Projection estimates of point processes boundaries

Journal of Statistical Planning and Inference 116 (2003) 1 – 15

www.elsevier.com/locate/jspi

Projection estimates of point processes boundaries St$ephane Girard, Pierre Jacob ∗ Laboratoire de Probabilit es et Statistique, Universit e Montpellier 2, place Eugene Bataillon, 34095 Montpellier cedex 5, France Received 22 March 2001; accepted 7 March 2002

Abstract We present a method for estimating the edge of a two-dimensional bounded set, given a .nite random set of points drawn from the interior. The estimator is based both on projections on C 1 bases and on extreme points of the point process. We give conditions on the Dirichlet’s kernel associated to the C 1 bases for various kinds of convergence and asymptotic normality. c 2002 We propose a method for reducing the negative bias and illustrate it by a simulation.  Elsevier B.V. All rights reserved. MSC: primary 60G70; secondary 62M30; 62G05; 62G20 Keywords: Projection on C 1 bases; Extreme values; Poisson process; Shape estimation

1. Introduction We address the problem of estimating a bounded set S of R2 given a .nite random set  of points drawn from the interior. This kind of problem arises in various frameworks such as classi.cation (Hardy and Rasson, 1982), image processing (Korostelev and Tsybakov, 1993) or econometrics problems (Deprins et al., 1984). A lot of di>erent solutions were proposed since Ge>roy (1964) and Renyi and Sulanke (1963) depending on the properties of the observed random set  and of the unknown set S. In this paper, we focus on the special case where  is the set of points of an homogeneous Poisson process whose support is S ={(x; y) ∈ R2 | 0 6 x 6 1; 0 6 y 6 f(x)}, where f is an unknown function. Thus, the estimation of subset S reduces to the estimation of ∗

Corresponding author. Tel.: +33-4-67-14-3510; fax: +33-4-67-14-3558. E-mail addresses: [email protected] (S. Girard), [email protected] (P. Jacob). c 2002 Elsevier B.V. All rights reserved. 0378-3758/03/$ - see front matter
PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 1 8 2 - 9

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S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

function f. Let us note that this kind of support was already considered in Ge>roy (1964). In the wide range of nonparametric functional estimators (Bosq, 1977), piecewise polynomials have been especially studied (Korostelev et al., 1995; Korostelev and Tsybakov, 1993; Mammen and Tsybakov, 1995; Gayraud, 1997; HGardle et al., 1995) and their asymptotic optimality was established under di>erent regularity conditions on f. The .rst support estimator based on orthogonal series appears in Abbar and Suquet (1993). Its properties are extensively studied in Jacob and Suquet (1995) in the case of Haar and C 1 bases. The expansion coeHcients estimation requires the knowledge of the process intensity. This is a limitation which is avoided in Girard and Jacob (1999) in the case of the Haar basis by considering a coeHcient estimation based on the extreme points of the sample. In this paper, a similar study is carried out in the case of C 1 bases. The estimator can be written as a linear combination of extreme values involving the Dirichlet’s kernel of the C 1 basis. A close study of the extreme values stochastic properties as well as a precise control of the Dirichlet’s kernel behavior allow one to establish general conditions for various convergences and asymptotic normality of the estimator. Our results are illustrated for the trigonometric basis case. Note that the model proposed here is well adapted to the estimation of a bounded starshaped subset of the plane. Let D be such a domain. Then, there exists a convex subset C of this domain, called the kernel of D, from which the whole boundary @D of D can be seen. If we assume an interior point of C to be known, the use of polar coordinates allows to reduce the problem of estimating @D to the problem considered here, that is the estimation of a function f, with the particularity that f(0) = f(1). In such a situation, extreme points observed in the neighborhood of x = 0 bring information on the behavior of f in the neighborhood of x = 1 and vice versa. Thus, an estimation relying on the trigonometric basis is specially well adapted. This paper is organized as follows. Section 2 is devoted to the de.nition of the estimator and Section 3 presents some basic results on extreme values and Dirichlet’s kernels. The mean integrated square convergence of the estimate is brieLy studied in Section 4 and the asymptotic normality is established in Section 5. In Section 6 a very simple bias correction is proposed, illustrated in Section 7 by a simulation.

