Lucien Birgé

How many bins should be put in a regular histogram

Keywords: classifications. Primary 62G05; secondary 62E25 Regular histogram, density estimation, penalized maximum likelihood, model selection

Given an n-sample from some unknown density f on [0, 1], it is easy to construct an histogram of the data based on some given partition of [0, 1], but not so much is known about an optimal choice of the partition, especially when the set of data is not large, even if one restricts to partitions into intervals of equal length. Existing methods are either rules of thumbs or based on asymptotic considerations and often involve some smoothness properties of f . Our purpose in this paper is to give a fully automatic and simple method to choose the number of bins of the partition from the data. It is based on a nonasymptotic evaluation of the performances of penalized maximum likelihood estimators in some exponential families due to Castellan and heavy simulations which allowed us to optimize the form of the penalty function. These simulations show that the method works quite well for sample sizes as small as 25.

0 AMS 1991 subject

Introduction

Among the numerous problems that have been considered for a long time in Statistics, a quite simple one is: "How many bins should be used to build a regular histogram?" Here, by regular histogram, we mean one which is based on a partition into intervals of equal length. One can of course argue that this question is of little relevance nowadays since the histogram is an old fashioned estimator and that much more sophisticated and better methods are now available, such as variable bandwiths kernels or all kinds of wavelets thresholding. This is definitely true. Nevertheless, histograms are still in wide use and one can hardly see any other density estimator in newspapers. It is by far the simplest density estimator and probably the only one that can be taught to most students that take Statistics at some elementary level. And when you teach regular histograms to students, you unavoidably end up with the same question: "how many bins?". The question was also repeatedly asked to one of the authors by colleagues who were not professional mathematicians but still did use histograms or taught them to students.

Unfortunately, when faced to such a question, the professional statistician has no really definitive answer or rather he has too many ones in view of the number of methods which have been suggested in the literature, none of them being completely convincing in the sense that it has been shown to be better than the others.

The purpose of this paper is to propose a new simple and fully automatic method to choose the number of bins to be used for building a regular histogram from data. The procedure is not based on any smoothness assumptions and works quite well for all kinds of densities, even discontinuous, and sample sizes as small as 25.

The limitation to densities with support on [0, 1] may seem restrictive, but real data can always be considered as generated from a compactly supported density. If the support is known, one can assume, by a suitable transformation of the data, that it is [0, 1]. If it is unknown, one can merely replace it by the range of the data since we cannot get any information of what happens outside this range without extra assumptions.

There have been many attempts in the past to solve the problem of choosing an optimal number of bins from the data and we shall recall a number of them in Section 4.1 below. Let us just mention here that, apart from some rules of thumbs like Sturges' rule (take approximately 1 + log 2 n bins) or recommendations of the type: "one should have at least k observations in each cell" (k depending on the author), all the methods we know about are based on some asymptotic considerations. Rules of thumbs are very simple and do not aim at any optimality property. More sophisticated rules are based on the minimization of some asymptotic estimate of the risk. This is the case of methods like cross-validation or those based on the evaluation of the asymptotically optimal binwidth under smoothness assumptions for the underlying density. Methods connected with penalized maximum likelihood estimation, like Akaike's criterion or rules based on stochastic complexity or minimum description length are also derived from asymptotic considerations. It follows that the main drawback of all these rules is their asymptotic nature which does not warrant good performance for small sample sizes. Moreover, many of them are based on prior smoothness assumptions about the underlying density.

Our estimator is merely a generalization of Akaike's. This choice was motivated by some considerations about the nonasymptotic performances of penalized maximum likelihood estimators derived by Barron, Birgé and Massart (1999). For the specific case of histogram estimators, their results have been substantially improved by [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] and our study is based on her theoretical work. Roughly speaking, she has shown that a suitably penalized maximum likelihood estimator provides a data-driven method for selecting the number of bins which results in an optimal value of the Hellinger risk, up to some universal constant κ. The proof does not require any smoothness assumption and allows to consider discontinuous densities. Unfortunately, although Castellan's study indicates which penalty structure is suitable to get such a risk bound, theoretical studies are not powerful enough to derive a precise penalty function that would minimize the value of κ for small or moderate sample sizes.

In order to solve this problem, we performed an extensive simulation study including a large variety of densities and sample sizes in order to determine by an optimization procedure a precise form of the penalty function leading to a small value of κ. The resulting estimator is as follows. Assume that we have at disposal an n-sample X 1 , . . . , X n from some unknown density f (with respect to Lebesgue measure) on [0, 1] and we want to design an histogram estimator fD based on some partition {I 1 , . . . , I D } of [0, 1] into D intervals of equal length. We then choose for D the value D(X 1 , . . . , X n ) which maximizes L n (D)pen(D) for 1 ≤ D ≤ n/ log n, where

L n (D) = D j=1 N j log(DN j /n) with N j = n i=1 1l I j (X i ), (1.1)
is the log-likelihood of the histogram with D bins and the penalty pen(D) is given by pen

(D) = D -1 + (log D) 2.5 for D ≥ 1. (1.2)
The resulting estimator f D will be denoted f1 from now on. A few simulation results describing graphically the performances of f1 are given in Figure 1 .

