Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries

DOI:10.1051/ps:2007039 www.esaim-ps.org

MULTIDIMENSIONAL LIMIT THEOREMS FOR SMOOTHED EXTREME VALUE ESTIMATES OF POINT PROCESSES BOUNDARIES

Ludovic Menneteau

Abstract. In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process.

We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.

Mathematics Subject Classification. Primary 60G70; Secondary 62M30, 62G05.

Received March 11, 2005. Revised June 26, 2006 and March 26, 2007.

Introduction

The problem of estimating a set S given a sequence of finite sets S_n of random points drawn from its interior arises in classification [15], clustering problems [21], discriminant analysis [2], outliers detection [6], image analysis [25] and in econometrics [4]. In many cases, the unknown support can be written

S={(x, y) :x∈E; 0≤y≤ f(x)}, (1) where f is an unknown function andE an arbitrary set (typically a subset of R^d). Then, the problem reduces to estimating f.

In econometrics, the data consist of pairs (X_i, Y_i) whereX_i represents the input, possibly multidimensional (labor, energy or capital), used to produce an outputY_iin a given firm. In such a framework, the valuef(x) can be interpreted as the maximum level of output which is attainable for the level of input x. Then, economical considerations suggest to suppose thatf is increasing and concave and an adapted estimator, called the DEA (Data Envelopment Analysis) has been developed and studied. Its asymptotic distribution is established by [10].

In the case where we do not assume that f is monotone, Geffroy [9] proposed an estimator off which is a kind of histogram based on the extreme values of the sample. Geffroy’s estimator has been improved in several directions. On one hand, piecewise polynomials estimators were introduced and their optimality in an asymptotic minimax sense was proved under different regularity conditions onf (see [17,20,25–27,33]).

On the other hand, regularization and bias correction of Geffroy’s estimate have been considered using different way of smoothing (see [8,11,12]). In [13], the multivariate central limit theorem has been obtained for a general class of estimates of this type including all the above. For technical reason (the maxima on disjoint

Keywords and phrases. Functional estimate, central limit theorem, moderate deviation principles, extreme values, shape estimation.

1 Place Eug`ene Bataillon, 34095 Montpellier Cedex 5, France;[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

cell of Poisson processes are independent) the results obtained in [8,11,12] and [13] were only given when S_n is generated by a Poisson point process. This paper encompasses the results obtained in [13] of several manners.

Central limit theorems but also moderate deviation principles are investigated for a wider class of estimators than in [13]. Moreover, we show that our results hold in the context of Poisson process but also when S_n is generated by an empirical point process.

In the next section, the general class of estimators is given. Section2contains our main results and provides some examples of possible bias reduction. Some applications, including kernel and projection estimators are provided in Section3. The proofs are given in Section4.

1. The boundary estimate

Letf : (E,E)→(R⁺,B(R⁺)) be a measurable function on a probability space (E,E, ν), whereB(R⁺) is the Borelσ-algebra onR⁺. Consider the set

S={(x, y)∈E×R,0≤y≤f(x)}.

Our aim is to estimateS from a sequence ofS-valued random vectors S_n={(X_n,i, Y_n,i), 1≤i≤N_n(S)}, with associated counting process

N_n =

N_n(D) :D∈ E ⊗ B R⁺

, n≥1, of mean measure

n c1_S(x, y) ν(dx) dy, (2)

wherec >0. Two cases are considered below:

(P)N_n is a Poisson point process,

(E)N_n is an (n-sample) empirical point process.

Letk_n↑ ∞and denote by {In,r: 1≤r≤k_n} a measurable partition ofE. For all 1≤r≤k_n, set ν_n,r=ν(I_n,r), D_n,r={(x, y) :x∈I_n,r, 0≤y≤f(x)} and N_n,r=N_n(D_n,r).

We also introduce

Y_n,r^∗ = max{Yn,i: (X_n,i, Y_n,i)∈D_n,r},

ifD_n,r=∅andY_n,r^∗ = 0 otherwise (in the sequel, we use the convention 0× ∞= 0) and f_n,r=ν⁻_n,r¹

In,r

f dν, denotes the mean value of f onI_n,r.