2. Denition of the estimator 2.1. Preliminaries Let N be a Poisson process with a mean measure =c where the intensity parameter c is unknown, is the Lebesgue measure, and the support of N is given by S = {(x; y) ∈ R2 | 0 6 x 6 1; 0 6 y 6 f(x)}:

(2.1)

We assume that f is measurable and satis.es 0 ¡ m = inf f 6 M = sup f ¡ + ∞; [0;1]

[0;1]

(2.2)

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

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which entails the square integrability of f on [0; 1]. In the sequel, we will introduce extra hypothesis on f as needed. Let (ei )i∈N be an orthonormal basis of L2 ([0; 1]). The expansion of f with respect to the basis is supposed to be both L2 and pointwise convergent to f on [0; 1] ∀x ∈ [0; 1];

f(x) =

+∞


ai ei (x)

(2.3)

i=1

with

 ∀i ¿ 0;

ai =

0

1

ei (t)f(t) dt:

(2.4)

We denote by Kn the Dirichlet’s kernel associated to the orthonormal basis (ei )i∈N de.ned by Kn (x; y) =

hn


ei (x)ei (y);

(x; y) ∈ [0; 1]2 ;

(2.5)

i=0

where (hn ) is an increasing sequence of integers. The trigonometric basis will provide us with an important example to illustrate our convergence results. It is de.ned by √ √ e0 (x) = 1; e2k−1 (x) = 2 cos 2kx; e2k (x) = 2 sin 2kx; k ¿ 1 (2.6) and we shall suppose for convenience that hn is even. This leads to  sin (1+hn )(x−y) ; x = y; sin (x−y) Kn (x; y) = 1 + hn ; x = y:

(2.7)

The speed at which the sequence (ak ) decreases to 0 is linked to the regularity of f. In the case of the trigonometric basis, if f is a function of class C 2 then ak = O(k −2 );

(2.8)

(see Gasquet and Witomski, 1990). The estimator is built in two steps. First, in Section 2.2, f is approximated by a sequence (fn ) obtained from its expansion with respect to the orthogonal basis. Then, in Section 2.3, an estimator fˆn of fn is proposed. 2.2. Approximation of f Let (kn ) be an increasing sequence of non-negative integers such that kn = o(n). Divide S into kn cells Dn; r where   r−1 r ; r = 1; : : : ; kn : Dn; r = {(x; y) ∈ S | x ∈ In; r }; In; r ; (2.9) kn k n Each coeHcient ai is approximated by discretizing (2.4) according to ai; kn =

kn
r=1

ei (xr ) (Dn; r );

xr =

2r − 1 : 2kn

(2.10)

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Then, expansion (2.3) is truncated to the hn .rst terms leading to
ai; kn ei (x); x ∈ [0; 1]; fn (x) =

(2.11)

i6hn

which can be written in terms of the Dirichlet’s kernel: fn (x) =

kn


Kn (xr ; x) (Dn; r );

x ∈ [0; 1]:

(2.12)

r=1

Let us emphasize that the approximation fn of f only depends on the basis (ei )i∈N through its Dirichlet’s kernel. The next step towards the de.nition of the estimator consists of estimating (Dn; r ). 2.3. Estimation of fn Let N ∗n denote the superposition N1 + · · · + Nn of n independent copies of the point process N , and n the set of points generated by N ∗n . For r = 1; : : : ; kn , consider the maximum Xn;?r of the second coordinates of the set of points n; r =n ∩Dn; r . Of course, if n; r = ∅, set Xn;?r = 0. Then, (Dn; r ) can be estimated by Xn;?r =kn . This leads to an estimate aˆi; kn of ai; kn de.ned as aˆi; kn =

kn


ei (xr )

r=1

Xn;?r ; kn

1 6 i 6 hn

(2.13)

and consequently to an estimate fˆn (x) of fn (x) via fˆn (x) =




aˆi; kn ei (x) =

i6hn

kn


Kn (xr ; x)

r=1

Xn;?r : kn

(2.14)

