Adaptive tests of homogeneity for a Poisson process

www.imstat.org/aihp 2011, Vol. 47, No. 1, 176–213

DOI:10.1214/10-AIHP367

© Association des Publications de l’Institut Henri Poincaré, 2011

Adaptive tests of homogeneity for a Poisson process

M. Fromont

, B. Laurent

and P. Reynaud-Bouret

aUniversité Européenne de Bretagne, CREST-ENSAI, IRMAR, Rennes, France. E-mail:mfromont@ensai.fr bInstitut de Mathématiques de Toulouse, INSA de Toulouse, Université de Toulouse, Toulouse, France.

E-mail:Beatrice.Laurent@insa-toulouse.fr

cCNRS ENS Paris et Université de Nice Sophia-Antipolis, Paris, France. E-mail:reynaudb@unice.fr Received 15 May 2009; revised 8 March 2010; accepted 6 April 2010

Abstract. We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates inL²-norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.

Résumé. Nous proposons de tester l’homogénéité d’un processus de Poisson observé sur un intervalle borné. Nous établissons tout d’abord des bornes inférieures pour les vitesses de séparation uniformes relativement à la normeL²sur des Besov bodies classiques ou faibles. De façon surprenante, nous obtenons des bornes inférieures sur les Besov bodies faibles qui coïncident avec les vitesses minimax d’estimation sur ce type de classe. Ensuite, nous construisons des procédures de tests non asymptotiques et non paramétriques qui sont adaptatives, au sens où elles atteignent, à un facteur logarithmique près dans certains cas, les vitesses de séparation optimales sur plusieurs classes d’alternatives simultanément. Ces procédures sont basées sur des méthodes de sélection de modèles et de seuillage. Enfin, nous complétons cette étude théorique par des simulations afin d’estimer par la méthode de Monte Carlo la puissance de nos tests sous diverses alternatives.

MSC:Primary 62G10; secondary 62G20

Keywords:Poisson process; adaptive hypotheses testing; uniform separation rate; minimax separation rate; model selection; thresholding rule

1. Introduction

Poisson processes have been used for many years to model a great variety of situations: machine breakdowns, phone calls, etc. Recently Poisson processes become popular for modeling occurrences of words or motifs on the DNA sequence (see [24]). In this context, it is particularly important to be able to detect abnormal behaviors.

With such applications in mind, we consider in this paper the question of testing the homogeneity of a Poisson processN. Since we can only observe a finite number of points of the process, this question has a sense only on a finite interval. For the sake of simplicity, we assume that the Poisson processNis observed on the fixed set[0,1], and that it has an intensityswith respect to some measureμon[0,1]with dμ(x)=Ldx.

Denoting byS0 the set of constant functions on[0,1], our aim is consequently to test the null hypothesis(H0)

“s∈S0,” against the alternative(H1)“s /∈S0.”

This problem of testing the homogeneity of a Poisson process has been widely investigated both from a theoretical and practical point of view (see [2] or [7] for a survey and [5] for a more recent work). In these papers, the alternative

intensities are monotonous. Another related topic is the problem of testing the simple hypothesis that a point process is a Poisson process with a given intensity. We can cite for instance the papers by Fazli and Kutoyants [12] where the alternative is also a Poisson process with a known intensity, Fazli [11] where the alternatives are Poisson processes with one-sided parametric intensities or Dachian and Kutoyants [10], where the alternatives are self-exciting point processes. The paper by Ingster and Kutoyants [18] is the closest one to the present work. The alternatives considered by Ingster and Kutoyants are Poisson processes with nonparametric intensities in a Sobolev or Besov B^(δ)_2,q(R)ball with 1≤q <+∞and known smoothness parameterδ.

However, in some practical cases like the study of occurrences of words or motifs on a DNA sequence, such smooth alternatives cannot be considered. The intensity of the Poisson process in these cases may burst at a particular position of special interest for the biologist (see [14] for more details). The question of testing the homogeneity of a Poisson process then becomes “how can we distinguish a Poisson process with constant intensity from a Poisson process whose intensity has some small localized spikes?.” This question has already been partially considered in the seventies in a precursory work by Watson [27]: he proposed a test based on the estimation of the Fourier coefficients of the intensity without evaluating the power of the resulting procedure.

In this paper, we focus on constructing adaptive testing procedures, i.e. which do not use any prior information about the smoothness of the intensitys, but which however have the best possible performances (in a minimax sense).