In that context,f can be estimated using Geffroy estimator

f_n(x) =

kn

r=1

1_In,r(x)Y_n,r^∗ . (3)

But several improvements of f_n can be considered. First, it is well-known that Y_n,r^∗ estimates f_n,r with a negative bias which reveals to be of the order (ncν_n,r)⁻¹. Hence, to reduce this bias we can introduce unbiased

versions of ( 3):

f_n^◦(x;c_n) =

kn

r=1

1_In,r(x)

Y_n,r^∗ + (nc_nν_n,r)⁻¹ , (4) wherec_n is a convenient estimator ofc. Now, the non random conterpoint off_n^◦ is the step function

f_n^◦(x) =

kn

r=1

1_In,r(x)f_n,r (5)

and iff is regular, it is clear that smoothed version of (5) can lead to more efficient estimation off. Therefore, we will consider estimators off of the form

f_n(x;c_n) =

kn

r=1

ν_n,rκ_n,r(x)

Y_n,r^∗ + (nc_n(x)ν_n,r)⁻¹ (x∈E), (6) whereκ_n,r:E→Ris a weighting function determining the nature of the smoothing introduced in the estimate.

In the next section, some general conditions are imposed onκ_n,r andc_n in order to obtain a central limit and a moderate deviation principle forf_n.

2. Main results

In the following, we consider theν-essential infimum and supremum off onE,

m= sup{α >0 :ν({f < α}) = 0} and M = inf{α >0 :ν({f > α}) = 0}

and, for all 1≤r≤k_n, we denote by

m_n,r= sup{α >0 :ν({f < α} ∩I_n,r) = 0}, M_n,r= inf{α >0 :ν({f > α} ∩I_n,r) = 0},

respectively the ν-essential infimum, supremum of f on I_n,r (in most of applications, E is a subset of R^d, ν is absolutely continuous with respect to Lebesgue measure and f is continuous, hence all ess-inf and ess-sup considered below are just classical inf and sup). For allx∈E, we set

f_n(x) =

kn

r=1

ν_n,rκ_n,r(x)f_n,r, κ_n(x) = _k

n

r=1

κ²_n,r(x) 1/2

w_n,r(x) =κ_n,r(x)/κ_n(x) and ν_n= min{νn,r, 1≤r≤k_n}.

In the sequel, (ε_n)_n≥1denotes a sequence of positive real numbers such that (ε_n)_n≥1≡1 orε_n↓ ∞.

The following assumptions will be needed below:

(H.1) k_n↑ ∞ and (nν_n)⁻¹max

log (n), ε⁻_n¹

→0 asn→ ∞.

(H.2) 0< m≤M <+∞and

δ_n := max

1≤r≤kn

ν_n,r(M_n,r−m_n,r) =o(1/n) asn→ ∞.

There existsF⊂E such that:

(H.3) For each{x1, ..., x_p} ⊂F, there exists a covariance matrix Σ(x1,...xp)= [σ(x_i, x_j)]1≤i,j≤p inR^p such that for all 1≤i,j≤p,

kn

r=1

w_n,r(x_i)w_n,r(x_j)→ σ(x_i, x_j) asn→ ∞.

(H.4) For allx∈F,

ε⁻_n^1/2 max

1≤r≤kn

|wn,r(x)| →0 asn→ ∞.

(H.5) For allx∈F,

ε¹_n^/2nκ_n(x)⁻¹|fn(x)−f(x)| →0 asn→ ∞.

(H.6) For allx∈F,

ε¹_n^/2

kn

r=1

|wn,r(x)| (nδ_n)²=o(1) asn→ ∞.

(C.1) For allx∈F, and anyη >0, lim sup

n→∞ ε_nlogP

ε¹_n^/2

kn

r=1

w_n,r(x)

c_n(x)⁻¹−c⁻¹≥η

=−∞.

(C.2) For allx∈F, and anyη >0, lim sup

n→∞ ε_nlogP(|c_n(x)−c| ≥η) =−∞.