Two remarks can be made. First, the estimator does not require knowledge of c, which ensures a wide range of applications. Second, the estimator is written as a linear combination of the maxima Xn;?r involving the Dirichlet’s kernel. Thus, the analysis of the behavior of fˆn will rely on both studies of the Dirichlet’s kernel features and of the maxima’s stochastic properties. This is the topic of the next section. 3. Basic results 3.1. Bounds on the Dirichlet’s kernel For x ∈ [0; 1] and j ∈ {1; 2; 3} de.ne Bn; j (x) =

 k n
r=1

1=j j

|Kn (xr ; x)|

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

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and Bn; ∞ (x) = max |Kn (xr ; x)|: 16r6kn

In what follows, some theorems involving conditions on Bn; j (x) for j ∈ {1; 2; 3; ∞} are given. Some of these conditions follow easily from the properties of Kn , see Jacob and Suquet (1995). In the following lemma, two properties which are less straightforward are given for the special case of the trigonometric basis. Lemma 1. Suppose Kn is the Dirichlet’s kernel associated to the trigonometric basis. (i) If hn = o(kn ) then supx Bn; 1 (x) = O(kn ln hn ); (ii) If hn ln hn = o(kn ) then for all x ∈ [0; 1]; Bn; 2 (x) ∼ (kn hn )1=2 . The proof is postponed to the Appendix. 3.2. Maxima stochastic properties In the sequel, we write

(Dn; r ) = n; r ;

min f(x) = mn; r ;

x∈In;r

maxf(x) = Mn; r : x∈In;r

Recall that Xn;?r is the maximum of the second coordinates of the set of points n; r . Noting that, for 0 6 x 6 mn; r , P(Xn;?r 6 x) = P(N ?n (Dn; r \(In; r × [0; x])) = 0); we easily obtain the distribution function Fn; r (x) = P(Xn;?r 6 x) on [0; mn; r ]:   nc Fn; r (x) = exp (x − kn n; r ) : kn

(3.1)

(3.2)

Straightforward calculations lead to the following expansions for the mathematical expectation and the variance of Xn;?r , where the knowledge of the C 1 -regularity of f compensates for the lack of precise expression for Fn; r on ]mn; r ; Mn; r [. Lemma 2. Suppose f is a function of class C 1 ; n = o(kn2 ) and kn = o(n=ln n). Then; (i) E(Xn;?r ) = kn n; r − (ii) Var(Xn;?r ) ∼

kn2 n2 c 2

kn nc

+ o( kn3 ); n

.

The proof of the following lemma, which is more diHcult, is postponed to the appendix. Lemma 3. Suppose f is a function of class C 1 and kn = o(n=ln n). Let (tn; r ) be a sequence such that tn; r = o(n=kn ) and tn; r = o(kn3 =n). Then; the characteristic function

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of (Xn;?r − mn; r ) can be written at point tn; r as 'n; r (tn; r ) =

1 + itn; r (kn =nc)FQ n; r (mn; r ) + o(|tn; r |n=kn3 ) + o(n−s ) ; 1 + itn; r kn =nc

with s arbitrary large; and FQ n; r = 1 − Fn; r .

