An adaptive wavelet shrinkage approach to the Spektor–Lord–Willis problem

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Journal of Multivariate Analysis

journal homepage:www.elsevier.com/locate/jmva

An adaptive wavelet shrinkage approach to the Spektor–Lord–Willis

problem

Bogdan Ćmiel

AGH University of Science and Technology, Faculty of Applied Mathematics, Al. Mickiewicza 30, 30-059 Cracow, Poland

a r t i c l e i n f o

Article history: Received 13 May 2009 Available online 29 January 2010 AMS subject classifications: 62G05 45Q05 Keywords: Spektor–Lord–Willis problem Inverse problem Rate of convergence Minimax risk Adaptive estimator

a b s t r a c t

The stereological problem of unfolding the sphere size distribution from linear sections is considered. A minimax estimator of the intensity function of a Poisson process that de-scribes the problem is introduced and an adaptive estimator is constructed that achieves the optimal rate of convergence over Besov balls to within logarithmic factors. The con-struction of these estimators uses Wavelet–Vaguelette Decomposition (WVD) of the oper-ator that defines our inverse problem.

©2010 Elsevier Inc. All rights reserved.

1. Introduction

Let us suppose that spheres of random radii are randomly distributed in some opaque medium inR3. The centers of the

spheres form a homogeneous Poisson process onR3, and the radii have a distributionQon

[

0

;

1

]

, independent of the centers

and absolutely continuous with respect to the Lebesgue measure with probability densityq. Letcdenote the expected number of sphere centers per unit volume. As the sphere radii cannot be observed directly, we take a linear section through the medium and observe the line segments that are intersections of the line and the spheres. From that observation we want to estimatef

:=

cq. Letnbe the ‘‘size of the experiment’’. We thus observe a Poisson processGn

gon

[

0

;

1

]

with intensityng and it can be shown (see [1]) that

g

(

u

)

=

2u

Z

1

u

f

(

x

)

dx

=:

(

Gf

)(

u

),

(1)

whereG

:

L2

(

[

0

;

1

]

,

dx

)

→

L2

(

[

0

;

1

]

,

du

)

. The problem of unfoldingf from linear sections is known in the literature as the Spektor–Lord–Willis (SLW) problem, and its degree of ill-posedness is higher than for the related and better-known Wicksell’s problem of unfolding the sphere size distribution from planar sections. The exact inverse of the operatorGleads to the equation f

(

x

)

=

1 2

g

(

x

)

x2

−

g0

₍

_x

₎

x

.

As noted in [1], inverse estimation off in L2

₍

_[

₀

_;

₁

_]

_,

_dx

₎

_{roughly corresponds to direct estimation of the intensityg in} L2

₍

_[

₀

_;

₁

_]

_,

_u−4_du

₎

_{and of its derivative inL}2

₍

_[

₀

_;

₁

_]

_,

_u−2_du

₎

_{, which explains the statistical difficulty of the problem. Eq.(1)}

E-mail address:[email protected].

0047-259X/$ – see front matter©2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2010.01.009

can be used as a model of some measurements in material sciences (see [2,3]) and in isotropic cases as a model of linear intercept measurements on polished metallographic sections (cf. sintering processes in [4] and methodological remarks in [5]). It can also be applied in metallurgy (see, e.g., [6]). The SLW problem was analysed in [1] where B-spline sieved quasi-maximum likelihood estimators were used. Construction of a spectral estimator that is, up to a logarithmic factor, asymptotically minimax over a Sobolev-type class of functions can be found in [7]. In this paper we fill a gap in the rates by exploiting the properties of the wavelets.

In many aspects, this paper is based technically on [8] where a two-dimensional problem of positron emission

tomography (PET) was considered. However, the main results of that paper could not be directly applied to the SLW problem. Firstly, the Wavelet–Vaguelette Decomposition (WVD) of the operatorGhad to be constructed. Secondly, a lower bound of the estimator risk had to be found. For the Radon operator in [8] it was possible to construct a class of the functions with the images separated from zero, which was essential in the construction of the lower bounds. For the operatorG, however,

g

(

1

)

=

0 for allf. This paper resolves that problem by using the Assouad’s cube technique to obtain a lower bound for the risk. A direct application of the methods from [8] for upper bounding the risk was not possible neither, because of the shape of the function

γ

, which is discussed later. This problem is resolved by restricting the domain to radii larger than some positive minimal detection level

ε

in which case exact minimaxity can be achieved. The influence of

ε

on the upper bound for the risk is studied in detail, which enables a construction of an almost minimax estimator for a model with

ε

=

0. In that case, however, there is still, as in [7], a logarithmic gap between the lower and the upper bound for the risk.

In the next section we will use a Wavelet–Vaguelette Decomposition to construct an estimator of the functionf. To circumvent some problems with the construction of the WVD, we change the dominating measure in the image space, which also changes the operator itself. Dividing both sides of(1)byu2_{, one obtains}

h

(

u

)

:=

g

(

u

)

u2

=

2 u

Z

1 u f

(

x

)

dx

=:

(

Kf

)(

u

),

whereK

:

L2

₍

_[

₀

_;

₁

_]

_,

_dx

₎

_→

_L2

₍

_[

₀

_;

₁

_]

_,

_d

_µ)

_{, d}

_µ

₌

_u2_{du. Note that the operatorsKandGare compact Hilbert–Schmidt}

operators. Consequently, their inverses are not bounded and the unfolding problem is ill-posed in the Hadamard sense. In the following sections we will denote by

h·

,

·i

and

[·

,

·]

the inner products inL2

₍

_dx

₎

_andL2

₍

_d

_µ)

_{, respectively.}

2. Wavelet–Vaguelette Decomposition

The Wavelet–Vaguelette Decomposition will be constructed along the lines described in [9], Sec. 5.2. For the construction of the WVD,Kwill be considered as an operator fromL2

₍

R

,

dx

)

toL2

(

R

,

d

µ)

and the domainRwill be omitted in the notation.