Before proceeding, let us comment on the assumptions. (H.1)–(H.4) are devoted to the control of the condition- ally centered estimatorf_n−E

f_n |N_n . Assumption (H.1) imposes that the number of terms of the partition goes to infinity not to quikly so that the mean number of points in each D_n,r goes to infinity. (H.2) requires the unknown functionf to be bounded away from 0. It also imposes that the mean number of points in D_n,r abovem_n,r converges to 0. Note that (H.1) and (H.2) force theν-essential oscillation off onI_n,rto converge uniformly to 0: δ_n → 0 asn→ ∞. (H.3) is devoted to the multivariate aspects of the limit theorems. (H.4) imposes to the weight functionsκ_n,r(x) in the linear combination (6) to be approximatively of the same order.

This is a natural condition to obtain an asymptotic gaussian behavior. These assumptions are easy to verify in practice since they involve either f or κ_n,r without mixing these two quantities. Assumptions (H.5) and (H.6) are devoted to the control of the residual conditional bias termE

f_n |N_n −f. (H.5) is natural since it implies that the non random part of the biasf_n−f vanishes. (H.6) control of the residual conditional bias term E

f_n|N_n −f_n. These two assumptions involve both the unknown functionf and the weight functionsκ_n,r and (H.6) can be looked at as a stronger version of (H.2). Condition (C.1) imposes the speed of convergence of c_n to the unknown parameter c to cancel the bias terms (ncν_n,r)⁻¹. It appears as a minimal condition needed to estimatec in the debiasing procedure without affecting the limit properties off_nin terms of central limit theorem and moderate deviation principle. Assumption (C.2) allows to replace c by its estimator in the asymptotic variance off_n.

2.1. Central limit theorem

Our first result states the multivariate central limit theorem forf_n:

Theorem 2.1. Assume that (H.1)−(H.6) hold for (ε_n)_n≥1≡1, that(C.1) holds forc1,n and that (C.2)holds for c2,n. Then, for all{x1, ..., x_p} ⊂F,

κ_n(x_j)⁻¹nc2,n(x_j)

f_n(x_j,c1,n)−f(x_j) : 1≤j≤p →

D N

0,Σ(x1,...xp) ,

where →

D denotes the convergence in distribution andN

0,Σ(x1,...xp)

is the centered gaussian law inR^p, with covariance matrix Σ(x1,...xp).

This result can be used to obtain explicit asymptoticγ% confidence interval forf(x) of the form:

f_n(x,c1,n)−z_γκ_n(x) (nc2,n(x))⁻¹, f_n(x,c1,n) +z_γκ_n(x) (nc2,n(x))⁻¹

,

wherez_γ is the (γ+ 1)/2th quantile of the N(0,1) distribution.

2.2. Moderate deviation principle

We will now establish a family of large deviation principles for f_n which is sometimes referenced in the litterature as a moderate deviation principle. Recall that a sequence of random variable (W_n)_n≥1 is said to follow the large deviation principle inR^p with speedε_n ↓0 and good rate function I :R^p →[0,∞], whenever, for every setA∈ B(R^p),

−inf

I (u) :u∈A^◦

≤ lim inf

n→∞ ε_nlogP(W_n ∈A)

≤ lim sup

n→∞ ε_nlogP(W_n∈A)≤ −inf

I (u) :u∈A ,

where A^◦ (resp. A) denotes the interior (resp. closure) of Ain R^p. This will denoted by (W_n)∈LDP (ε_n,I) in the sequel. We refer to [7] for general informations about large deviation theory.

Theorem 2.2. Assume that (H.1)–(H.6) hold for some ε_n ↓ 0, that(C.1) holds for c1,n and that(C.2) holds for c2,n. Then, for all{x1, ..., x_p} ⊂F such thatΣ_(x1,...,xp) is regular,

ε¹_n^/2κ_n(x_j)⁻¹nc2,n(x_j)

f_n(x_j,c1,n)−f(x_j) : 1≤j≤p

∈LDP

ε_n,I(x₁,...xp) where

I(x1,...xp):u∈R^p→2^−1tuΣ⁻_(x¹

1,...xp)u.