From a theoretical point of view, we evaluate the performances of the tests in terms of uniform separation rates with respect to some prescribed distanced over various classes of functions. Givenβ∈ ]0,1[, a class of functionsS1, and a levelαtestΦ_α with values in{0,1}(rejecting(H0)whenΦ_α=1), the uniform separation rateρ(Φ_α,S1, β) ofΦα over the classS1is defined as the smallest positive numberρsuch that the test has an error of second kind at most equal toβfor all alternativessinS1at anL²-distanceρfromS0. More precisely, ifPs denotes the distribution of the Poisson processNwith intensitys,

ρ(Φ_α,S1, β)=inf

ρ >0, sup

s∈S1,d(s,S0)>ρ

Ps(Φ_α=0)≤β

(1.1)

=inf

ρ >0, inf

s∈S1,d(s,S0)>ρPs(Φ_α=1)≥1−β

. (1.2)

In view of the practical situations of our interest, we study some classes of alternatives that can be very irregular, for instance that can have some localized spikes. We then consider some classical Besov bodies and also some spaces that can be viewed as weak versions of these classical Besov bodies and that are defined precisely in the following.

The interested reader may find in [23] some illustrations of functions in weak Besov spaces and how the smoothness parameters of the functions govern the proportion and amplitude of their spikes.

As a first step, we evaluate the best possible value of the uniform separation rate over these spaces. In other words, we give a lower bound for

ρ(S1, α, β)=inf

Φα

ρ(Φ_α,S1, β), (1.3)

where the infimum is taken over all level αtestsΦα, and whereS1 can be either a Besov body or a weak Besov body. This quantity introduced by Baraud [3] as the(α, β)-minimax rate of testing over S1 or the minimax sepa- ration rate over S1 is a stronger version of the (asymptotic) minimax rate of testing usually considered. The key reference for the computation of minimax rates of testing in various statistical models is the series of papers due to Ingster [16]. Concerning the Poisson model, Ingster and Kutoyants [18] give the minimax rate of testing for Sobolev or BesovB^(δ)_2,q(R)balls with 1≤q <+∞and smoothness parameter δ >0. They find that this rate of testing for the Sobolev or Besov norm or semi-norm is of orderL^{−2δ/(4δ+1)}. Let us note that we find here lower bounds for the classical Besov bodies similar to Ingster and Kutoyants’ ones. Furthermore, our lower bounds for the weak Besov bodies are larger than the ones for classical Besov bodies. Alternatives in weak Besov bodies are in fact so irreg- ular that it is as difficult to detect them as to estimate them. The problem of estimation in weak Besov spaces is solved by using thresholding procedures: indeed the weak Besov spaces are closely related to the maxisets of those procedures (see [19] in the Gaussian framework and [22] in a Poisson model). To our knowledge, no previous re- sults of this kind exist for weak Besov bodies in testing problems, even in more classical statistical models, like the density model. Despite the similarity of both models, our lower bounds over weak Besov bodies cannot however

be straightly transposed to the density model since our proofs heavily rely on the Poissonian independence proper- ties.

As a second and main step, we construct non asymptotic levelαtests which achieve, up to a possible logarithmic factor, the minimax separation rates over many Besov bodies and weak Besov bodies simultaneously, whereas using no prior information about the smoothness of the intensity s. Our idea here is to combine some model selection methods that are effective for alternatives in classical Besov bodies and a thresholding type approach, inspired by the thresholding rules used for adaptive estimation in weak Besov bodies. Key tools in the proofs of our results are exponential inequalities for U-statistics of order 2 due to Houdré and Reynaud-Bouret [15].

Of course, both model selection and thresholding approaches have already been used to construct adaptive tests in various statistical models. One can cite among others the papers by Spokoiny [25,26] in Gaussian white noise models or by Baraud, Huet and Laurent [4] in a Gaussian regression framework. These papers propose adaptive tests which combine methods closely related to both model selection and thresholding ones. As for the density framework, adaptive tests were proposed by Ingster [17] or Fromont and Laurent [13], using model selection type methods and by Butucea and Tribouley [6] using thresholding type methods.

The present work is organized as follows. In Section2, we provide lower bounds for the minimax separation rates over various Besov bodies. Our testing procedures are defined in Section3, and their uniform separation rates over Besov bodies are established in Section4. We carry out a simulation study in Section5to illustrate these theoretical results, and the proofs are postponed to the last section.

2. Lower bounds for the minimax separation rates over Besov bodies

We consider the Poisson processNwith intensityswith respect to some measureμon[0,1], with dμ(x)=Ldx. In the following, we assume thatsbelongs toL²([0,1]), and·,·, · andd respectively denote the scalar product

f, g =

[0,1]f (x)g(x)dx, theL²-norm

f²=

[0,1]f²(x)dx, and the associated distance.

Let us denote the Haar basis ofL²([0,1])by{φ₀, φ_(j,k), j ∈N, k∈ {0, . . . ,2^j−1}}with φ0(x)=1_[0,1](x)

and

φ_(j,k)(x)=2^j/2ψ

2^jx−k

, (2.1)

whereψ (x)=1_[0,1/2[(x)−1_[1/2,1[(x).

