Deconvolution for an atomic distribution Shota Gugushvili Peter - - PowerPoint PPT Presentation

Deconvolution for an atomic distribution

Shota Gugushvili Peter Spreij Bert van Es

Universiteit van Amsterdam

Stochastic processes: theory and applications A conference in honor of the 65th birthday of Wolfgang J. Runggaldier Bressanone, July 16 - 20, 2007

Something completely different Introduction Estimation procedure Conditions Results

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Conference dinner at the 23rd EMS, Madeira, 2001

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Hiking, Porto Santo, 2001

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Swimming, Porto Santo, 2001

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

The ”better” picture, Porto Santo, 2001

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Filtering

The problem Given X = Y + Z

• bservation = signal + noise,

find characteristics of Y given X Recursive filtering Update the conditional distribution (expectation) of Yt given X0, . . . Xt.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Filtering

The problem Given X = Y + Z

• bservation = signal + noise,

find characteristics of Y given X Recursive filtering Update the conditional distribution (expectation) of Yt given X0, . . . Xt.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Deconvolution

The rough problem Given Xi = Yi + Zi

• bservation = signal + noise,

estimate the distribution of the Yi given X1, . . . , Xn. Basic assumptions The Yi are iid, the Zi are iid and the Yi are independent from the Zi.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Deconvolution

The rough problem Given Xi = Yi + Zi

• bservation = signal + noise,

estimate the distribution of the Yi given X1, . . . , Xn. Basic assumptions The Yi are iid, the Zi are iid and the Yi are independent from the Zi.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Deconvolution - model assumptions

Classical case The Yi have a density f and the noise variables Zi have a known distribution (standard normal). Non-classical case The Yi are distributed according to Y = UV , where U is Bernoulli with P(U = 0) = p, V has a density f , and U and V are independent.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Deconvolution - model assumptions

Classical case The Yi have a density f and the noise variables Zi have a known distribution (standard normal). Non-classical case The Yi are distributed according to Y = UV , where U is Bernoulli with P(U = 0) = p, V has a density f , and U and V are independent.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Estimation problem

Aim Estimate p f (infinite dimensional parameter) based on the observations X1, . . . , Xn. Nonparametric tools Kernel smoothing Fourier inversion

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Estimation problem

Aim Estimate p f (infinite dimensional parameter) based on the observations X1, . . . , Xn. Nonparametric tools Kernel smoothing Fourier inversion

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Estimation problem

Aim Estimate p f (infinite dimensional parameter) based on the observations X1, . . . , Xn. Nonparametric tools Kernel smoothing Fourier inversion

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Motivation

Let yt = xt + zt, where x is a compound Poisson process (xt = Nt

k=1 Vi) and z an

independent Brownian motion (y is a L´ evy process). Assume that y is observed at times 1, 2, . . .. Let Yi = yi − yi−1, then the Yi are of the above type. Related problem Estimate the common density of the Vi.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Characteristic functions

For φX and φf , the ch.f.’s of X and V , respectively, one has φX(t) = [p + (1 − p)φf (t)]e−t2/2, Assuming that φf is integrable, we have Inversion formula f (x) = 1 2π ∞

−∞

e−itx φX(t) − pe−t2/2 (1 − p)e−t2/2 dt.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Empirical c.f. and kernel

Basic idea Replace in the inversion formula φX by its empirical counterpart and apply some smoothing. By φemp we denote the empirical characteristic function, φemp(t) = 1 n

n

• j=1

eitXj. We also use w, a kernel function with compact support [−1, +1], and h a bandwidth.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Estimator

The basic idea results in Kernel type estimator for f fnh(x) = 1 2π ∞

−∞

e−itx φemp(t) − pe−t2/2 (1 − p)e−t2/2 φw(ht)dt,

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Bias

We have, with wh(x) = 1

hw(x h),

E[fnh(x)] − f (x) = 1 2π ∞

−∞

e−itxφf (t)(φw(ht) − 1)dt = f ∗ wh(x) − f (x), which vanishes for h → 0, similar to ordinary kernel estimation.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Estimation of p