Let

ψ

be a smooth mother wavelet that satisfies the conditions: supp

ψ

= [

0

;

N

]

,

R

_−∞∞

ψ(

x

)

dx

=

0

,

R

_−∞∞

ψ

2

₍

_x

₎

_dx

₌

₁

_{, ψ}

_∈

C2,

ψ

0

L2_(dx)

<

∞

and

ψ

(−1)

L2_(dx)

<

∞

, where

ψ

( −1)

₍

_u

₎

_:=

R

u

−∞

ψ(

x

)

dx

.

The set of functions

ψjk(

x

)

=

2

j/2

_ψ(

₂j_x

₋

_k

₎

_, withjandkintegers, is a complete, orthonormal system inL2

₍

_dx

₎

_{. Define}

εjk(

u

)

:=

(

K

ψjk)(

u

)

=

2 u

Z

∞ u

ψjk(

x

)

dx

= −

2 u

ψ

(−1) jk

(

u

),

and

γjk(

u

)

:=

ψ

0 jk(u

)

2u

=

2 3 2j

ψ

0

₍

₂j_u

₋

_k

₎

2u

.

(2)

It is easy to see that for anyjandk,

εjk

∈

L2

₍

_d

_µ)

_and

_γjk

_∈

_L2

₍

_d

_µ)

_{. Write}

[

γjk, εj

0_k0

] = −

Z

∞ −∞

ψ

jk(0 _u

)

2u 2 u

Z

u −∞

ψj

0_k0

(

x

)

dx

u2du

.

Integrating by parts, we have

[

γjk, εj

0_k0

] =

Z

∞

−∞

ψjk(

u

)ψj

0_k0

(

u

)

du

= h

ψjk(

u

), ψj

0_k0

(

u

)

i =

δ

_(jk),(j0_k0₎

,

where

δ

denotes the Kronecker delta. Let us observe that

2

−j

γjk

L2_(dµ)

=

1 4

ψ

0

L2_(du)

=

Const and

2

j

εjk

L2_(dµ)

=

4

ψ

( −1)

L2_(du)

=

Const

.

Now, withujk

:=

2−j

γjk

and

νjk

:=

2j

εjk

one has the WVD of the operatorK(see [9], sec. 1.5)

K

ψjk

=

2−j

νjk,

The functionsujkand

νj

0_k0are biorthogonal:

[

ujk, νj0_k0

] =

δ

_(jk),(j0_k0₎

.

For completeness, one can also prove (see [9], sec. 4) the near orthogonality relations

X

jk ajkνjk

L2_(dµ)

=

4

X

jk ajk2 j 2

ψ

(−1)

(

₂j_u

−

k

)

L2_(du)

k

(

ajk)

k

_l 2

,

X

jk ajkujk

L2_(dµ)

=

1 4

X

jk ajk2 j 2

ψ

0

(

₂j_u

−

k

)

L2_(du)

k

(

ajk)

k

_l2

,

which are, however, not necessary for our purpose. Considering the WVD for the original operatorGinL2

₍

_dx

₎

_{, one can see}

that the function

γjk

for the operatorGis the same as for the operatorKbut, unfortunately, for somek

∈

_Z,

γjk

is not anL2

₍

_du

₎

object in this case. The second point is that

k

γjk

k

_L2_(du)depends onk, which results in problems with finding quasi-singular

values of the operatorGdepending only on resolution levelj.

Let

φ

be a father wavelet that satisfies the conditions: supp

φ

= [

0

;

N

]

,

R

_−∞∞

φ(

x

)

dx

=

1 and

φ

∈

C2_{. The functionfhas}

the following inhomogeneous wavelet expansion (cf. [10] Ch. 3.2):

f

=

X

k∈Z

h

f

, φj

₁k

i

φj

1k

+

∞

X

j=j1

X

k∈Z

h

f

, ψjk

i

ψjk,

wherej1is any fixed integer and

φjk

=

2j/2

φ(

2jx

−

k

)

. Let us define functionalscjk(

·

)

as cjk(g

)

= [

γjk,

g

]

,

so that

γjk

is the Riesz representer of the functionalcjk. Observe that

cjk(Kf

)

= [

γjk,

Kf

] = h

K∗

γjk,

f

i = h

ψjk,

f

i

.

(3) Since

φj

₁k1

∈

L

2

₍

_dx

₎

_fork

1

∈

Z, it has a homogeneous wavelet expansion

φj

1k1

=

X

j,k

a₍j1k1)jkψjk.

Define a linear functional by

bj1k1

(

·

)

=

X

j,k

a₍j1k1)jkcjk(

·

)

and observe that

bj1k1

(

Kf

)

=

X

j,k a₍j1k1)jkcjk(Kf

)

=

X

j,k a₍j1k1)jk

h

ψjk,

f

i = h

φj

1k1

,

f

i

.

(4)

Now, using(3)and(4), we have the reproducing formula

f

=

X

k∈Z bj1k(Kf

)φj

1k

+

∞

X

j=j1

X

k∈Z cjk(Kf

)ψjk.

As discussed in [9], p. 111, althoughK

φ

6∈

L2

₍

_d

_µ)

_{the reproducing formula remains valid at least for functionsf with only}

finite number of nonzero terms in the inhomogeneous wavelet expansion. Let

γj

˜

_1kbe the Riesz representer of the functional

bj1k. Then f

=

X

k∈Z

[

Kf

,

γj

˜

1k

]

φj

1k

+

∞

X

j=j1

X

k∈Z

[

Kf

, γjk

]

ψjk.

(5) BecauseK∗

_γj

_˜

1k

=

φj

1kand

(

K ∗

_γj

_˜

1k)(x

)

=

2

R

x −∞u

γj

˜

1k(u

)

duwe have

˜

γj

1k(u

)

=

φ

0 j1k(u

)

2u

=

2 3 2j1

φ

0

₍

₂j1_u

₋

_k

)

2u

.

(6)

3. The minimax risk

Besov spaces admit a characterization in terms of wavelet coefficients. A functionf

=

P

k∈Z

αj

1kφj1k

+

P

∞

j=j1

P

k∈Z

βjkψjk

k

αj

₁·

k

lp

+

∞

X

j=j1

(

2j(σ+1/2−1/p)

k

βj

·

k

lp

)

q

!