We setα₀= s, φ₀and for everyj∈N, k∈ {0, . . . ,2^j−1},α_(j,k)= s, φ_(j,k). We can now introduce the Besov bodies defined forδ >0,R >0 by

B^δ_2,∞(R)=

s≥0, s∈L² [0,1]

, s=α₀φ₀+

j∈N 2^j−1

k=0

α_(j,k)φ_(j,k),

∀j∈N,

2^j−1 k=0

α²_(j,k)≤R²2^{−2j δ} , (2.2)

and more generally forp≥1,R >0 andδ >max(0,1/p−1/2), by Bp,^δ∞(R)=

s≥0, s∈L² [0,1]

, s=α₀φ₀+

j∈N 2^j−1

k=0

α_(j,k)φ_(j,k),

∀j∈N,

2^j−1 k=0

|α(j,k)|^p≤R^p2^{−pj (δ+1/2−1/p)} . (2.3)

As in [22], we also introduce some weaker versions of the above Besov bodies given forγ >0 andR>0 by

W_γ R

=

s≥0, s∈L² [0,1]

, s=α₀φ₀+

j∈N 2^j−1

k=0

α_(j,k)φ_(j,k),

∀t >0,

j∈N 2^j−1

k=0

α_(j,k)² 1_α2

(j,k)≤t≤R²t^{2γ /(1+2γ )} . (2.4)

Fixing some levels of error αandβ in ]0,1[, and denoting byL^∞(R)the set of functions bounded by R, our purpose in this section is to find sharp lower bounds forρ(B_2,^δ_∞(R)∩W_γ(R)∩L^∞(R), α, β), whereρ is defined by (1.3).

Starting from a general idea developed by Ingster [16], we obtain the following result.

Theorem 1. Assume thatR >0,R>0,andR≥2,and fix some levelsαandβ in]0,1[such thatα+β≤0.59.

Let us consider the different cases that are represented in Fig.1.

In case(i)whereδ≥max(γ /2, γ /(1+2γ )),then lim inf

L→+∞L^2δ/(1+4δ)ρ

B_2,^δ_∞(R)∩W_γ R

∩L^∞ R

, α, β

>0.

In case(ii)whereδ < γ /2andγ >1/2,then lim inf

L→+∞

L lnL

γ /(1+2γ )

ρ

B2,^δ∞(R)∩W_γ R

∩L^∞ R

, α, β

>0.

Fig. 1. The set of possible parameters (δ, γ): visualization of the different cases appearing in Theorem1.

In case(iii)whereδ < γ /(1+2γ )andγ≤1/2,then lim inf

L→+∞

L^1/4∧L^{2γ /((1+4δ)(1+2γ ))} ρ

B^δ2,∞(R)∩W_γ R

∩L^∞ R

, α, β

>0.

Comments:

(1) For the whole set of parameters(δ, γ )such thatδ≥γ /(1+2γ ), we prove in Section4that these lower bounds are actually sharp.

(2) We have in case (i) lower bounds which coincide with the minimax rates of testing obtained by Ingster and Kutoyants [18] when testing that a periodic Poisson process has a given intensity in the Besov spacesB_2,^(δ)_∞(R).

We know (see [13,17], for instance) that such rates can be achieved by some multiple testing procedure based on model selection type methods. This is the principle of our first procedure described in Section3.1.

(3) We notice that the lower bounds obtained in case (ii) are equal to the minimax estimation rates on the maxisets of the thresholding estimation procedure, namelyB^{κγ /(1}_2,∞ ^{+2γ )}(R)∩W_γ(R)with some constantκ <1 (see [19, 22,23] for more details). This means that it is at least as difficult to test as to estimate over such classes of functions, phenomenon which is quite unusual. Since the minimax estimation rates on these classes are achieved by thresholding rules, it will be natural to construct a testing procedure based on thresholding methods: this is the idea that originated our second procedure described in Section3.2.

(4) As for the case (iii), our present results do not allow us to say anything about the possible optimality of the obtained lower bound.

3. Two tests of homogeneity

Let us recall thatS0denotes the set of constant functions on[0,1]and that we assume thatsbelongs toL²([0,1]).

In this section, we construct levelαtests of the null hypothesis(H0)“s∈S0,” against the alternative(H1)“s /∈S0,”

from the observation of the Poisson processN, or the points{X_l, l=1, . . . , NL}of the Poisson process.

We introduce two testing procedures that come from two different statistical approaches. The first one originates in general model selection methods, while the second one is closer to the thresholding type methods.

In order to understand the global ideas of these procedures, let us notice that the squaredL²-distanced²(s,S0) betweensand the set of constant functionsS0can be rewritten as

d²(s,S0)=

[0,1]

s(x)−

[0,1]s(y)dy 2

dx

= s²−α²₀

=

λ∈Λ_∞

α²_λ,

whereα₀= s, φ₀, and for allλ∈Λ_∞= {(j, k), j∈N, k∈ {0, . . . ,2^j−1}},α_λ= s, φ_λ. For everyλ∈Λ_∞,α_λcan be estimated by

α_λ= 1

L

[0,1]φ_λ(x)dNx, which is also equal to

α_λ= 1

L

N_L

l=1

φ_λ(X_l).