4. Estimate convergences We refer to Jacob and Suquet (1995) for a careful study of the bias convergences fn − f2 → 0 and fn − f∞ → 0. Then, we just have to consider (fˆn − fn ). A complete investigation for mean integrated convergence, mean uniform convergence, L2 -almost complete convergence and uniform almost complete convergence is available at our website. In order to avoid lengthy developments, we just propose the following basic result. Theorem 1. Suppose f is a C 1 function. If fn − f2 = o(1) and if supx Bn; 1 (x) = o(n2 =kn ); then E(fˆn − f2 ) = o(1). Proof. Introducing the random variable Yn; r = (Xn;?r =kn ) − n; r ; we have  
E(fˆ − fn 22 ) = E Yn; r Kn (xr ; xs )Yn; s n

(4.1)

r; s

62


r; s

|Kn (xr ; xs )|(E(Yn;2 r ) + E(Yn;2 s ))

6 4 sup Bn; 1 (x) x

kn


E(Yn;2 r )

(4.2)

(4.3)

r=1

and the result follows from Lemma 2. Corollary 1. If Kn is the Dirichlet’s kernel associated to the trigonometric basis and f is a C 1 function then hn ln1=2 hn = o(kn ) and kn (ln hn )1=2 = o(n) are su
and the conclusion follows.

(4.4)

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

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5. Asymptotic distribution In this section, we present a limit theorem for the distribution of (fˆn − E fˆn ). A similar result is not available for (fˆn − f) without reducing the bias, which is done in the next section. Theorem 2. Suppose that f is a function of class C 1 . If kn = o(n=ln n); n = o(kn3=2 ) and Bn; ∞ (x) = o(Bn; 2 (x)); then (nc=Bn; 2 (x))(fˆn (x) − E(fˆn (x))) converges in distribution to a standard Gaussian variable for all x ∈ [0; 1]. Proof. Denote *n; r =mn; r −kn =(nc) and for all x ∈ [0; 1] introduce function of (nc=Bn; 2 (x))(fˆn (x) − E(fˆn (x))). It expands as   kn
nc ? Kn (xr ; x)(*n; r − E(Xn; r )) n; x (t) = exp it kn Bn; 2 (x)

n; x

the characteristic

(5.1)

r=1





E exp it



k

n
nc Kn (xr ; x)(Xn;?r − *n; r ) kn Bn; 2 (x)

:

(5.2)

r=1

Consider .rst the argument of (5.1): k

Tn (x) =

n
nc Kn (xr ; x)(*n; r − E(Xn;?r )) kn Bn; 2 (x)

(5.3)

r=1

and show it converges to 0. By the Cauchy–Schwartz inequality: |Tn (x)| 6

nc Bn; 1 (x) nc max |*n; r − E(Xn;?r )| 6 1=2 max|*n; r − E(Xn;?r )|: r r kn Bn; 2 (x) kn

Now Lemma 2 entails





 n n n |Tn (x)| 6 1=2 max (kn n; r − mn; r ) + o 3 = o 3=2 r kn kn kn

(5.4)

(5.5)

which converges to 0. Thus; introducing tn; r = ncKn (xr ; x)t=kn Bn; 2 (x); we obtain     kn 1
nc Kn (xr ; x) E exp it n; x (t) ∼ exp it Bn; 2 (x) kn Bn; 2 (x) r=1

kn
× Kn (xr ; x)(Xn;?r − mn; r ) r=1

= exp it





 k kn n 1
Kn (xr ; x) 'n; r (tn; r ): Bn; 2 (x) r=1

r=1

(5.6)

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S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

To apply Lemma 3; we have to verify that tn; r = o(n=kn ) and tn; r = o(kn3 =n). The .rst condition is satis.ed since



tn; r kn 6 |t| Bn; ∞ (x) = o(1): (5.7)

nc Bn; 2 (x) The second condition is satis.ed as well

2





tn; r n = tn; r kn n = o(1):

k3

k4 n n n

(5.8)

The characteristic function can be seen to be of the order exp it n; x (t)

∼



1 Bn; 2 (x)

kn  r=1

×

kn

r=1

kn  r=1



Kn (xr ; x)

(5.9)

kn (1 + itn; r nc )

1 + itn; r



 kn Q n F n; r (mn; r ) + o |tn; r | 3 + o(n−s ) : kn nc

(5.10)

Consider .rst the logarithm of term (5.9). A second-order Taylor expansion yields  

 kn 
Bn;3 3 (x) t2 kn kn (1) : (5.11) − ln 1 + itn; r =− +O itn; r Jn (x) = nc nc 2 Bn;3 2 (x) r=1