1/q 6M

.

Usually one cannot observe spheres with radiir

< ε

. We call

ε

the minimal detection radius, assume

ε

∈

(

0

;

1

/

2

)

and restrict the domain of the radii to

[

ε

;

1

]

. Define

K_jε

(φ)

:=

k

∈

_Z

:

supp

φjk

∩ [

ε

;

1

] 6= ∅

,

K_jε

(ψ)

:=

k

∈

_Z

:

supp

ψjk

∩ [

ε

;

1

] 6= ∅

,

Fσpq(M

,

[

ε

;

1

]

)

=

f

·

1[

ε;1]

:

f

∈

Bσpq(M

),

f

·

1[

ε;1]>0

.

Then, forf

∈

Fσpq(M

,

[

ε

;

1

]

)

,f

=

P

k∈K_jε 1(φ)

αj

1kφj1k

+

P

∞ j=j1

P

k∈K_jε(ψ)

βjkψjk

on the interval

[

ε,

1

]

. It is easy to see that

|

K_jε

(φ)

|

6 AN2jand

|

Kjε

(ψ)

|

6 AN2j, whereAN is a constant independent ofj. The following theorem is an immediate consequence of propositions proved in Sections3.1and3.2.

Theorem 1.Let p>1. Assume that

σ >

3

(

1

/

p

−

1

/

2

)

if p

<

4

/

3and

σ >

1

/

p if p>4

/

3. Then

∀

ε

∈

0

;

1 2

inf ˆ fn sup f∈Fσpq(M,[ε;1]) Ef

kˆ

fn

−

f

k

2_L2_([ε;1])

n −_2σ2σ₊₃

,

where

ˆ

fndenotes any estimator of the intensity function f .

The theorem remains valid iff

∈

g0

+

Fσpq(M

,

[

ε

;

1

]

)

, whereg0>0 is a known and fixed function with support in

[

ε

;

1

]

.

Note that we estimate the functionf on a support separated from zero. This is forced by the shape of the functions

γjk

. If supp

ψjk

contained zero as an interior point, then

k

γjk

k

∞would be infinite (cf.(2)), which would cause problems with the upper bound for the risk. We avoid that by takingj1large enough. The functionsfinTheorem 1may by quite irregular. The

assumptions are satisfied by, e.g., the sample paths of the Brownian motion supported on

[

ε

;

1

]

. It can be shown that they belong toB1/2p∞for any 16p

<

∞

(see [10], Ch. 9). It should be remarked that, as in [8], forp

∈ [

1

;

2

)

the minimax linear risk onFσpq(M

,

[

ε

;

1

]

)

is higher than the minimax risk for all estimators. It can be shown that

sup f∈Fσpq(M,[ε;1])

Ef

kˆ

fL

−

f

k

2_L2_([ε;1])>Cn

− 2σ0

2σ0 +3

,

where

σ

0

=

σ

+

1

/

2

−

1

/

min

(

2

,

p

)

, and

ˆ

fLdenotes any linear estimator of the intensity functionf.

3.1. The lower bound

In this section we use the Assouad’s cube technique to prove the following proposition.

Proposition 1. Let p>1,

σ >

1

/

p, and06

ε <

1

/

2. Then for any estimatorf

ˆ

nof function f , there exists some constant C such

that sup f∈Fσpq(M,[ε;1]) Ef

kˆ

fn

−

f

k

2_L2_([ε;1])>Cn −_2σ2σ₊₃

_.

Proof. Let Gεj

(ψ)

:=

k

∈

_Z

:

supp

ψjk

⊂

ε,

1 2

,

and Gε_σpq(j

)

=







f_ω>0

:

f_ω

=

f0

+

δj

X

k∈Gε_j(ψ)

ωkψjk







,

where

ωk

∈ {

0

,

1

}

,f0

∈

Fσpq(M

/

2

,

[

ε

;

1

]

)

,

R

11/2f0

(

x

)

dx

=

C1

>

0 and

δj

6 min

2−j(σ+1/2)_M

_/

₂

_,

₂−j/2_C

1

/(

2ANN

k

ψ

k

∞

)

. It is easy to see that, for any constantC1,Gε_σpq(j

)

⊂

Fσpq(M

,

[

ε

;

1

]

)

. Denote asL

(

GKf

)

the distribution ofGKf—the Poisson process with the intensity functionKf with respect to

µ

. The Hellinger affinity between the distributions takes the form

ρ(

L

(

GKfω

),

L

(

GKf_ω0

))

=

exp

[−

H 2

₍

_Kf ω

,

Kfω0

)

]

,

whereH2

(

Kf_ω

,

Kf_ω0

)

=

1 2

R

1 0

√

Kf_ω

−

√

Kf_ω0

2d

µ

(cf. [11], Ch. 3.2).

Letf_ω

,

f_ω0

∈

Gε_σpq(j

)

and∆

(ω, ω

0

)

=

1, where∆

(

·

,

·

)

denotes the Hamming distance. Notice that, by the construction of Gε_σpq(j

)

, the support of the function

(

Kf_ω

−

Kf_ω0

)

is contained in

[

ε

;

1

/

2

]

which will ease the evaluation ofH2

(

Kf_ω

,

Kf_ω0

)

:

H2

(

Kf_ω

,

Kf_ω0

)

=

1 2

Z

1 0 u2

₍

_Kf ω

−

Kfω0

)

2

√

Kf_ω

+

√

Kf_ω0

2 du

=

δ

_j2

Z

1/2 ε u

R

1 u

ψjk(

x

)

dx

2

R

1 u f0

(

x

)

dx





v

u

u

t

R

1 u fω

(

x

)

dx

R

1 u f0

(

x

)

dx

+

v

u

u

t

R

1 u fω0

(

x

)

dx

R

1 u f0

(

x

)

dx





−2 du

.

