θ q

  • θ|Xn

dθ, where the posterior density q

  • ·|Xn

is given by: q

  • θ|Xn

= L(θ, Xn) q(θ)  

β

  • α

L(θ, Xn) q(θ) dθ  

−1

.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

The “cusp” case. — We introduce the stochastic process (on R)

Z1(u) = exp

  • Γθ W p+1/2(u) − 1

2 Γ2

θ |u|2p+1

  • where

Γ2

θ = B

  • p + 1 , p + 1
  • Ψ(0, 0)

a2 + b2 cos(πp) − 2ab

  • ,

0 < Γ2

θ < +∞,

and W H(·) is a fractional Brownian motion (fBm) of Hurst parameter H, that is, a centered Gaussian process with covariance E

  • W H (u1) W H (u2)
  • = 1

2

  • |u1|2H + |u2|2H − |u1 − u2|2H

. We introduce equally the random variables ξ1 and ζ1 by Z1(ξ1) = sup

u∈R

Z1(u) and ζ1 =

+∞

  • −∞

u Z1(u) du  

+∞

  • −∞

Z1(u) du  

−1

.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

  • Theorem. — In the case of “cusp”, we have the following lower bound on the

risks of all the estimators of θ: lim

δ→0

lim

n→∞

inf

θn

sup

|θ−θ0|<δ

Eθ

  • n1/(2p+1)

θn − θ 2 Eζ2

1

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

lim

δ→0

lim

n→∞

sup

|θ−θ0|<δ

Eθ

  • n1/(2p+1)

θn − θ 2 = Eζ2

1

for all θ0 ∈ Θ.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

  • Theorem. — In the case of “cusp”, the BE

θn and the MLE θn have uniformly in θ ∈ K (for any compact K ⊂ Θ) the following properties:

θn and θn are consistent, that is,

  • θn

Pθ

− → θ and

  • θn

Pθ

− → θ,

  • the limit distributions of

θn and θn are given by n1/(2p+1) θn − θ

  • =

⇒ ζ1 and n1/(2p+1) θn − θ

  • =

⇒ ξ1,

  • for any k > 0 the convergence of moments equally holds:

lim

n→∞ Eθ

  • n1/(2p+1)

θn − θ

  • k

= E |ζ1|k , lim

n→∞ Eθ

  • n1/(2p+1)

θn − θ

  • k

= E |ξ1|k . Moreover, the BE θn are asymptotically efficient.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

The “0”-type and “∞”-type singularity cases. — We introduce

the stochastic process (on R) Z2(u) = exp

  • p

+∞

  • −∞

ln

  • 1 − u

z

  • ν(dz) − Eν(dz)
  • − a − b

p + 1 |u|p+1 sign(u) + + ln a b

u

  • ν(dz) −

+∞

  • −∞
  • 1 − u

z

  • p

− 1 − p ln

  • 1 − u

z

  • d(z) |z|p dz
  • ,

where d(z) = a, if z 0 b, if z 0 and ν is a Poisson process of intensity d(z) |z|p. We introduce equally the random variables ξ2 (in the case of “0”-type singularity

  • nly) and ζ2 (in both cases) by

Z2(ξ2) = sup

u∈R

Z2(u) and ζ2 =

+∞

  • −∞

u Z2(u) du  

+∞

  • −∞

Z2(u) du  

−1

.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

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: lim

δ→0

lim

n→∞

inf

θn

sup

|θ−θ0|<δ

Eθ

  • n1/(p+1)

θn − θ 2 Eζ2

2

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

lim

δ→0

lim

n→∞

sup

|θ−θ0|<δ

Eθ

  • n1/(p+1)

θn − θ 2 = Eζ2

2

for all θ0 ∈ Θ.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

✬ ✫ ✩ ✪

Theorem. —

The BE θn (in both cases) and the MLE θn (in the case of “0”-type singularity only) have uniformly in θ ∈ K (for any compact K ⊂ Θ) the following properties:

θn and θn are consistent, that is,

  • θn

Pθ

− → θ and

  • θn

Pθ

− → θ,

  • the limit distributions of

θn and θn are given by n1/(p+1) θn − θ

  • =

⇒ ζ2 and n1/(p+1) θn − θ

  • =

⇒ ξ2,

  • for any k > 0 the convergence of moments equally holds:

lim

n→∞ Eθ

  • n1/(p+1)

θn − θ

  • k

= E |ζ2|k , lim

n→∞ Eθ

  • n1/(p+1)

θn − θ

  • k

= E |ξ2|k . Moreover, the BE θn are asymptotically efficient.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

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Ideas of the proof

We use the Ibragimov and Khasminskii method which consist in studying the nor- malized likelihood ratio process Zn(u) = L (θu, Xn) L (θ, Xn) , u ∈ Un, where we denote θu = θ + u n−1/ν (with ν = 2p + 1 or ν = p + 1) and the set Un =

  • n1/ν(α − θ) , n1/ν(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. – Eθ

  • Z1/2

n

(u1) − Z1/2

n

(u2)

  • 2

C |u1 − u2|ν uniformly in θ ∈ K. – Eθ Z1/2

n

(u) exp

  • −c |u|ν

uniformly in θ ∈ K.

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

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The simulations

– The model. — Sθ(t) = 2 − |t − θ|1/10 – 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. — n5/6 θn − θ

  • =

⇒ ζ1 = 1 γ ζ ≈ 14ζ and n5/6 θn − θ

  • =

⇒ ξ1 = 1 γ ξ ≈ 14ξ, where γ = Γ2/(2p+1)

θ

≈ 0,07 and the random variables ξ and ζ are defined by Z(ξ) = sup

u∈R

Z(u) and ζ =

+∞

  • −∞

u Z(u) du  

+∞

  • −∞

Z(u) du  

−1

, where Z(u) = exp

  • W p+1/2(u) − 1

2 |u|2p+1

  • = exp
  • W 3/5(u) − 1

2 |u|6/5

  • .
  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

0.5 1 1.5 2 0.5 1 1.5 2 2.5 0.5 1 1.5 2 0.5 1 1.5 2 2.5

The intensity function Sθ(t) = 2 − |t − θ|1/10 for θ = 1,5

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

0.5 1 1.5 2

  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

5 10 15 20

A realization of the log-likelihood for n = 1000

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

1.47 1.48 1.49 1.5 1.51 1.52 1.53 15 15.5 16 16.5 17 17.5

A “zoom” of the log-likelihood in the vicinity of the true value θ = 1,5

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

  • 0.2
  • 0.1

0.1 0.2

The error θn − 1.5 of the MLE for n = 50, 100, . . . , 10000

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

  • 100

100

The normalized error n5/6( θn − 1.5) of the MLE for n = 50, 100, . . . , 10000

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

  • 0.2
  • 0.1

0.1 0.2

The error θn − 1.5 of the BE for n = 100, 200, . . . , 10000

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

  • 100

100

The normalized error n5/6( θn − 1.5) of the BE for n = 100, 200, . . . , 10000

  • S. Dachian, Clermont-Ferrand (France)

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Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002

100 200 300 400 500 600 700 800 900 1000 1100 100 200 300 400 500 600 700 800 900 1000

The empirical variances of n5/6( θn − 1.5) for BE and n5/6( θn − 1.5) for MLE

  • S. Dachian, Clermont-Ferrand (France)

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