Let us mention some applications of Theorem2.2. First, it entails that for alls >0 and allx∈F, P

ε^1/2_n nκ_n(x)⁻¹c2,n(x)

f_n(x,c1,n)−f(x) ≥s exp

−2⁻¹s²ε⁻_n¹

(here u_nv_n means that logu_n/logv_n →1 asn→ ∞). This fact and the Borel-Cantelli Lemma can be used to prove that under (H.1)–(H.6) withε_n= log (n)⁻¹, we have, for allx∈F,

lim sup

n→∞

(2 log (n))^1/2κ_n(x) ⁻

1

nc2,n(x)f_n(x,c1,n)−f(x)≤1 a.s.

Moreover, it is well known that moderate deviation principle is a key tool to prove laws of the iterated logarithm (see e.g. [5] Th. 1-4-1). In terms of confidence intervals, Theorem 2.2 can also be useful to compute the logarithmic asymptotic level of confidence intervals with asymptotic level 0. More precisely, for fixedx∈Eand t >0, consider the interval

In(x, t) =

f_n(x,c1,n)−tε⁻_n^1/2κ_n(x) (nc2,n(x))⁻¹, f_n(x,c1,n) +tε⁻_n^1/2κ_n(x) (nc2,n(x))⁻¹ whereε_n =o(1). Then, Theorem2.1entails that

P(f(x)∈/In(x, t))→0 asn→ ∞.

Now, if conditions of Theorem2.2hold for (ε_n) we readily obtain that lim sup

n→∞ ε_nlogP(f(x)∈/ In(x, t)) =−2⁻¹t².

In other words, (In(x, t))_n≥1 is a sequence of confidence intervals for f(x) with logarithmic level 2⁻¹t² and speed (ε_n) (see [29] Def. 1).

Finally, in estimation theory, Theorem2.2can be used to compute the Kallenberg efficiency off_n(see [22,23]).

2.3. Estimation of c

At this stage, we present two estimates of f which belong to our general class (6) with some particular estimations ofc.

The first example has been previously introduced and studied in term of central limit theorem in case (P) in [13]

and is defined by

f_n^loc(x) =

kn

r=1

κ_n,r(x)ν_n,rY_n,r^∗

1 +N_n,r⁻¹ .

It is readily seen that f_n^loccan be written as in (6) with the localized estimator ofc:

c_n(x) =c^loc_n (x) :=k_nκ_n(x)

n

kn

r=1

κ_n,r(x)N_n,r⁻¹ν_n,rY_n,r^∗ −1

,

where

κ_n(x) :=k⁻_n¹

kn

r=1

κ_n,r(x).

Now, asc is constant, it may appear interesting to use a more global estimation of it. To this aim, we consider the global estimator

c^glo_n =k_n

n

kn

r=1

ν_n,rY_n,r^∗ N_n,r⁻¹ −1

,

leading to

f_n^glo(x) :=f_n x;c^glo_n

=

kn

r=1

κ_n,r(x) +κ_n(x)N_n,r⁻¹

ν_n,rY_n,r^∗ .

The following corollary shows that Theorems2.1and2.2can be applied to f_n^locandf_n^glo. Corollary 2.1. Theorems2.1and2.2 hold whenc_n,1∈

c^loc_n ;c^glo_n

andc_n,2=c^glo_n .

Remark 2.1. For the limit theorems considered here, f_n^loc and f_n^glo are equivalent. Nethertheless, it can be shown thatc^glo_n is a better estimator ofc than c^loc_n and thereforef_n^glo may be prefered to f_n^loc. On the other hand, if the intensity (2) of the point process is not totally uniform, it is clear thatf_n^loc, which is based on a local estimation ofc, is more robust thanf_n^glo.

3. Applications

We first introduce a general class of kernel estimators which will be shown to satisfy our main results given in Theorems2.1and2.2. Then, we focus on the particular cases of Parzen-Rosenblatt and Dirichlet kernels.