From this variable, we deduce an unbiased estimator ofα²_λgiven by T_λ=α_λ²− 1

L²

[0,1]φ_λ²(x)dNx= 1 L²

N_L

l=l=1

φ_λ(X_l)φ_λ(X_l). (3.1)

Our first approach will consist in constructing estimators ofd²(s,S0)=

λ∈Λ_∞α²_λ based on a combination of the T_λ’s, and in rejecting the null hypothesis when one of these estimators is too large. This was already the spirit of Watson’s procedure (see [27]). Our second approach is related to the test considered in [4] to detect local alternatives.

It will consist in considering a set ofT_λ’s and rejecting the null hypothesis directly when one of theT_λ’s is too large.

Let us now precisely define both procedures.

3.1. A first procedure based on model selection

Assuming that s∈L²([0,1]), a natural idea is to construct a testing procedure from an estimation of the squared L²-distanced²(s,S0).

In order to estimate this functional of s, following the ideas of Laurent [20] and Fromont and Laurent [13], we introduce embedded finite-dimensional linear subspaces of L²([0,1]). We choose here to consider forJ ≥1 the subspacesS_J generated by the subsets{φ₀, φ_λ, λ∈Λ_J}of the Haar basis defined by (2.1), withΛ_J = {(j, k),j∈ {0, . . . , J−1},k∈ {0, . . . ,2^j−1}}. Each subspaceS_J is called amodel. We denote byD_J=2^Jthe dimension ofS_J, and bys_J the orthogonal projection ofsonto the modelS_J.

Focusing on one model SJ, we estimate d²(s,S0)= s²−α₀² by the unbiased estimator of sJ²−α₀²=

λ∈Λ_Jα_λ²given by T_J =

λ∈Λ_J

T_λ, (3.2)

withT_λdefined by (3.1). The estimatorT_J obviously depends on the choice of the modelS_J.

Since we do not want to choose a priori such a model, we consider a collection of models{S_J, J∈J}whereJ is a finite subset ofN^∗, and the corresponding collection of estimators{T_J, J∈J}.

The procedure that we introduce here then consists in rejecting(H0)“s∈S0” when there existsJ inJ such that the estimatorT_J given by (3.2) is too large.

At this point there are several ways to decide whenT_J is too large.

In all cases, we use the well-known argument that, conditionally on the event “the number of pointsN_L falling into [0,1] isn,” the points of the process obey the same law as ann-sample (X˜₁, . . . ,X˜_n)with common density s/

[0,1]s(x)dx. It follows that for alln∈N, Ps

T_J > q|N_L=n

=P

1 L²

λ∈Λ_J

n l=l=1

φ_λ(X˜_l)φ_λ(X˜_l) > q

.

Under the null hypothesis, the intensitysis constant on[0,1], and theX˜_l’s are i.i.d., with uniform distribution on [0,1]. This distribution is free from the parameters. As a consequence, for everyu∈ ]0,1[, we can introduce and estimate by Monte Carlo experiments the(1−u)quantile of the distribution ofT_J|N_L=nunder the null hypothesis, that we denote byq_J⁽ⁿ⁾(u).

We now consider the test statistics:

Tα⁽¹⁾=sup

J∈J

T_J −q_J^(NL) u_J,α^(NL)

, (3.3)

withu_J,α^(NL)to be correctly chosen.

Finally, we define the corresponding test function:

Φ_α⁽¹⁾=1_T(1)

α >0. (3.4)

And our first test consists in rejecting the null hypothesis(H0)whenΦα⁽¹⁾=1.

Let us see how we can chooseu_J,α^(NL)so that our test has a levelα.

An obvious possibility is to set u_J,α⁽ⁿ⁾= α

|J| for everyJinJ andninN.

This choice corresponds to a Bonferroni procedure andΦ_α⁽¹⁾actually defines a levelαtest. Indeed, fors∈S0, Ps

sup

J∈J

T_J−q_J^(NL) u_J,α^(NL)

>0

=

n∈N

Ps

sup

J∈J

T_J−q_J⁽ⁿ⁾ α

|J|

>0N_L=n

Ps(N_L=n)

≤

n∈N

J∈J

α

|J|Ps(N_L=n)

≤α.

Another choice foru_J,α⁽ⁿ⁾, inspired by Fromont and Laurent [13], consists in setting u_J,α⁽ⁿ⁾=e^−WJsup

u∈ ]0,1[,sup

s∈S0

Ps

sup

J∈J

T_J−q_J⁽ⁿ⁾

ue^−WJ

>0N_L=n ≤α

, (3.5)

where{W_J, J∈J}is a collection of positive weights such that

J∈J

e^−WJ ≤1.