Sinceu

∈ [

ε

;

1

/

2

]

, it is clear that

R

_u1f0

(

x

)

dx>C1and

R

1 ufω

(

x

)

dx

R

1 u f0

(

x

)

dx

=

1

+

δj

R

1 u

P

k∈Gε_j(ψ)

ωkψjk(

x

)

dx

R

1 u f0

(

x

)

dx >1

−

δj

P

k∈Gε_j(ψ)

R

1 u

|

ψjk(

x

)

|

dx

R

1 u f0

(

x

)

dx > 1

−

δj

AN2jN2−j/2

k

ψ

k

∞

/

C1> 1 2

.

We have H2

(

Kf_ω

,

Kf_ω0

)

₆ 1 2

δ

2 j

Z

suppψjk u

R

1 u

ψjk(

x

)

dx

2

R

1 u f0

(

x

)

dx du 6C2

δ

j2N2 −j

_k

_ψ

_k

∞N2−j/2

2 6C32−j(2σ+3)

.

Let 2j

_n1/(2σ+3)_{. Then, using the Assouad Lemma (see [7]), we get the result.}

3.2. The upper bound

In this section we construct an estimator that achieves the optimal rate of convergence. The construction of that estimator and the evaluation of its risk are much the same as in [8] with two important differences: the dominating measure in the image space is not the Lebesgue measure and there is an additional parameter

ε

which influences the support of the function

f and, consequently, the number of wavelets used for the estimation. LetGn

hdenote the observed Poisson process with intensity functionnhwith respect to d

µ

, and let

ν

n

hdenote the intensity measure of that process. Withn

=

1, we write

νh

rather than

ν

_h1. We consider the following estimator offon the interval

[

ε

;

1

]

:

ˆ

f_nε

((λj),

j1

(ε),

j2

(

n

))

=

X

k∈K_jε 1(ε)(φ)

ˆ

αj

1(ε)kφj1(ε)k

+

j2(n)

X

j=j1(ε)

X

k∈K_jε(ψ)

δS(

_{βjk, λj)ψjk}

ˆ

₍₇₎ where

ˆ

αj

1(ε)k

=

1 n

Z

1 0

˜

γj

1(ε)kdG n h,

_βjk

ˆ

₌

1 n

Z

1 0

γjk

dGnh, (8)

and the nonnegative sequence

(λj)

defines a soft-threshold rule:

δS

(

_{βjk, λj)}

ˆ

₌

_sgn

₍

_βjk)(

ˆ

_{| ˆ}

_βjk

_{| −}

_λj)

+

.

Proposition 2. Let p>1and0

< ε <

1

/

2. Assume that

σ >

3

(

1

/

p

−

1

/

2

)

if p

<

4

/

3and

σ >

1

/

p if p>4

/

3. Then for any f

∈

F_σpq(M

,

[

ε

;

1

]

)

, there exist some constant C and sequences

(

j2

(

n

))

and

(λj)

such that

E

kˆ

f_nε

((λj),

j1

(ε),

j2

(

n

))

−

f

k

2_L2_([ε;1])6Cn

−_2σ2σ₊₃

_.

Proof. Let us assume that

ε

4 6N2

−j1(ε)₆

ε

2

.

(9)

It is easy to see that

∀

k

∈

Kjε1(ε)

(φ)

supp

γj

˜

1(ε)k

⊂

h

ε

2

;

1

+

ε

2

i

and

∀

j>j1

(ε)

∀

k

∈

Kjε

(ψ)

supp

γjk

⊂

h

ε

2

;

1

+

ε

2

i

.

It is known (see [12], Ch. 3.2) that

E

αj

ˆ

_1(ε)k

=

1 n

Z

1 0

˜

γj

1(ε)kd

ν

n h

=

bj1(ε)k(Kf

)

= h

f

, φj

1(ε)k

i :=

αj

1(ε)k, E

βjk

ˆ

=

1 n

Z

1 0

γjk

d

ν

_hn

=

cjk(Kf

)

= h

f

, ψjk

i :=

βjk,

Var

βjk

ˆ

=

1 n

Z

1 0

γ

2 jkd

νh

:=

σ

2 jk n

.

With ljk

:=

γjk/σjk,

one has

Z

1 0 l2_jkd

νh

=

1

.

(10)

Sincek

∈

K_jε

(ψ)

, using(2)we have

k

ljk

k

∞6 232j_C1

εσjk

.

(11)

In order to evaluate the risk of the estimator(7), a Gaussian approximation will be constructed in the sequence space. Let

ˆ

ηjk

=

βjk

+

n−12

σjk

_zjk, ₍₁₂₎

wherezjkis a Gaussian variable specified below. We evaluateE

[ ˆ

βjk

− ˆ

ηjk

]

2by considering two cases. First, let us assume that for some constantC2

σ

2 jk> 1 nC2

ε

−2₂3j_log3_n

_.

₍₁₃₎ Then E

[ ˆ

βjk

− ˆ

ηjk

]

2

=

σ

2 jk n E

n−12

Z

1 0 ljkd

(

Gnh

−

ν

n h

)

−

zjk

2

.

DefineVn

=

R

1 0 ljkd

(

G n h

−

ν

n h

)

:=

R

1 0 ljkdG

¯

n h. We have E

[ ˆ

βjk

− ˆ

ηjk

]

2

=

σ

2 jk n E

h

n−12_Vn

−

zjk

i

2

.

We now need the following lemma (see [8], Lemma V.1):

Lemma 1. Suppose that

νh(

[

0

;

1

]

)

=

1and

R

₀1l2d

νh

=

1. Let

k

l

k

∞ 6 L and Vn

=

R

1

0 l

(

dG

n

h

−

d

ν

h)n , where Gnhis a Poisson

process with intensity measure

ν

n

h

=

n

νh

. Then, there exist absolute constants D1and D2such that, whenever L2n−1log3n6D1, there exists a random variable Z_∼N

(

0

,

1

)

such that

E

n−12_Vn

−

Z

2

6D2L2n−1

.

Lemma 1and formulas(10),(11)and(13)prove the existence ofzjk∼N

(

0

,

1

)

such that E

[ ˆ

βjk

− ˆ

ηjk

]

26C3

ε

−2

23j

n2

.