3.1. General kernel estimates

Here, in order to smooth (4), a sequence K_n : E×E → R, of general smoothing kernels is introduced.

Conditions on this sequence will be imposed later. The general integrated kernel estimate is defined by f_n(x;c_n) =

E

K_n(x, t)f_n^◦(t;c_n)ν(dt)

=

kn

r=1

In,r

K_n(x, t)ν(dt) Y_n,r^∗ + (nc_nν_n,r)⁻¹ . (7) It appears that (7) is a particular case of (6) with κ_n,r(x) = ν_n,r⁻¹

In,rK_n(x, t)ν(dt). In the case where the calculation of this mean value is computationally expansive, it can be approximated byK_n(x, x_n,r) for some x_n,r∈I_n,r, leading to the simplified estimate

f_n(x;c_n) =

kn

r=1

ν_n,rK_n(x, x_n,r)

Y_n,r^∗ + (nc_nν_n,r)⁻¹ , (8) which is still a particular case of (6) withκ_n,r(x) =K_n(x, x_n,r).

In order to introduce the assumptions needed onK_n, we set, for allx∈E, Γ_n,r(x) = sup{Kn(x, t)−K_n(x, s) : (s, t)∈I_n,r×I_n,r}, Ξ_n(x) =k_n

kn

r=1

In,r×In,r

K_n(x, t) (f(s)−f(t))ν(dt)ν(ds) , and

Ψ_n(x) =

E

K_n(x, t)f(t)ν(dt)−f(x) .

For the sake of simplicity, assume that, for alln≥1, the partitions{In,r: 1≤r≤k_n}are such thatν_n,r=k⁻_n¹ for allr≤k_n. We also set

∆_n = max

r≤kn

(M_n,r−m_n,r) and, for all function g:E→R, we note

g_p =

E

|g(t)|^pν(dt) 1/p

, and g_I = sup

t∈I|g(t)| (I⊂E). In this context, the general assumptions (H.1)−(H.6) can be expressed as:

(H.1) k_n↑ ∞ andn⁻¹k_nmax

log (n), ε⁻_n¹

→0 asn→ ∞.

(H.2) 0< m≤M <+∞andnk_n⁻¹∆_n→0 asn→ ∞.

(K.1) For alln≥1,

E×E|Kn(x, t)|ν(dx)ν(dt)<∞.

(K.2) For all (x1, x2)∈F×F,

kn

r=1

Γ_n,r(x1)

In,r

|Kn(x2, t)|ν(dt) =o(Kn(x1, .)₂Kn(x2, .)₂) asn→ ∞.

(K.3) For all (x1, x2)∈F×F,

Kn(x1, .), K_n(x2, .)₂(Kn(x1, .)₂Kn(x2, .)₂)⁻¹→ σ(x1, x2) asn→ ∞.

(K.4) For allx∈F,

(ε_nk_n)^−1/2Kn(x, .)⁻₂¹ Kn(x, .)_E→0 asn→ ∞.

(K.5) For allx∈F,

ε^1/2_n nk_n^−1/2K_n(x, .)⁻₂¹ max (Ψ_n(x) ; Ξ_n(x))→0 asn→ ∞.

(K.6) For allx∈F,

ε^1/4_n nk_n^−3/4K_n(x, .)⁻2^1/2K_n(x, .)¹1^/2∆_n→0 asn→ ∞. (K.7) For allx∈F,

ε¹_n^/2nk_n^−1/2Kn(x, .)⁻₂¹ _k

n

r=1

In,r

(K_n(x, t)−K_n(x, x_n,r))ν(dt)

→0 asn→ ∞.

In the following, we set

v_n(x) =nk_n^−1/2Kn(x, .)⁻₂¹, and

Σ(x1,...xp)= [σ(x_i, x_j)]1≤i,j≤p. The results established in Section2yield:

Theorem 3.1. a) If (H.1), (H.2) and (K.1)–(K.6) hold with(ε_n)_n≥1≡1, then, for all c1,n (resp. c2,n) such that (C.1) (resp. (C.2)) hold and all (x1, ...x_p)⊂F,

c2,n(x_j)v_n(x_j)

f_n(x_j;c1,n)−f(x_j) : 1≤j≤p →

D N

0,Σ(x1,...,xp)

. (9)

b) If, moreover, (K.7)holds with (ε_n)_n≥1≡1, then for all {x1, ..., x_p} ⊂F, c2,n(x_j)v_n(x_j)

f_n(x_j;c1,n)−f(x_j) : 1≤j≤p →

D N

0,Σ(x1,...,xp)

. (10)

Theorem 3.2. a) For all ε_n ↓ 0 such that (H.1), (H.2) and (K.1)–(K.6) hold, for all c1,n (resp. c2,n) such that (C.1) (resp. (C.2)) hold and all (x1, ...x_p)⊂F such that Σ(x1,...,xp)is regular,

ε¹_n^/2c2,n(x_j)v_n(x_j)

f_n(x_j;c1,n)−f(x_j) : 1≤j≤p

∈LDP

ε_n,I(x₁,...xp)

(11) with

I(x1,...,xp):u∈R^p→2^−1tuΣ⁻_(x¹

1,...,xp)u.

b) If, moreover, (K.7)holds, then

ε¹_n^/2c2,n(x_j)v_n(x_j)

f_n(x_j;c1,n)−f(x_j) : 1≤j≤p

∈LDP

ε_n,I(x1,...xp)

. (12)

Some illustrations of this result are now provided.

Remark 3.1. In all the applications below involvingf_n,x_n,r will be defined as the center ofI_n,r.

3.2. Parzen kernel estimates

Here, we takeE= [0,1]^d(d∈N^∗),ν is the Lebesgue measure onEand{In,r: 1≤r≤k_n}is an equidistant partition of E such that I_n,r = _d

j=1J_n,r,j where the J_n,r,j are intervals of [0,1] of length k⁻_n^1/d, leading to ν_n,r=k⁻_n¹for all 1≤r≤k_n.

The multivariate Parzen kernel estimate is defined by K_n^{P R}(x, t) =h^−d_n K

h⁻_n¹(x−t) ,

where K :R^d →R⁺ is a Parzen-Rosenblatt kernel such thatK ∈L² R^d

, and (h_n) is a sequence of positive real numbers tending to zero (seee.g.[18]).

In a first time, we will assume that there existsα∈(0,1] such that:

(PR.1a)f isα-Lipschitz,

(PR.1b) Kis uniformly continuous except on a finite setD⊂R^d and

R^du^α_RdK(u) du <∞.

Denote by I_p the identity matrix of R^p. We are now in position to prove a central limit theorem and a moderate deviation principle for f_n.

Corollary 3.1. Assume that(PR.1)holds and

(i) n⁻¹k_nlog (n) =o(1), (ii) h^d_nk_n → ∞ and (iii) nk_n^−1/2h^α_n^+d/2=o(1). Then, (9) holds for F = (0,1)^d,v_n=nh^d/_n²k_n^−1/2 andΣ_(x1,...,xp)=K²2 I_p.

The best rate of convergence is obtained for h_n =n^−α+d1 and k_n=n^α+dd u²_n, whereu_n → ∞arbitrarily slowly.

In this case, v_n =n^α+dα u⁻_n¹.

Corollary 3.2. Assume that(PR.1)holds and for someε_n →0, (i) n⁻¹k_nmax

log (n), ε⁻_n¹

=o(1), (ii) h^d_nk_nε_n→ ∞ and (iii) nk_n^−1/2h^α_n^+d/2ε¹_n^/2=o(1). (13) Then, (11) holds for F= (0,1)^d,v_n=nh^d/n ²kn^−1/2 andΣ(x1,...,xp)=K²₂ I_p.

In particular, if ( 13) holds forε_n = log (n)⁻¹, then for all,x∈F, lim sup

n→∞ (2 log (n))^−1/2nh^d/2_n k_n^−1/2c2,n(x)f_n(x,c1,n)−f(x)≤ K²2 a.s.