Since Bn;3 3 (x)=Bn;3 2 (x) 6 Bn; ∞ (x)=Bn; 2 (x) = o(1); it follows that Jn(1) (x) → −t 2 =2 as n → ∞. Finally, consider the logarithm of (5.10): 

kn
kn Q n F n; r (mn; r ) + o |tn; r | 3 Jn(2) = ln (1 + un; r ); with un; r = itn; r nc kn r=1

+ o(n−s ):

(5.12)

Observe that maxr |un; r | converges to 0 for (5.7) and (5.8). Thus, for n large enough |un; r | ¡ 1=2 uniformly in r and the classical identity |ln(1 + un; r )| ¡ |un; r | yields 

Bn; 1 (x) nc Bn; 1 (x) n2 (2) +o + o(kn n−s ): (5.13) |Jn | 6 |t| Bn; 2 (x) kn2 Bn; 2 (x) kn4 Therefore, the Cauchy–Schwartz inequality leads to Jn(2) → 0 and n → ∞.

n; x (t)

→ e−t

2

=2

as

Corollary 2. Suppose Kn is the Dirichlet’s kernel associated to the trigonometric basis and f is a C 1 function. If hn = o(kn ); kn = o(n=ln n) and n = o(kn3=2 ); then; for all x ∈ [0; 1]; nc(hn kn )−1=2 (fˆn (x) − E(fˆn (x))) converges in distribution to a standard Gaussian variable.

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

9

Proof. From (2.7); Bn; ∞ (x) 6 Kn ∞ = 1 + hn ; and from Lemma 1(ii); Bn; 2 (x)∼ (hn ; kn )1=2 so that Bn; ∞ (x)=Bn; 2 (x) = o((hn =kn )1=2 ) = o(1). Possible choices of kn and hn sequences in Corollary 2 are kn = n2=3 (ln n)/ and hn = (ln n)/ for / ¿ 0 arbitrary small. These choices entail n(hn kn )−1=2 = n2=3 (ln n)−/ .

6. Reducing the bias The bias can be decomposed as follows: E(fˆn − f) = (E fˆn − fn ) + (fn − f);

(6.1)

where the .rst term in the sum is the statistical part of the bias, and the second term is the systematic part of the bias. First, consider the irreducible bias (fn − f). In order to obtain a limit distribution for (nc=Bn; 2 (x))(fˆn (x) − f(x)), we need to satisfy the condition nc (fn (x) − f(x)) = 0: lim (6.2) n→∞ Bn; 2 (x) hn Introduce Sn (f) = i=0 ai ei . Eq. (3.12) in Jacob and Suquet (1995) provides sharp bounds for (Sn (f) − fn ), so that the question reduces to considering (Sn (f) − f), which only depends on the basis. We shall see that the trigonometric basis satis.es (6.2) under reasonable conditions on hn and kn . In the case of a general C 1 basis, we shall take (6.2) as a condition. Now, it follows from Lemma 2(i) that   kn
E(Xn;?r ) E fˆn (x) − fn (x) = (6.3) Kn (xr ; x) − n; r kn r=1

presents a negative component which should be eliminated. To this end, for r=1; : : : ; kn , de.ne Zn;?r by Zn;?r = 0 if n; r = ∅ and Zn;?r is the in.mum of the second coordinates of the points of n; r otherwise. Then, the random variable Zn =

kn 1
Zn;?r kn

(6.4)

r=1

has a mathematical expectation

 kn n E(Zn ) = +o 3 nc kn

(6.5)

and a variance Var(Zn ) ∼

kn ; n2 c 2

(6.6)

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S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

(use Lemma 2 for Zn;?r ). We de.ne a corrected estimate by   kn
Xn;?r + Zn ˜ = fˆn (x) + gˆn (x): Kn (xr ; x) fn (x) = kn