(14)

We still have to check the case when condition(13)is not valid, i.e. when

σ

2

jk

<

1

nC2

ε

−2₂3j_log3_n

_.

In that case, we take anyzjk

∼

N

(

0

,

1

)

and use the inequality

(

a

−

b

)

262a2

+

2b2to obtain

E

[ ˆ

βjk

− ˆ

ηjk

]

2 6 4

σ

_jk2n−1

<

C2n

−2

_ε

−2

Formulas(14)and(15)show that for fixedj

,

kthere exists a Gaussian variablezjksuch that

E

[ ˆ

βjk

− ˆ

ηjk

]

26C4n−2

ε

−223jlog3n

.

(16)

We now evaluate the risk of the estimator(7)forf

∈

Fσpq(M

,

[

ε

;

1

]

)

:

E

kˆ

f_nε

((λj),

j1

(ε),

j2

(

n

))

−

f

k

2_L2_([ε;1])6

X

k∈Kε_j 1(ε)(φ) E

[ ˆ

αj

_1(ε)k

−

αj

_1(ε)k

]

2

+

j2(n)

X

j=j1(ε)

X

k∈Kε_j(ψ) E

[

δS(

βjk, λj)

ˆ

−

βjk

]

2

+

∞

X

j=j2(n)+1

X

k∈K_jε(ψ)

β

2 jk

:=

Ln(f

)

+

Sn(f

)

+

Tn(f

).

Using(6)and(9)we obtain

Ln(f

)

6C5 1 n

X

k∈K_jε 1(ε)(φ) 22j1(ε)

ε

2 6C6n −1

_ε

−5

_.

₍₁₇₎

For the last term we have

Tn(f

)

6sup

{

Tn(f

),

f

∈

Fσpq(M

,

[

ε

;

1

]

)

} =

C72 −2j2(n) σ+1₂−1_p

.

(18)

To evaluateSn(f

)

we will use the random variable

ηjk

ˆ

defined in(12):

Sn(f

)

6 j2(n)

X

j=j1(ε)

X

k∈Kjε(ψ) 2E

[

δS(

βjk, λj)

ˆ

−

δS(

ηjk, λj)

ˆ

]

2

+

j2(n)

X

j=j1(ε)

X

k∈Kjε(ψ) 2E

[

δS

(

ηjk, λj)

ˆ

−

βjk

]

2

.

Since for every

λ

the mappingy

→

δS

(

y

, λ)

is a contraction, i.e.

|

δS

(

y1

, λ)

−

δS

(

y2

, λ)

|

<

|

y1

−

y2

|

, we have Sn(f

)

6 j2(n)

X

j=j1(ε)

X

k∈Kjε(ψ) 2E

[ ˆ

βjk

− ˆ

ηjk

]

2

+

j2(n)

X

j=j1(ε)

X

k∈Kjε(ψ) 2E

[

δS

(

ηjk, λj)

ˆ

−

βjk

]

2

:=

an(f

)

+

bn(f

).

Using(16), we obtain an(f

)

64C4AN

ε

−2n−2log3n24j2(n)

.

(19)

To evaluatebn(f

)

we will use the following lemma (see [13], Lemma 3).

Lemma 2. If

σjk

62j

ε

−1C8for all j

,

k, then E

h

δS

βjk

+

n−12

σjk

_zjk, λj

−

βjk

i

2 62E

h

δS

βjk

+

n−12₂j

ε

−1_C 8zjk, λj

−

βjk

i

2

.

We have bn(f

)

6 j2(n)

X

j=j1(ε)

X

k∈K_jε(ψ) 4E

h

δS

βjk

+

n−12₂j_C8

ε

−1_zjk, λj

−

βjk

i

2

.

For an appropriate choice of

(λj)

it can be shown (see [9] sec. 8) that

bn(f

)

6C9n −_2σ2σ₊₃

ε

−_2σ4σ₊₃ 6C9n −_2σ2σ₊₃

ε

−2

_.

₍₂₀₎ Using(17)–(20)we obtain E

kˆ

f_nε

((λj),

j1

(ε),

j2

(

n

))

−

f

k

2_L2_([ε;1]) 6C6

ε

−5n−1

+

C72 −2j2(n) σ+1₂−1_p

+

C10

ε

−2n−2log3n24j2(n)

+

C9

ε

−2n −_2σ2σ₊₃

.

(21)

We can choosej2

(

n

)

such that

σ

log₂n

(

2

σ

+

3

)(σ

+

1

/

2

−

1

/

p

)

j2

(

n

)

σ

+

3 2

(

2

σ

+

3

)

log2n

−

3 4log2logn

,

wherean

bnmeans that limn→∞bn

−

an

= ∞

. This is possible when

σ

(

2

σ

+

3

)(σ

+

1

/

2

−

1

/

p

)

<

σ

+

3

2

(

2

σ

+

3

)

.

Sincep>1 it is easy to check that this condition is true if we assume that

σ >

1

/

p. With that choice ofj2

(

n

)

we have C₇2−2j2(n) σ+1₂−1_p

<

C₇22−2σ+3σ log2n

₌

_C 7n −_2σ2σ₊₃ and C10

ε

−2n−2log3n24j2(n)

<

C10

ε

−2n− 2σ 2σ+3

.

Using this in(21)we finally obtain

E

kˆ

f_nε

((λj),

j1

(ε),

j2

(

n

))

−

f

k

2_L2_([ε;1])6C11

ε

−2_n−_2σ2σ₊₃

₊

_ε

−5_n−1

_.

₍₂₂₎

Since

ε

is fixed, this completes the proof ofProposition 2.

It should be remarked that, in some special cases, the estimatorf

ˆ

_nεcan achieve good rates of convergence also without the restriction of the domain to

[

ε

;

1

]

. Let us evaluate

E

kˆ

f_nε

−

f

k

2

L2_([0;1])

=

E

kˆ

fnε

−

f

k

2L2_([0;ε])

+

E

kˆ

fnε

−

f

k

2L2_([ε;1])

.

Iff

∈

Fσpq(M

,

[

0

;

1

]

)

thenf

·

1[

_ε;1]

∈

Fσpq(M

,

[

ε

;

1

]

)

and using(22)we have E

kˆ

f_nε

−

f

k

2 L2_([0;1])6C11

ε

−2_n−_2σ2σ₊₃

+

ε

−5n−1

+

C12

ε

max x∈[0;ε]f 2

₍

_x

_).

If we assume that, for some

α >

0, lim x→0+f

(

x

)

e 1 xα

_<

_∞

_,

₍₂₃₎ then E

kˆ

f_nε

−

f

k

2 L2_([0;1])6C11

ε

−2_n−_2σ2σ₊₃

+

ε

−5n−1

+

C13

ε

e −_εα2

.

Now, if we take

ε

=

(

logn

)

−_α1 _{and denote the correspondingf}

ˆ

ε

n asf

ˆ

n0, then

E

kˆ

f_n0

−

f

k

2

L2_([0;1])6C14n

−_2σ2σ₊₃

(

logn

)

α2

.

Combining this withProposition 1we conclude that

ˆ

f_n0achieves almost the optimal rate of convergence to within logarithmic terms.

4. The adaptive estimator

The form of the minimax estimator from the previous section depends on the parameters of the spaceF_σpqthat our intensity function belongs to (consider the choice ofj2

(

n

)

, for example). In practical estimation problems we do not know

the parameters of the spaceFσpq. Because of that, we need an estimator, with the best possible rate of convergence, that does not use the values of those parameters. We will call it an adaptive estimator. As in Section3.2we closely follow the derivation from [8], with modifications forced by the changed dominating measure and the restricted domain.

Fix an integerr0

>

3

/

2 and suppose that the parameters

(σ,

p

,

q

)

belong to the class J

=

(σ ,

p

,

q

)

:

max

1 p

,

3

1 p

−

1 2

< σ <

r0

,

16p

,

q6

∞

.

Consider the following estimator:

˜

f_nε

=

X

k∈K_jε 3(n)(φ)

ˆ

αj

3(n)kφj3(n)k

+

j4(n)

X

j=j3(n)

X

k∈K_jε(ψ)

δH

(

_βjk,

ˆ

_T

_(ε)

_{cj)ψjk, (24)}

where the coefficients

αj

ˆ

_3(n)kand

βjk

ˆ

are defined in(8),

cj

=

2j

r

j n

,

2 j3(n)

_n1/(2r0+3)

,

₂j4(n)

_n

/

_log 2n

and

δH

(

_βjk,

ˆ

_T

_(ε)

_cj)

₌

_ˆ

βjk,

if

| ˆ

βjk

|

>

T

(ε)

cj 0

,

if

| ˆ

βjk

|

6T

(ε)

cj

is a hard-threshold rule with a constantT

(ε)

. We will prove the following theorem:

Theorem 2. Let

(σ ,

p

,

q

)

∈

J. Then

∀

ε

∈

0

;

1 2

sup f∈Fσpq(M,[ε;1]) Ef

k˜

fnε

−

f

k

L22_([ε;1])6C

logn n

_2σ2σ₊₃

.

The theorem says that the estimator

˜

f_nεachieves almost the optimal rate of convergence to within logarithmic terms. Since

˜

f_nεdoes not depend on the parameters

σ ,

p

,

q, it is an adaptive estimator.

Proof of Theorem 2. Recall that supp

ψ

= [

0

;

N

]

and assume that

N2−j3(n)

6

ε

2

.

It is easy to see that

∀

k

∈

K_jε_3(n)

(φ)

supp

γj

˜

_3(n)k

⊂

h

ε

2

;

1

+

ε

2

i

and

∀

j>j3

(

n

)

∀

k

∈

Kjε

(ψ)

supp

γjk

⊂

h

ε

2

;

1

+

ε

2

i

.

We write some useful inequalities. Using(2)we have

Z

1 0

|

γjk(

u

)

|

md

µ

=

Z

1 0 232jm

ψ

0

₍

₂j_u

₋

_k

₎

2u

m d

µ

6C1

ε

−m2j( 3 2m−1)

,

₍₂₅₎

and from(25)we obtain

E

[ ˆ

βjk

−

βjk

]

2

=

1 n

Z

1 0

γ

2 jkd

νh

6C2

ε

−2 22j n

.

(26)

From the equation (see [8] Lemma A.1)

E

Z

1 0

γjk

d

(

Gn_h

−

ν

nh)

4

=

Z

1 0

γ

4 jkd

ν

n h

+

3

Z

1 0

γ

2 jkd

ν

n h

2

and from(25)we have

E

[ ˆ

βjk

−

βjk

]

4

=

E

1 n

Z

1 0

γjk

d

(

Gn_h

−

ν

h)n

4 6C3

ε

−4

25j n3

+

24j n2

.

(27)

We now use the following lemma (see [8] Lemma A.2):

Lemma 3. If

R

₀1

γ

_jk2d

ν

_hn6V and

k

γjk

k

∞6H then P

Z

1 0

γjk

d

(

Gn_h

−

ν

nh)

>

λ

62 exp

−

1 2

λ

2 V

+

H

λ/

3

.

Since

R

₀1

γ

2 jkd

ν

hn6C4

ε

−222jnand

k

γjk

k

∞6C4

ε

−12 3 2j_{we have} P

ˆ

βjk

−

βjk

>

T

(ε)

2 cj

62 exp

−

1 8

ε

T2

(ε)

j22j C4

ε

−122j

+

C422jT

(ε)/

6

.

LetT

(ε)

=

C5

η(ε)

, whereC52>8C4

(

1

+

C5

/

6

)

log 2. Then,

P

ˆ

βjk

−

βjk

>

T

(ε)

2 cj

62 exp

−

εη(ε)

j 1

+

C5

/

6 1

/(εη(ε))

+

C5

/

6 log 2

.

If we now choose

η(ε)

>

ε

−1, we obtain

∀

η(ε)

_>

ε

−1 P

ˆ

βjk

−

βjk

>

T

(ε)

2 cj

62−εη(ε)j+1

.

(28) Let us define Ejf

=

X

k∈Kεj(φ)

αjkφjk,

Dj3(n)j4(n)f

=

j4(n)

X

j=j3(n) Djf

=

j4(n)

X

j=j3(n)

X

k∈K_jε(ψ)

βjkψjk,

ˆ

Ej3(n)

=

X

k∈K_jε 3(n)(φ)

ˆ

αj

3(n)kφj3(n)k,

ˆ

Dj3(n)j4(n)

=

j4(n)

X

j=j3(n)

X

k∈Kε_j(ψ)

δH

(

_βjk,

ˆ

_T

_(ε)

_cj)ψjk.

Sincef

=

Ej3(n)f

+

Dj3(n)j4(n)f

+

f

−

Ej4(n)fandf

˜

nε

= ˆ

Ej3(n)

+ ˆ

Dj3(n)j4(n), we have

E

k˜

f_nε

−

f

k

2 L2_([ε;1]) 63E

k ˆ

Ej3(n)

−

Ej3(n)f

k

2 L2_([ε;1])

+

3E

k ˆ

Dj3(n)j4(n)

−

Dj3(n)j4(n)f

k

2 L2_([ε;1])

+

3E

k

f

−

Ej4(n)f

k

2 L2_([ε;1])

.

(29)

First let us evaluateE

k

f

−

Ej4(n)f

k

2 L2_([ε;1]):

k

f

−

Ej4(n)f

k

2 L2_([ε;1])

=

∞

X

j=j4(n) Djf

2 L2_([ε;1]) 6 ∞

X

j=j4(n)

k

Djf

k

L2_([ε;1])

!

2

.

SinceB_σpq

⊂

Bσ0_2∞

(σ

0

=

σ

+

1

/

2

−

1

/

p

)

, we have

k

f

k

_σ0_2∞6

k

f

k

_σpq, so 2jσ0

k

Djf

k

L2_([ε;1])6 sup j>j4(n) 2jσ0

k

Djf

k

L2_([ε;1])

+ k

E_j4(n)f

k

_L2_([ε;1])6

k

f

k

_σpq.

From this we conclude that

k

f

−

Ej4(n)f

k

2 L2_([ε;1])6C6

k

f

k

2_σpq2 −2j4(n)σ0_6C 7n −_2σ2σ₊₃

.

(30) Now we evaluate E

k ˆ

Ej3(n)

−

Ej3(n)f

k

2 L2_([ε;1])6 C8 n

X

k∈Kε_j 3(n)(φ)

Z

1 0

˜

γ

2 j3(n)kd

νh.

Using(6)we can show, exactly as in(25), that

Z

1

0

| ˜

γjk(

u

)

|

md

µ

6C9

ε

−m2j( 3 2m−1)

,

and from this we conclude that

E

k ˆ

Ej3(n)

−

Ej3(n)f

k

2 L2₍[_ε;1]₎ 6 C10 n

ε

2

X

k∈K_jε 3(n)(φ) 22j3(n) 6C11

ε

−2n− 2σ 2σ+3

.

₍₃₁₎ To evaluateE

k ˆ

Dj3(n)j4(n)

−

Dj3(n)j4(n)f

k

2 L2_([ε;1])let us define

ˆ

Bj

= {

k

∈

Kjε

(ψ)

: | ˆ

βjk

|

>

T

(ε)

cj

}

,

S

ˆ

j

= ˆ

Bcj Bj

= {

k

∈

Kjε

(ψ)

: |

βjk

|

>

T

(ε)

cj/2

}

,

Sj

=

Bcj B0_j

= {

k

∈

K_jε

(ψ)

: |

βjk

|

>

2T

(ε)

cj

}

,

Sj0

=

B 0cj ebs

=

j4(n)

X

j=j3(n)

X

k∈ ˆBjSj

(

_βjk

ˆ

−

βjk)ψjk

ebb

=

j4(n)

X

j=j3(n)

X

k∈ ˆBjBj

(

_βjk

ˆ

−

βjk)ψjk

esb

=

j4(n)

X

j=j3(n)

X

k∈ˆSjB0j

βjkψjk

ess

=

j4(n)

X

j=j3(n)

X

k∈ˆSjSj0

βjkψjk.

We have

ˆ

Dj3(n)j4(n)

−

Dj3(n)j4(n)f

=

ebs

+

ebb

−

esb

−

ess. (32)

First we evaluateE

k

ebs

k

2_L2_([ε;1]). Let

Gj

= {

k

∈

Kjε

(ψ)

: | ˆ

βjk

−

βjk

|

>

T

(ε)

cj/2

}

.

SinceB

ˆ

jSj

⊂

Gjwe have E

k

ebs

k

2_L2_([ε;1])6 j4(n)

X

j=j3(n)

X

k∈K_jε(ψ)

[

E

(

_βjk

ˆ

₋

_βjk)

4

_]

12

_[

_P

(

_{| ˆ}

βjk

₋

βjk

_|

>

_T

(ε)

_cj/₂

)

_]

1 2

.

We now use(27)and(28)and we get

E

k

ebs

k

2_L2_([ε;1])62 C12AN

ε

2_n j4(n)

X

j=j3(n) 2(3−εη(ε)/2)j

.

If we now choose

η(ε)

>8

ε

−1_{we conclude that} E

k

ebs

k

2_L2_([ε;1])6

C13

ε

2_n

.

(33)

Let us evaluateE

k

esb

k

2_L2_([ε;1]). SinceS

ˆ

jB0j

⊂

Gjwe get

E

k

esb

k

2_L2_([ε;1])6 j4(n)

X

j=j3(n)

X

k∈Kjε(ψ)

β

2 jkP

(

| ˆ

βjk

−

βjk

|

>

T

(ε)

cj/2

).

Using(28)we have E

k

esb

k

2_L2_([ε;1])62 j4(n)

X

j=j3(n)

X

k∈K_jε(ψ)

β

2 jk2 −εη(ε)j

₌

₂ j4(n)

X

j=j3(n) 2−εη(ε)j

k

βj

·

k

2_l2

.

Since 22j(σ0+1/2−1/p)

_k

_βj

·

k

2_l2 6C14

k

f

k

2_σ0_2∞6C14

k

f

k

2_σpq, then E

k

esb

k

2_L2_([ε;1])62C14

k

f

k

2_σpq j4(n)

X

j=j3(n) 2−j(2σ0+1−2p+εη(ε))

.

We notice that 2

σ

0

₊

₁

₋

2

p

+

εη(ε) >

0, and from this it follows that

E

k

esb

k

2_L2_([ε;1])6C152 −j3(n)(2σ0+1−2p+εη(ε))_6C 16n −2σ0 +1−2_2r/p+εη(ε) 0+3

.

If we take

η(ε)

>

ε

−1

(

2r0

+

2

)

, we obtain E

k

esb

k

2_L2_([ε;1])6C16n− 2σ 2σ+3

.

₍₃₄₎

Let us now evaluateE

k

ebb

k

2_L2_([ε;1]). We choosej0

(

n

)

such that 2j0(n)

n1/(2σ+3). We notice that

2j3(n)

_n 1 2r0+3 _6n 1 2σ+3

<

n log₂n

2 j4(n)

,

soj3

(

n

)

6j0

(

n

)

6j4

(

n

)

. We have E

k

ebb

k

2_L2_([ε;1])6 E

j0(n)

X

j=j3(n)

X

k∈ ˆBjBj

(

_βjk

ˆ

₋

_βjk)ψjk

2 L2_([ε;1])

+

E

j4(n)

X

j=j0(n)

X

k∈ ˆBjBj

(

_βjk

ˆ

₋

_βjk)ψjk

2 L2_([ε;1])

=

E

k

ebba

k

2_L2_([ε;1])

+

E

k

ebbb

k

2_L2_([ε;1])

.

Fork

∈

Bj,

|

2

βjk/(

T

(ε)

cj)

|

p

>

1, so using(26)we get

E

k

ebbb

k

2_L2_([ε;1])6 C2

ε

2_n1−p/2 j4(n)

X

j=j0(n) 2j(2−p)j−p/2

k

βj

·

k

p_lp

.

Sincej−p/261 and 2jσ0p

k

βj

·

k

p lp6

k

f

k

p σpq, we have E

k

ebbb

k

2_L2_([ε;1])6 C17

ε

2_n1−p/22 −j0(n)(σ0p+p−2)_6C 18

ε

−2n −_2σ2σ₊₃

.

(35)

To evaluateE

k

ebba

k

2_L2_([ε;1])we use(26):

E

k

ebba

k

2_L2_([ε;1])6C2 j0(n)

X

j=j3(n)

X

k∈K_jε(ψ) 22j

ε

2_n 6C19 23j3(n)

ε

2_n 6C20

ε

−2_n−_2σ2σ₊₃

.

(36)

From(35)and(36)we obtain

E

k

ebb

k

2_L2_([ε;1])6C21

ε

−2n− 2σ

2σ+3

.

₍₃₇₎

We now evaluateE

k

ess

k

2_L2_([ε;1]). Let

τjk

=

2

−j

_βjk

_and

τ

=

(τjk),

β

=

(βjk),

j>j3

(

n

),

k

∈

Kjε

(ψ).

If

β

∈

B_σpq(M

)

, then

τ

∈

Bσ¯pq(M

)

, where

σ

¯

=

σ

+

1. We have

k

ess

k

L2_([ε;1])6

k{

τjk

:

j3

(

n

)

6j6j4

(

n

),

k

∈

Sj0

}k

122

.

Fork

∈

S0 j, 2j

|

τjk

|

62T

(ε)

2j

√

j

/

n, so

|

τjk

|

₆2T

(ε)

r

j n 62T

(ε)

r

j4

(

n

)

n

:=

Λn. (38) Let Ω

(

Λ

; k · k;

A

)

=

sup

{k

τ

k :

τ

∈

A

,

|

τjk

|

<

Λ

}

and Ωn

=

Ω

(

Λn

; k · k

122

;

Bσ¯pq(M

)).

(39) Using(38),(39)and ([14], Theorem 3) we get

Ωn6M1− ¯α 2T

(ε)

r

j4

(

n

)

n

!

α¯

,

where

¯

α

=

σ

¯

−

1

¯

σ

+

1

/

2

=

σ

σ

+

3

/

2

=

2

σ

2

σ

+

3

.

Sincej4

(

n

)

6C22lognandT

(ε)

=

C5max

{

8

ε

−1

, (

2r0

+

2

)ε

−1

}

we have E

k

ess

k

2_L2_([ε;1])6Ωn26C23

logn

ε

2_n

_2σ2σ₊₃

.

(40)

Using(33),(34),(37)and(40)we finally obtain sup f∈Fσpq(M,[_ε;1]₎ Ef

k˜

fnε

−

f

k

2 L2_([ε;1])6C24

"

logn

ε

2_n

_2σ2σ₊₃

+

ε

−2n−2σ2σ+3

#

.

(41)

Since

ε

is fixed we have sup f∈Fσpq(M,[ε;1]) Ef

k˜

fnε

−

f

k

L22_([ε;1])6C

logn n

_2σ2σ₊₃

,

which provesTheorem 2.

As in Section3, the applicability of the estimator can be extended to the case with minimal detection radius equal to zero. Forf

∈

Fσpq(M

,

[

0

;

1

]

)

, we assume(23), then we define

˜

f0

n by taking

ε

=

(

logn

)

−_α1 _{and using(41)we have}

Ef

k˜

fn0

−

f

k

2

L2_([0;1])6C25n− 2σ

2σ+3

(

_logn

)

s

,

wheres

=

max

{

(

1

+

2

/α)(

2

σ/(

2

σ

+

3

)),

2

/α

}

. Combining this withProposition 1we conclude thatf

˜

_n0achieves almost the optimal rate of convergence to within logarithmic terms.

Acknowledgments

The author wishes to express his gratitude to Professor Zbigniew Szkutnik for suggesting the problem and for many stimulating conversations.

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