Remark 3.2. a) Corollary 3.1 shows that, when f is α-lipschitzian, the speed of convergence of f_n can be chosen arbitrarily close to the minimax speed n^−α+dα (see [17]).

b) In the case of Poisson point process withd= 1,f_n with Parzen Rosenblatt kernel has been studied in [12].

In particular [12] Theorem 6-2, gives a central limit theorem forf_n with optimal ratev_n=n^α+5/4α u⁻_n¹. Hence, from asymptotical point of view,f_n is better thanf_n.

Since our approach involves regularization of Geffroy’s estimates it is natural to study the case wheref is more regular than justα-lipschitzian. In the following, for simplicity, we takeE= [0,1] and we only deal with the central limit theorem. Assume that:

(PR.2a) f is inC²(E)

(PR.2b), There exists a finite set Dsuch that

K isC¹onD^c, K∈L¹(R) and

Ru²K(u) du <∞. (14)

Corollary 3.3. a) If (PR.2) holds and

(i)n⁻¹k_nlog (n) =o(1), (ii)h_nk_n→ ∞, (iii)nk_n^−7/4h¹_n^/4=o(1) and (iv)nk_n^−1/2h⁵_n^/2=o(1), then, (9) is true for F= (0,1),v_n=nh^1/2_n k⁻_n^1/2 andΣ(x1,...,xp)=K²₂ I_p.

The choices ofh_n=n^−5/17 andk_n =n^9/17u²_n, whereu_n→ ∞arbitrarily slowly, lead to v_n=n^10/17u⁻_n¹. b) If moreover,K is even,1-Lipschitzian andC² onD^c with K∈L¹(R), then, under (i)–(iv) and

(v) nk⁻_n^5/2h⁻_n^3/2=o(1), (10) is true for F,v_n andΣ(x1,...,xp) defined as above.

The choices ofh_n=n^−2/7 andk_n =n^4/7u²_n, whereu_n→ ∞arbitrarily slowly, lead to v_n=n^4/7u⁻_n¹.

3.3. Projection estimates: Dirichlet kernels

Letf ∈L²(E, ν) and (e_j)_j∈N be an orthonormal basis ofL²(E, ν). The expansion off on this basis is f(x) =

∞ j=0

a_je_j(x), x∈E,

where eacha_j=

Ee_j(t)f(t)ν(dt) can be estimated by

a_j,kn=

kn

r=1

In,r

e_j(t)ν(dt) Y_n,r^∗ + (nc_nν_n,r)⁻¹ , 1≤j≤b_n.

This leads to an estimation off(x)via:

f_n(x;c_n) =

ln

j=0

a_j,kne_j(x) =

kn

r=1

In,r

K_n^D(x, t)ν(dt) Y_n,r^∗ + (nc_nν_n,r)⁻¹ , (15)

where (l_n) is a sequence of integers tending to infinity andK_n^Dthe Dirichlet’s kernel associated to the orthonor- mal basis (e_j)_j∈N defined by

K_n^D(x, t) =

ln

j=0

e_j(x)e_j(t), (x, t)∈E². (16)

It appears that (15) is a particular case of (7t) with K_n =K_n^D. Of course, the, sometimes easier to handle, estimate

f_n(x;c_n) =

kn

r=1

ν_n,rK_n^D(x, x_n,r)

Y_n,r^∗ + (nc_nν_n,r)⁻¹ , (17) can also be defined. Below, we focus on the trigonometric basis onE= [0,1], ν is the Lebesgue measure onE, and for all 1≤r≤k_n, I_n,r are disjoint intervals of [0,1] of length k_n⁻¹ (and thereforeν_n,r= 1/k_n).

This basis is defined forx∈[0,1] by

e0(x) = 1, e2j−1(x) = 2^1/2cos (2jπx), e2j(x) = 2^1/2sin (2jπx), j≥1.

It is easily seen in that case that the Dirichlet kernel is

K_n^D(x, t) = sin ((1 +l_n)π(x−t))

sin (π(x−t)) forx=t (18)

= 1 +l_n ifx=t.