‘Nonparametric’ estimator png = g 2 1/g

−1/g

φemp(t)φk(gt) e−t2/2 dt, where the number g > 0 denotes the bandwidth and φk denotes a Fourier transform of a kernel k. The definition is motivated by another basic idea and the fact lim

g→0

g 2 1/g

−1/g

φX(t) e−t2/2 dt = lim

g→0

g 2 1/g

−1/g

φY (t)dt = p.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results when p is known with unknown p

Estimation of f when p is unknown

Plug-in kernel estimator f ∗

nhg(x) = 1

2π ∞

−∞

e−itx φemp(t) − ˆ pnge−t2/2 (1 − ˆ png)e−t2/2 φw(ht)dt, where ˆ png = min(png, 1 − ǫn). Here 0 < ǫn < 1 and ǫn ↓ 0 at a suitable rate, which will be specified below. The truncation in is introduced due to technical reasons.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Condition on f

Integrability condition Let the true density f be such that u2+γφf (u) is integrable. Here γ is some strictly positive number.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Conditions on the kernels

Condition on w Let φw be real valued, symmetric and have support [−1, 1]. Let φw(0) = 1 and let φw(1 − t) = Atα + o(tα), as t ↓ 0 (1) for some constants A and α ≥ 0. Moreover, we assume that α < γ/2.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Conditions on the kernels

Condition on k Let φk be real valued, symmetric and have support [−1, 1]. Let φk integrate to 2 and let φk(1 − t) = Atα + o(tα), φk(t) = Bt2+γ + o(t2+γ). as t ↓ 0. Here B is some constant, and A, γ and α are as above.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Examples of kernels

Example Sinc kernel: w(x) = sin x πx . Its characteristic function is given by φw(t) = 1[−1,1](t). Example The kernel k has an awful expression, but is such that φk(t) = 1[−1,1](t)585 32 t4(1 − t4)2.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Examples of kernels

Example Sinc kernel: w(x) = sin x πx . Its characteristic function is given by φw(t) = 1[−1,1](t). Example The kernel k has an awful expression, but is such that φk(t) = 1[−1,1](t)585 32 t4(1 − t4)2.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Conditions on the bandwidths

h and g depend on n h = hn = ((1 + ηn) log n)−1/2 g = gn = ((1 + δn) log n)−1/2, where ηn and δn are such that ηn → 0, δn → 0, and (ηn − δn) log n → ∞. Example ηn = 2log log n log n , δn = log log n log n .

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Conditions on the bandwidths

h and g depend on n h = hn = ((1 + ηn) log n)−1/2 g = gn = ((1 + δn) log n)−1/2, where ηn and δn are such that ηn → 0, δn → 0, and (ηn − δn) log n → ∞. Example ηn = 2log log n log n , δn = log log n log n .

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Condition on truncation variable

Rate of ǫn Let ǫn be such that − log ǫn ≪ log n(ηn − δn). Example With ηn and δn as previously given, take ǫn = (log log n)−1.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Condition on truncation variable

Rate of ǫn Let ǫn be such that − log ǫn ≪ log n(ηn − δn). Example With ηn and δn as previously given, take ǫn = (log log n)−1.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Outline

1 Something completely different 2 Introduction 3 Estimation procedure

when p is known with unknown p

4 Conditions 5 Results

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Asymptotic normality of fnh

Theorem (Case p is known) Assume the conditions on f , w, and h. Suppose that E [X 2] < ∞. Then, as n → ∞ and h → 0, √n h1+2αe1/2h2 (fnh(x) − E [fnh(x)]) D → N

• 0, σ2

f

• ,

where σ2

f = A2 2π2(1−p)2 (Γ(α + 1))2.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Asymptotic normality of png

Theorem Assume the conditions on f , w, and g. Then we have √n g2+2αe1/2g2 (png − E [png]) D → N

• 0, σ2

p

• ,

with σ2

p = A2(Γ(1+α))2 2

.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution

Something completely different Introduction Estimation procedure Conditions Results

Asymptotic normality of f ∗

nhg(x)

Theorem (Case p is unknown) Assume the conditions on f , w, h, g and k. Then, as n → ∞ and h → 0, g → 0, we have √n h1+2αe

1 2h2 (f ∗

nhg(x) − E [f ∗ nhg(x)]) D

→ N

• 0, σ2

f

• ,

where σ2

f is as in the previous case.

Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution