Nonparametric estimation of the mark density of Exponential Shot-Noise

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Nonparametric estimation of the mark density of Exponential Shot-Noise

Paul Ilhe, Eric Moulines, François Roueff, Antoine Souloumiac

To cite this version:

Paul Ilhe, Eric Moulines, François Roueff, Antoine Souloumiac. Nonparametric estimation of the mark density of Exponential Shot-Noise. 10e Journées du groupe Modélisation Aléatoire et Statistique de la SMAI, Aug 2014, Toulouse, France. 2014. �hal-02437229�

Nonparametric estimation of the mark density

of Exponential Shot-Noise

P. Ilhe, E. Moulines, F. Roueff (ilhe,moulines,roueff@telecom-paristech.fr)

A. Souloumiac (antoine.souloumiac@cea.fr)

I

NTRODUCTION

Shot-noise processes - or Filtered Marked Point Processes - are natural models for phenomenons that can be represented as a linear filtration of sums of independent jumps (marks) (Yi)_i∈Z

arrived at random times (Ti)_i∈Z. There are particularly adapted to model data transfers in telecommunication or interacting particles in physics.Based on a low-frequency sample of the

Shot-Noise (X1, · · · , Xn), we introduce a novel nonparametric estimator of the mark density when arrival times follow a Poisson Process with intensity λ and the filter is of the form

h(t) = e−αt _{using a "plug-in" method with the empirical characteristic function ˆ}ϕ_n(u) = n∆ −1 Pn_k=1 eiuXk .

P

ROBLEM

We consider the stationary process (X

t

)

t≥0

defined by:

X

_t

=

X

T_k≤t

Y

_k

e

−α(t−Tk)

(1)

where:

1.

P

_k

δ

_Tk,Yk

_{is a Poisson random measure on R}

2

_{with intensity measure λLeb ⊗ F}

2. λ, α > 0

3. Y

0

has a p.d.f. θ which belong to the class:

Θ

_C,K,L,a,β

= {θ : θ

_{p.d.f. ,}

R

R

x

8

_{θ(x)dx ≤ K,}

R

R

|u|

β

_|ϕ

Y

(u)|du ≤ L, |ϕ

Y

(u)| ≥ Ce

−au

2

}

From its definition and under Assumptions 1-3, the characteristic function ϕ

X₀

of X

0

is non

zero,differentiable and given by:

ϕ

_X0

(u) = exp λ

R

₀∞

(ϕ

_Y

(ue

−αs

) − 1) ds

_,

_{u ∈ R}

S

YNTHETIC SCHEME

Figure 1:

Simulation of a Shot-Noise sample path

I

NVERSE PROBLEM AND ESTIMATOR

Inversion formula:

Differentiating the characteristic function ϕ

X₀

gives:

ϕ

Y

(u) = 1 + u

α_λ ϕ0

X0(u)

ϕ_X0(u)

,

u ∈ R

Estimation issues

:

Tempted to construct a "plug-in" estimator using the empirical characteristic (ECF) function but:

• ˆ

ϕ

_n

(u)

_{might take arbitrary small values}

• The "pluged-in" integrand might diverge

"Plug-in" estimator

We then propose a mark density estimator ˆ

θ

_n

_{given by:}

ˆ

θ

_n

(x)

= 2 max

∆

R

Z

h−1_n 0

e

−ixu



1 + u

α

λ

ˆ

ϕ

0_n

(u)

ˆ

ϕ

_n

(u)

1

| ˆϕn(u)|≥κn



!

, 0

!

(2)

where

h

_n

=



_log(n)6a



1/2

and κ

n

=

C₂

e

−ah −2 n

=

C 2

n

−1/6

Theoritical result :

sup

θ∈Θ_C,K,L,a,β

E

θ









ˆ

θ

_n

− θ







2 ∞



. log(n)

−β

N

UMERICAL

R

ESULTS

Illustration of our algorithm with two

numeri-cal examples with λ = 4.10

5

, α = 3, 5.10

5

:

• Marks

follows

a

gaussian

mixture

P

3

i=1

p

i

N

µ_i,σ_i2

(x)

with:

p = [0.3 0.5 0.2] _, µ = [4 12 22] _, σ = [1 1 0.5]

Figure 2:

Gaussian mixture case

• Marks

follows

a

gamma

mixture

P

3

i=1

p

i

G

a,βi

(x)

with :

a = 2 _, p = [0.2 0.3 0.5] _, β = [3 6 12]

Figure 3:

Gamma mixture case

A

LGORITHM

Algorithm 1:

Nonparametric mark density estimator

Input: Shot-noise observations X1, · · · , Xn

1. Choose ∆ > 0.

2. Compute the histogram (Hi)_i∈Z of observations with bins of size ∆ :

H_i = {X_k ∈ [∆i; ∆(i + 1)]}/n 3. Set (dHi)_i∈Z by dHi = ∆iHi

4. Set N = 2∆ 16 _{and compute Xφ = FFT(H, N ), dXφ = FFT(dH, N )} It gives for k = 1, · · · , N , Xφk ' ˆϕn



− 2π(k−1)_{N ∆}  _{and dXφk} ' −i ˆϕ0_n



− 2π(k−1)_{N ∆}  5. Compute the ECF estimator : ˆϕ_{Y n}  2π(k−1)

N ∆  = 1 + 2πα(k−1) N ∆λ dXφ_k dXφ_k

6. Compute the density estimator by : ˆθ_n = _{N ∆}2 F F T ( ˆϕ_{Y n}, N )

R

EMARKS

• The rate of convergence depends on the smoothness of the mark distribution

• The distribution F

X

of X

0

is regularly varying at 0 with index λ/α

,→

_{One can use Hill’s estimator to estimate λ/α}

• When the mark distribution is a mixture of exponential laws, a parametric approach is possible

R

EFERENCES

[1] Lowen, Steven B and Teich, Malvin C, Power-law shot noise, Information Theory, IEEE Transactions on, 36(6) (1990) 1302–1318.

[2] Iksanov A. and Jurek Z., Shot noise distributions and selfdecomposability, Taylor & Francis, (2003).

[3] Neumann M. and Reiß M., Nonparametric estimation for Lévy processes from low-frequency observations, Bernoulli, 15(1) (2009) 223–248.

[4] Gugushvili S., Nonparametric estimation of the char-acteristic triplet of a discretely observed Lévy process, Journal of Nonparametric Statistics, (2009).