(6.7)

r=1

Lemma 4. Suppose f is a function of class C 1 ; n = o(kn2 ) and kn = o(n=ln n). Then; nc |E(f˜n (x)) − fn (x)| = o(n2 =kn7=2 ); Bn; 2 (x)

∀x ∈ [0; 1]:

If; moreover; for all x ∈ [0; 1]; kn = o(Bn;2 2 (x)) and (6.2) holds; then nc (gˆ (x) − E(gˆn (x))) = op (1): Bn; 2 (x) n Proof. We have nc nc Bn; 1 (x) max |E(Xn;?r ) + E(Zn ) − kn n; r | |E(f˜n (x)) − fn (x)| 6 kn Bn; 2 (x) 16r6kn Bn; 2 (x)

2 

 nc Bn; 1 (x) n n = O 3 = O 7=4 kn Bn; 2 (x) kn kn

(6.8) (6.9)

from (6.5); Lemma 2 and the Cauchy–Schwartz inequality. Now, applying (6.2) to the constant function f = 1 yields kn 1
Kn (xr ; x) → 1; kn

(6.10)

r=1

as n → ∞. Therefore, Var(gˆn (x)) ∼ Var(Zn ) ∼ kn =(n2 c2 ) and then 

kn nc Var (gˆn (x) − E(gˆn (x)) ∼ 2 Bn; 2 (x) Bn; 2 (x)

(6.11)

which converges to 0. Theorem 3. Suppose that f is a function of class C 1 . If the following conditions are veri=ed: (i) kn = o(n=ln n); n = o(kn3=2 ) ; (ii) for all x ∈ [0; 1]; max(kn1=2 ; Bn; ∞ (x)) = o(Bn; 2 (x)); (iii) for all x ∈ [0; 1]; Bn;nc2 (x) |fn (x) − f(x)| = o(1); then; for all x ∈ [0; 1]; (nc=Bn; 2 (x))(f˜n (x) − f(x)) converges in distribution to a standard Gaussian variable.

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

11

The proof is a simple consequence of the expansion (f˜n − f) = (fˆn − E(fˆn )) + (gˆn − E(gˆn )) + (E(f˜n ) − fn ) + (fn − f)

(6.12)

and of Theorem 2 and Lemma 4. Corollary 3. Suppose Kn is the Dirichlet’s kernel associated to the trigonometric basis and f is a C 2 function. If hn ln hn = o(kn ); n = o(hn3=2 kn1=2 ); nhn1=2 ln hn = o(kn3=2 ) and kn = o(n=ln n) then; for all x ∈ [0; 1]; nc(hn kn )−1=2 (f˜n (x) − f(x)) converges in distribution to a standard Gaussian variable. Proof. Conditions (i); (ii) of Theorem 3 are veri.ed. Consider (iii). On using (2.8) we have
nc(hn kn )−1=2 |Sn (f)(x) − f(x)| 6 nc(hn kn )−1=2 |ai | |ei (x)| i¿hn

6

√

2nc(hn kn )−1=2




|ai |

i¿hn

6

√

2nc(hn kn )−1=2




i−2 :

(6.13)

i¿hn

A straightforward calculation yields nc |Sn (f)(x) − f(x)| = O(nhn−3=2 kn−1=2 ): Bn; 2 (x) From Jacob and Suquet (1995); Eqs. (3.11) and (3.12);  



1 1

@Kn |Sn (f)(y) − fn (y)| = O (v; y)

dv

kn 0 @x and 

1

0



@Kn



@x (v; y) dv = O(hn ln hn ):

(6.14)

(6.15)

(6.16)

Therefore; nc |Sn (f)(y) − fn (y)| = O(nkn−3=2 hn1=2 ln hn ) Bn; 2 (y)

(6.17)

and (6.14) with (6.17) conclude the proof. Possible choices of kn and hn sequences in Corollary 3 are kn = n4=5 (ln n)3=5 (ln ln n)/ and hn =n2=5 (ln n)−1=5 (ln ln n)/ for / ¿ 0 arbitrary small. These choices entail n(hn kn )−1=2 = n2=5 (ln n)−1=5 (ln ln n)−/ .