Since this new definition ofu_J,α⁽ⁿ⁾leads to a less conservative procedure, this is the one that we use in the rest of the paper. We notice thatu_J,α⁽ⁿ⁾≥αe^−WJ for everyninN, and that in practice,u_J,α⁽ⁿ⁾can be estimated by Monte Carlo experiments.

3.2. A second procedure based on a thresholding approach

Let us recall here that the squaredL²-distanced²(s,S0)betweensand the set of constant functionsS0is equal to

λ∈Λ_∞α_λ² and that T_λ defined by (3.1) is an unbiased estimator of α²_λ. Based on general thresholding ideas, our second procedure consists in fixing someJ¯≥1 and rejecting the null hypothesis(H0)when there existsλinΛ_J¯such thatT_λis too large.

Let us now see what we mean by “Tλis too large.” We can still use the fact that Ps(T_λ> q|N_L=n)=P

1 L²

n l=l=1

φ_λ(X˜_l)φ_λ(X˜_l) > q

,

and that under the null hypothesis, theX˜_l’s are i.i.d., with uniform distribution on[0,1]. We can therefore introduce and estimate by Monte Carlo experiments the(1−u)quantile of the distribution ofT_λ|N_L=nunder the null hy- pothesis, that we denote byq_λ⁽ⁿ⁾(u), for everyu∈ ]0,1[. Notice that forλ=(j, k)∈Λ_J¯,q_λ⁽ⁿ⁾(u)does not depend onk.

We set Tα⁽²⁾= sup

λ∈Λ_J¯

T_λ−q_λ^(NL) u^(N_λ,α^L)

, (3.6)

withu^(N_λ,α^L)to be correctly chosen.

We also define Φ_α⁽²⁾=1_T(2)

α >0. (3.7)

Our test consists in rejecting the null hypothesis(H0)whenΦ_α⁽²⁾=1.

Let us now see how we chooseu⁽ⁿ⁾_λ,α. An obvious choice corresponding to the Bonferroni procedure would be u⁽ⁿ⁾_λ,α= α

2^jJ¯ for everyλ=(j, k)inΛ_J¯andninN. To obtain a less conservative procedure, we prefer setting

u⁽ⁿ⁾_λ,α=u⁽ⁿ⁾_α

2^jJ¯ for everyλ=(j, k)inΛ_J¯andninN, (3.8)

with

u⁽ⁿ⁾_α =sup

u∈ ]0,1[,sup

s∈S0

Ps

sup

(j,k)∈Λ_J¯

T(j,k)−q_(j,k)⁽ⁿ⁾ u

2^jJ¯

>0NL=n

≤α

. (3.9)

Whens∈S0, Ps

sup

λ∈Λ_J¯

Tλ−q_λ^(NL) u^(N_λ,α^L)

>0

=Ps

sup

(j,k)∈Λ_J¯

T(j,k)−q_(j,k)^(NL) u^(N_α ^L)

2^jJ¯

>0

=

n∈N

Ps

sup

(j,k)∈Λ_J¯

T_(j,k)−q_(j,k)⁽ⁿ⁾ u⁽ⁿ⁾α

2^jJ¯

>0N_L=n

P(N_L=n)

≤α,

which means thatΦ_α⁽²⁾defines a levelαtest.

Note thatu⁽ⁿ⁾_α satisfiesu⁽ⁿ⁾_α ≥αand that it can be estimated in practice by Monte Carlo experiments.

Comments:

(1) Though the two testing procedures defined by (3.4) and (3.7) are very different in their spirit, they can formally be written in a common way. For any subsetΛofΛ_∞, we denote byS_Λthe subspace generated by{φ₀, φ_λ, λ∈ Λ}, bys_Λ the orthogonal projection ofsontoS_Λ, and we introduce the unbiased estimatorT_Λ=

λ∈ΛT_λ of s_Λ²−α₀²=

λ∈Λα_λ². Then our test functions can be written as

Φ_α=1_Tα>0, (3.10)

where Tα=sup

Λ∈C

T_Λ−t_Λ,α ^(NL) ,

andCis a finite collection of subsets ofΛ_∞. Noticing thatT_J=T_Λ

J, we can easily see that our first test amounts in taking a collectionCequal to{Λ_J, J∈ J}, andt_Λ

J,α

(N_L)=q_J^(NL)(u_J,α^(NL)). Furthermore, our second test amounts in taking a collectionCcomposed of all subsets ofΛ_J¯, and forΛ⊂Λ_J¯,t_Λ,α ^(NL)=

λ∈Λq_λ^(NL)(u^(N_λ,α^L)). Indeed, there exists a subsetΛofΛ_J¯such that

λ∈ΛTλ>

λ∈Λq_λ^(NL)(u^(N_λ,α^L))if and only if there existsλinΛ_J¯such thatTλ> q_λ^(NL)(u^(N_λ,α^L)).