Besides, we assume that:f is inC²([0,1]), f(0) =f(1) andf(0) =f(1). In particular,

∆_n =O k_n⁻¹

. (19)

Corollary 3.4. a) Assume that

(i)n⁻¹k_nlog (n) =o(1), (ii)k_n⁻¹l_nlog (l_n) =o(1), (iii)nk_n^−1/2l_n⁻²=O(1), (iv) nk⁻_n^5/2l^1/2_n log (l_n) =o(1) and (v) nl⁻_n^1/4k_n^−7/4log (l_n)^1/2=o(1). Then (9) holds forF = [0,1],v_n=n(l_nk_n)^−1/2 andΣ(x1,...,xp)=I_p.

In particular, the choices of l_n=n^10/27 andk_n =n^14/27log (n)^2/7u²_n, whereu_n→ ∞arbitrarily slowly, lead tov_n =n^5/9log (n)^−1/7u⁻_n¹.

b) If (i)–(iii) and (iv’) nk_n^−5/2l_n^3/2log (l_n) =o(1) are true, then (10) holds for F, v_n andΣ(x1,...,xp) defined as above.

The choices of l_n = n^8/23 and k_n = n^14/23log (n)^2/5u²_n, where u_n → ∞ arbitrarily slowly, lead to v_n =n^12/23log (n)^−1/5u⁻_n¹.

Remark 3.3. a) From the asymptotical point of view,f_n is still better thanf_n. Nevertheless, since whenf is C², the minimax speed of convergence isn^−2/3(see [17]), the above estimates are suboptimal.

b) In the case of Poisson point process,f_nwith the trigonometric Dirichlet kernel has been introduced by Girard and Jacob [11]. In that context, they get a pointwise central limit theorem (see [11] Cor. 3) but their result is not sharp and Corollary3.4 (b) constitutes a significant improvement since, instead of n^12/23log (n)^−1/5u⁻_n¹, they obtain an “optimal” speed of convergence ofn^2/5log (n)^−1/5(log log (n))^ε (ε >0).

3.4. Concluding remarks

In terms of applications, we have presented examples of integrated smoothed kernel estimates f_n and of discrete smoothed kernel estimatesf_n. In all the cases considered,f_nis strictly better thanf_nfrom asymptotical point of view (i.e. (K.7) is not implied by the other assumptions of Ths. 3.1and 3.2). When f ∈C², Parzen kernel leads to better speed of convergence than trigonometric Dirichlet kernel but it does not reach the minimax speed.

Of course, other kernels are allowed, in particular wavelet kernels can be treated in the framework of Theo- rems3.1 and3.2. This is part of our future work. More generaly, the following problem should be of interest:

given a class of functionF, and assuming thatf ∈ F, what is the best estimate of type (7) and in which case could we raise the minimax speed of convergence?

4. Proofs 4.1. Proofs of the results of Section 2

Let us first introduce some notations useful for the sequel. For all 1≤r≤k_n, we define:

• λ_n,r=ncν_n,rm_n,r, Λ_n,r=ncν_n,rM_n,r, µ_n,r=ncν_n,rf_n,r,

• b_n,r=b_n,r(ε_n) =

3µ⁻_n,r¹

ε⁻_n^1/2(nν_n,r)^1/2+ log

ν_n,r^{−1 1/2}

, b_n=b_n(ε_n) = max

r≤kn

b_n,r,

• B_n,r=B_n,r(ε_n) ={|Nn,r−µ_n,r| ≤b_n,rµ_n,r}, B_n=B_n(ε_n) = ∩

r≤kn

B_n,r,

• Z_n,r^∗ =ncν_n,rY_n,r^∗ , E_n,r=E

Z_n,r^∗ |N_n,r

, Z_n,r^◦ =Z_n,r^∗ −E_n,r,

• β_n,r=N_n,r⁻¹(N_n,r+ 1)E_n,r−µ_n,r, ζ_n,r=N_n,r⁻¹(N_n,r+ 1)Z_n,r^◦ ,

• Θ_n(x) =_kn

r=1w_n,r(x) ζ_n,r.