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S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

Fig. 1. Function f to estimate.

7. Illustration on a simulation We consider the following simple illustration. The sum of n = 100 independent Poisson processes with the same intensity rate c = 3 is simulated on the set S de.ned as in (2.1) with f(x) = 3 +

sin(2x) : 1:1 + cos(2x)

(7.1)

The function is represented in Fig. 1. We consider an expansion of order hn = 16 of f in the trigonometric basis, where each coeHcient is approximated by dividing [0; 1] into kn = 32 intervals. The estimate fˆn is represented in Fig. 2 where it is compared with f˜n obtained by reducing the bias as described in the previous section.

8. Conclusion and further developments In this paper, we showed how the convergence results established in Girard and Jacob (1999) in the case of the Haar basis can be adapted for any C 1 basis under some assumptions on the Dirichlet’s kernel behavior. We have emphasized that the estimator and these assumptions only depend on the Dirichlet’s kernel of the basis. This suggests to de.ne a new estimator based on a Parzen–Rosenblatt kernel.

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

13

Fig. 2. Comparison between the two estimates fˆn (grey line) and f˜n (dark line).

Appendix. Proof of Lemma 1. (i) In the sequel; we shall use the inequality found in Jacob and Suquet (1995); Eq. (2.11):

sin(pu) 

(A.1)

sin(u) 6 p1[0; 5] (|u|) + 2|u| 1[5; =2] (|u|); for all p ¿ 0; 0 ¡ 5 ¡ =2 and |u| ¡ =2. Taking account of the periodicity and symmetry properties of the trigonometric kernel; it suHces to study [kn =2]+1 2
|Kn (xr ; x)|; sup x∈[0;1=kn ] kn

(A.2)

r=1

where [u] denotes the integer part of u. Let us write [kn =2]+1 [6n ] 1
1
1 Kn (xr ; x) = Kn (xr ; x) + kn kn kn r=1

r=1

[kn =2]+1




Kn (xr ; x)

r=[6n ]+1

with 6n = kn =(hn + 1); and consider the two terms separately.

(A.3)

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S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

• Introduce 5 = (=kn )([6n ] − 1=2). For r = 1; : : : ; [6n ]; we have (xr − x) 6 5 and thus (A.1) yields |Kn (xr ; x)| 6 1 + hn which gives in turn [6n ] 1
|Kn (xr ; x)| 6 1: kn

(A.4)

r=1

• For r = [6n ] + 1; : : : ; [kn =2] + 1; we have (xr − x) ¿ 5 and consequently (A.1) yields 1 kn

[kn =2]+1




|Kn (xr ; x)| 6

r=[6n ]+1

1 kn

[kn =2]+1




r=[6n ]+1

[kn =2] 1 1 1
6 2(xr − x) 2 kn

1 (r r=[6n ] kn

1 : − 1=2)

(A.5)

Therefore, 1 kn

[kn =2]+1




|Kn (xr ; x)| 6

r=[6n ]+1

1 2



(1=2)+1=2kn 5 

1 du 6 ln(4(hn + 1)) u 2

(A.6)

for kn ¿ 2(hn + 1). Finally, collecting (A.4) and (A.6), we obtain kn 1
|Kn (xr ; x)| 6 2 + ln(4(hn + 1)): x∈[0;1] kn

sup

(A.7)

r=1

(ii) From Jacob and Suquet (1995), Eq. (4.14),

 1

B2 (x)

@Kn

Kn ∞ 1

n; 2

sup − 1 6 (x; v)

dv:

kn Kn (x; x)

Kn (x; x) kn x 0 @y

(A.8)

In the case of the trigonometric basis Kn (x; x) = Kn ∞ = 1 + hn (see (2.7)) and from Jacob and Suquet (1995), Eq. (3.12) we have

 1

@Kn

dv = O(hn ln hn ) = o(kn ): (A.9) (x; v)