Such a common expression will be particularly useful to derive the properties of the tests.

It also allows us to see our tests as multiple testing procedures. Indeed, we can consider that for eachΛinC, we construct a test rejecting the null hypothesis whenT_Λ−t_Λ,α^(NL)>0. We thus obtain a collection of tests and we finally decide to reject the null hypothesis when it is rejected for at least one of the tests of the collection.

(2) Both procedures have a specific interest to prove the optimality of lower bounds obtained in Theorem1. We will actually prove in the next section that the first one achieves the lower bounds obtained in case (i) of Theorem1(up to a possible logarithmic factor) whereas the second one achieves lower bounds obtained in case (ii) of Theorem1.

However, if we want a procedure that achieves lower bounds of cases (i) and (ii) simultaneously, we will have to consider the test which consists in mixing the two procedures. In this case, we reject the null hypothesis(H0) when sup{Φ_α/2⁽¹⁾, Φ_α/2⁽²⁾} =1.

4. Uniform separation rates

In this section, we evaluate the performances of our new testing procedures from a theoretical point of view. More precisely, we prove that our procedures are optimal in the sense that their uniform separation rates over Besov bodies are of the same order as the lower bounds forρ obtained in Section2. These results justify the construction of our procedures as well as they provide the upper bounds needed for the exact evaluation of the minimax separation rates over weak and classical Besov bodies in the Poisson framework.

In the following, the expressionC(α, β, R, R, R, δ, γ , . . .)orCk(α, β, R, R, R, δ, γ , . . .)is used to denote some constant which only depends on the parametersα, β, R, R, R, δ, γ , . . . ,and which may vary from line to line.

4.1. Uniform separation rates of the first procedure 4.1.1. The error of second kind

The aim of the following theorem is to give a condition on the alternative so that our first levelαtest has a prescribed error of second kind.

Theorem 2. Assume thats∈L^∞([0,1]),and thatL≥1.Fix some levelsαandβ in]0,1[,and letΦ_α⁽¹⁾ be the test function defined by(3.4).There exist some positive constantsC1(α, β,||s||_∞),C2(α, β),C3,C4(β)andC5such that whenssatisfies

d²(s,S0) > inf

J∈J

||s−s_J||²+C₁

α, β,||s||∞√ D_J

L +C₂(α, β)D_J L² +

C₃

[0,1]s(x)dx+C₄(β) √

D_JW_J L +W_J

L

+C₅D_JW_J² L²

(4.1) then

Ps

Φ_α⁽¹⁾=0

≤β.

Comment:Considering here a multiple testing procedure instead of a simple one allows to obtain in the right-hand side of the inequality(4.1)an infimum over allJ inJ at the only price of introducing some terms withW_J. These last terms will appear in the following uniform separation rates over classical Besov bodies as a ln lnLfactor, which is now known to be the price to pay for adaptivity in some classical statistical models. As a consequence, our multiple testing procedure will be proved to be adaptive over classical Besov bodies in Proposition1, which would not occur with a simple testing procedure.

4.1.2. Uniform separation rates over Besov bodies

In this section, we evaluate the uniform separation ratesρ(Φα⁽¹⁾,B^δ_2,∞(R)∩L^∞(R), β)whereρis defined by (1.1), andB_2,^δ_∞(R)is any Besov body defined by(2.2).

Let us first notice that the functions ofB^δ_2,∞(R)are well approximated by their projections onto subspaces of the collection{S_J, J∈J}considered in our first procedure, in the sense that ifs∈B_2,^δ_∞(R), then

s−sJ²≤c(δ)R²D⁻_J^2δ.

As a consequence we can use Theorem2to obtain upper bounds for the uniform separation rates of our test.

We denote byxthe integer part ofx.

Proposition 1. Assume thatln lnL≥1. Given some levelsαand β in]0,1[,letΦ_α⁽¹⁾ be the test function defined by(3.4)withJ = {1, . . . ,log₂(L²/(ln lnL)³)}andW_J =ln|J|for everyJ inJ.

For everyδ >0,R >0andR>0,there exists some positive constantC(α, β, R, δ)such that whensbelongs to B_2,^δ_∞(R)∩L^∞(R)and satisfies

d²(s,S0) > C

α, β, R, δ

R^2/(4δ+1) √

ln lnL L

4δ/(4δ+1)

+R²

(ln lnL)³ L²

2δ

+ln lnL L

, then

Ps

Φ_α⁽¹⁾=0

≤β.