@y

0 The conclusion follows from (A.8) and (A.9). Proof of Lemma 3. Consider the expansion  mn;r  ? E(eitn; r Xn; r ) = P(Xn;?r = 0) + eixtn; r Fn; r (x) d x + 0

Mn;r

mn;r

eixtn; r Fn; r (d x):

(A.10)

The .rst and second term can be computed explicitly since (3.2) provides an expression of Fn; r on [0; mn; r ]:  Mn;r eitn; r mn; r Fn; r (mn; r ) − e−nc n; r itn; r Xn;?r −nc n; r E(e )=e + eixtn; r Fn; r (d x): + 1 + itn; r kn =nc mn;r (A.11)

S. Girard, P. Jacob / Journal of Statistical Planning and Inference 116 (2003) 1 – 15

The third term can be expanded as   Mn;r eixtn; r Fn; r (d x) = eitn; r mn; r FQ n; r (mn; r ) +

Mn;r

mn;r

mn;r

15

(eixtn; r − eitn; r mn; r )Fn; r (d x); (A.12)

with |eixtn; r − eitn; r mn; r | 6 (Mn; r − mn; r )|tn; r |. Thus



Mn;r

(eixtn; r − eitn; r mn; r )Fn; r (d x) 6 (Mn; r − mn; r )|tn; r |FQ n; r (mn; r )

mn;r

6 (Mn; r − mn; r )|tn; r |

n = O |tn; r | 3 kn

 :

nc (kn n; r − mn; r ) kn (A.13)

Collecting (A.11) – (A.13); and remarking that kn = o(n=ln n) yields |(kn =nc)e−nc n; r | = O(n−s ) concludes the proof. References Abbar, H., Suquet, Ch., 1993. Estimation L2 du contour d’un processus de Poisson homogSene sur le plan. Pub. IRMA Lille 31, II. Bosq, D., 1977. Contribution aS la th$eorie de l’estimation fonctionnelle. Publ. Inst. Statist. Univ. Paris XIX (2), 1–96. Deprins, D., Simar, L., Tulkens, H., 1984. Measuring labor eHciency in post oHces. In: Pestieau, P., Tulkens, H. (Eds.), The Performance of Public Enterprises: Concepts and Measurements. North-Holland, Amsterdam. Gasquet, C., Witomski, P., 1990. Analyse de Fourier et Applications. Masson, Paris. Gayraud, G., 1997. Estimation of functionals of density support. Math. Meth. Statist. 6 (1), 26–46. Ge>roy, J., 1964. Sur un problSeme d’estimation g$eom$etrique. Publ. Inst. Statist. Univ. Paris XIII, 191–200. Girard, S., Jacob, P., 1999. Extreme values and Haar series of point processes boundaries. Technical Report ENSAM-INRA-UM2, pp. 99-08. HGardle, W., Park, B.U., Tsybakov, A.B., 1995. Estimation of a non sharp support boundaries. J. Multivariate Anal. 43, 205–218. Hardy, A., Rasson, J.P., 1982. Une nouvelle approche des problSemes de classi.cation automatique. Statist. Anal. Donn$ees 7, 41–56. Jacob, P., Suquet, P., 1995. Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference 46, 215–234. Korostelev, A., Simar, L., Tsybakov, A.B., 1995. EHcient estimation of monotone boundaries. Ann. Statist. 23, 476–489. Korostelev, A.P., Tsybakov, A.B., 1993. Minimax linewise algorithm for image reconstruction. In: HGardle, W., Simar, L. (Eds.), Computer Intensive Methods in Statistics. Statistics and Computing. Physica Verlag, Springer, Berlin. Mammen, E., Tsybakov, A.B., 1995. Asymptotical minimax recovery of set with smooth boundaries. Ann. Statist. 23 (2), 502–524. Renyi, A., Sulanke, R., 1963. Uber die konvexe HGulle von n zufGalligen gewGahlten Punkten. Z. Wahr. Verw. Geb. 2, 75–84.