In particular,there exist some positive constantsL₀(δ)andC(α, β, R, R, δ)such that ifL > L₀(δ),then ρ

Φ_α⁽¹⁾,B^δ_2,∞(R)∩L^∞ R

, β

≤C

α, β, R, R, δ√ ln lnL

L

2δ/(4δ+1)

. Comments:

(1) Our first testing procedure is therefore adaptive: indeed, for largeL, it achieves the lower bounds for the minimax separation rates over all the spacesB_2,^δ_∞(R)∩W_γ(R)∩L^∞(R)withδ≥max(γ /2, γ /(1+2γ ))simultaneously up to a possible ln lnLfactor (see Theorem1).

However it does not achieve the optimal separation rates obtained in the case whereδ < γ /2 andγ >1/2. In this range of parameters, the regularity inγis higher than the regularity inδ, meaning that the weak Besov body governs the separation rate. That is the reason why we introduced the thresholding type procedure.

(2) The upper bound for the uniform separation rate obtained here is exactly of the same order as the (asymptotic) adaptive minimax rate of testing obtained by Ingster [17] in the density model, replacing the numbernof obser- vations in the density model by the parameterLof the Poisson model. In particular the ln lnn, replaced here by a ln lnLfactor, is proved to be necessary in the density model for adaptive procedures.

(3) It is easy to see that B^δp,∞(R)⊂B^δ_2,∞(R)when p >2. So the result of Proposition 1 directly leads to upper bounds for the uniform separation ratesρ(Φ_α⁽¹⁾,B^δ_p,∞(R)∩L^∞(R), β)whenp >2. These rates, obtained in the Poisson framework, correspond to the ones in some Gaussian models (see [25], for instance) or in the density model (see [17]).

(4) Note that one could also consider some tests based on the Fourier basis as well as the Haar basis, as Fromont and Laurent [13] did in the density model. The theoretical results would remain unchanged, and the practical performances of the procedure would be better when considering smooth alternatives (see [13] for more details and Section5). We have only considered here tests based on the Haar basis for the sake of simplicity.

4.2. Uniform separation rates of the second procedure 4.2.1. The error of second kind

From the common expression (3.10) of the test function for the two procedures, we obtain here a result similar to Theorem2for the error of second kind of our second test.

Theorem 3. Assume thats∈L^∞([0,1]),and thatL≥1.Fix some levelsαandβ in]0,1[,and letΦ_α⁽²⁾ be the test function defined by(3.7).Recall that for any subsetΛofΛ_∞,SΛandsΛrespectively denote the subspace generated by{φ0, φ_λ, λ∈Λ}and the orthogonal projection ofsontoS_Λ.Denoting byD_Λthe dimension ofS_Λ,there exist some positive constantsC₁(α, β,||s||_∞),C₂(β),C₃(α),C₄(α, β)andC₅(α)such that whenssatisfies

d²(s,S0) > inf

Λ⊂Λ_J¯

||s−s_Λ||²+C₁

α, β,||s||∞√ D_Λ L +2^{J /2¯}

L^3/2

+C₂(β)2^J¯ L² +

C₃(α)

[0,1]s(x)dx+C₄(α, β)

D_Λln(2^J¯J )¯

L +C₅(α)D_Λ2^J¯ln²(2^J¯J )¯ L²

(4.2) then

Ps

Φ_α⁽²⁾=0

≤β.

4.2.2. Uniform separation rates over Besov bodies

Proposition 2. Assume thatlnL≥1.Given some levelsαandβ in]0,1[,letΦ_α⁽²⁾ be the test defined by(3.7)with J¯= log2(L/lnL).

For everyδ >0andγ >0,R >0,R>0andR>0,there exists some positive constantC(α, β, R, δ, γ )such that ifsbelongs toB_2,^δ_∞(R)∩W_γ(R)∩L^∞(R)and satisfies

d²(s,S0) > C

α, β, R, δ, γlnL L +R²

lnL L

2δ

+R² lnL

L

2γ /(1+2γ )

+R^2+4γ lnL

L 2γ

, then

Ps

Φ_α⁽²⁾=0

≤β.

In particular,whenδ≥γ /(1+2γ ),there exist some positive constantsL0(δ, γ )andC(α, β, R, R, R, δ, γ )such that ifL > L0(δ, γ ),then

ρ

Φ_α⁽²⁾,B^δ2,∞(R)∩W_γ R

∩L^∞ R

, β

≤C

α, β, R, R, R, δ, γlnL L

γ /(1+2γ )

. Comments:

(1) Our second testing procedure is still adaptive in the sense that for large L, it achieves the lower bounds for the minimax separation rates over all the spaces B_2,^δ_∞(R)∩Wγ(R)∩L^∞(R) withγ /(1+2γ )≤δ < γ /2 simultaneously (see Theorem1). In this case, we also remark that these rates are so large that there is no further price to pay for adaptivity in the sense that the upper bound does not involve any extra logarithmic factor. To our knowledge, this phenomenon is completely new for nonparametric testing procedures.

(2) Our second procedure achieves the lower bounds for the minimax separation rates over all the spacesB_2,^δ_∞(R)∩ W_γ(R)∩L^∞(R)withγ /(1+2γ )≤δ < γ /2 simultaneously, but it does not whenδ≥max(γ /2, γ /(1+2γ )).

To obtain a test that achieves the minimax separation rates in both cases, our two procedures need to be combined.

4.3. Uniform separation rates of the combined procedure

Corollary 1. Assume thatln lnL≥1.Fix some levelsαandβin]0,1[.LetΦ_α/2⁽¹⁾ be the levelα/2test defined by(3.4) withJ = {1, . . . ,log₂(L²/(ln lnL)³)}andWJ =ln|J| for everyJ inJ.LetΦ_α/2⁽²⁾ be the levelα/2test defined by(3.7)withJ¯= log₂(L/lnL).We considerΦα⁽³⁾=sup{Φ_α/2⁽¹⁾, Φ_α/2⁽²⁾}.

(i) For all δ >0 and γ >0, R >0, R >0 and R >0, there exist some positive constants L₀(δ) and C(α, β, R, R, δ)such that ifL > L₀(δ),then

ρ

Φ_α⁽³⁾,B^δ_2,∞(R)∩W_γ R

∩L^∞ R

, β

≤C

α, β, R, R, δ√ ln lnL

L

2δ/(4δ+1)

.

(ii) For all(δ, γ )such thatδ≥γ /(2γ+1),R >0,R>0andR>0,there exist some positive constantsL0(δ, γ ) andC(α, β, R, R, R, δ, γ )such that ifL > L0(δ, γ ),then

ρ

Φ_α⁽³⁾,B^δ2,∞(R)∩Wγ

R

∩L^∞ R

, β

≤C

α, β, R, R, R, δ, γlnL L

γ /(1+2γ )

. Comments:Since

Ps

Φ_α⁽³⁾=0

≤inf Ps

Φ_α/2⁽¹⁾ =0 ,Ps

Φ_α/2⁽²⁾ =0 , the proof of this result directly comes from Propositions1and2.

The upper bounds for the uniform separation rates of the above combined procedure over B_2,^δ_∞(R)∩W_γ(R)∩ L^∞(R)obtained here actually coincide with the lower bounds of Theorem1for the whole set of parameters(δ, γ ) such thatδ≥γ /(2γ+1)(up to a ln lnLfactor whenδ≥γ /2). This proves that our combined procedure is conse- quently adaptive over the spaces B^δ_2,∞(R)∩W_γ(R)∩L^∞(R), and also that the lower bounds of Theorem1are sharp for this set of parameters.

We however do not obtain such a result for the set of parameters(δ, γ )such thatδ < γ /(2γ+1). Hence we do not know the exact order of the uniform separation rate of our combined procedure overB_2,^δ_∞(R)∩Wγ(R)∩L^∞(R) whenδ < γ /(2γ+1), and we cannot say here whether our lower bounds for these spaces are optimal or not.

5. Simulation study

We aim in this section at studying the performances of our tests from a practical point of view. We consider several intensitiessdefined on[0,1]such that1

0s(x)dx=1.N denotes here a Poisson process with intensityLs on[0,1] with respect to the Lebesgue measure, andPs the distribution of this process. We denote bys₀the intensity which is constant (equal to 1) on[0,1]. We chooseL=100 and a level of testα=0.05.

Let us now recall that our first procedure may be based on the test statistics Tα⁽¹⁾=sup

J∈J

T_J −q_J^(NL)

u_α^(NL)/|J| ,

whereq_J⁽ⁿ⁾(u)denotes the(1−u)quantile ofT_J|NL=nunder the hypothesis thats=s0, anduα⁽ⁿ⁾is chosen such that

u_α⁽ⁿ⁾=sup

u∈ ]0,1[,Ps0

sup

J∈J

T_J−q_J⁽ⁿ⁾ u/|J|

>0N_L=n ≤α

. The null hypothesis(H0)“s=s₀” is rejected whenTα⁽¹⁾>0.

We chooseJ = {1, . . . ,6}. For 40≤n≤160, we estimate the quantitiesu_α⁽ⁿ⁾and the quantilesq_J⁽ⁿ⁾(u_α⁽ⁿ⁾/|J|) for allJinJ. These estimations are based on the simulation of 200 000 independent samples with sizen, uniformly distributed on [0, 1]. Half of the samples is used to estimate the quantilesq_J⁽ⁿ⁾(u/|J|)foruvarying on a grid over [0,1], and the other samples are used to estimate the probabilities occurring in the definition ofu_α⁽ⁿ⁾. Finally,u_α⁽ⁿ⁾is estimated by the largest value on the grid such that these estimated probabilities are smaller thanα.

Let us also recall that our second procedure is based on the test statistics T_α⁽²⁾= sup

λ∈Λ_J¯

T_λ−q_λ^(NL) u^(N_λ,α^L)

,