In order to test this new procedure, we conducted another simulation study involving a large family of densities, sample sizes ranging from 25 to 1000 and different loss functions, in order to compare f1 with a number of existing methods. The conclusion of this large scale empirical study, which is given in Section 4, is that our method, on the whole, outperforms all the others, although one of them, namely the one based on Rissanen's minimum complexity ideas [START_REF] Rissanen | Stochastic complexity and the MDL principle[END_REF] and introduced, in the context of histogram estimation, by [START_REF] Hall | On stochastic complexity and nonparametric density estimation[END_REF], is almost as good, in many cases. This is not so surprising since Rissanen's method and our approach are based on similar theoretic arguments.

The next section recalls the theoretical grounds on which our method is based while Section 3 describes the details of our simulation study. The results of the comparison with previous methods are given in Section 4. The Appendix contains some additional technical details.

2 Some theoretical grounds

Histograms and oracles

Let us first describe more precisely what is the mathematical problem to be solved. Let X 1 , . . . , X n be an n-sample from some unknown distribution with density f with respect to Lebesgue measure on [0, 1]. The histogram estimator of f based on the regular partition with D pieces, i.e. the partition

I D of [0, 1] consisting of D intervals I 1 , . . . , I D of equal length 1/D is given by fD = fD (X 1 , . . . , X n ) = D n D j=1 N j 1l I j , with N j = n i=1 1l I j (X i ). (2.1)
It is probably the oldest and simplest nonparametric density estimator. It is called the regular histogram with D pieces and it is the maximum likelihood estimator with respect to the set of piecewise constant densities on I D . In order to measure the quality of such an estimator, we choose some loss function ℓ and compute its risk

R n (f, fD , ℓ) = E f ℓ f, fD (X 1 , . . . , X n ) . (2.2)
From this decision theoretic point of view, the optimal value

D opt = D opt (f, n) of D is given by R n (f, fD opt , ℓ) = inf D≥1 R n (f, fD , ℓ).
Unfortunately, no genuine statistical procedure can tell us what is the exact value of D opt (f, n) because it depends on the unknown density f to be estimated. This is why the procedure fD opt is called an oracle and R n (f, fD opt , ℓ) is the risk of the oracle. Obviously, an oracle is of no practical use but its risk can serve as a benchmark to evaluate the performance of any genuine data driven selection procedure D(X 1 , . . . , X n ). The quality of such a procedure at f can be measured by the value of the ratio

R n (f, f D, ℓ) R n (f, fD opt , ℓ) = R n (f, f D, ℓ) inf D≥1 R n (f, fD , ℓ) , (2.3)
where f D denotes the histogram estimator based on the regular partition with D pieces. Ideally, one would like this ratio to be bounded uniformly with respect to f by some constant C n tending to one when n goes to infinity and that C n -1 stay reasonably small even for moderate values of n. This is unfortunately impossible since, when f = 1l [0,1] , D opt = 1, f1 = f and the ratio (2.3) is infinite. Even if f is different from 1l [0,1] but is very close to it, R n (f, f1 , ℓ) may be arbitrarily close to zero and there is not hope to get a small value for (2.3). A more precise discussion of this problem in the context of Gaussian frameworks can be found in Section 2.3.3 of Birgé and Massart (2001). This implies that, in order to judge the quality of a procedure D, we should only consider the ratio (2.3) for densities which are far enoughsee the precise condition (3.1) below -from the uniform.

Loss functions

Clearly, the value of D opt and the performances of a given selection procedure D depend on the choice of the loss function ℓ. Popular loss functions include powers of L p -norms, for 1 ≤ p < +∞ or the L ∞ -norm, i.e.

ℓ(f, g) = f -g p p or ℓ(f, g) = f -g ∞ ,
(with a special attention given to the cases p = 1 and 2), the squared Hellinger distance

h 2 (f, g) = 1 2 1 0 f (y) -g(y)
2 dy, (2.4) and the Kullback-Leibler divergence (which is not a distance and possibly infinite) given by

K(f, g) = 1 0 log f (y) g(y) f (y) dy ≤ +∞. (2.5)
This last loss function is definitely not suitable to judge the quality of classical histograms since, as soon as D ≥ 2, there is a positive probability that one of the intervals I j be empty, implying that K(f, fD ) = +∞. A similar problem occurs with K( fD , f ) when f is not bounded away from 0.

Since we want to be able to deal with discontinuous densities f , the L ∞ -norm is also inappropriate as a loss function since discontinuous functions cannot be properly approximated by piecewise constant functions on fixed partitions in L ∞ -norm. By some continuity argument, large values of p should also be avoided and we shall restrict ourselves hereafter to Hellinger distance and L p -norms for moderate values of p.

The most popular loss function in our context is probably the squared L 2 -loss for the reason that it is more tractable. Indeed,

E f f -fD 2 = E f fD -fD 2 + f -fD 2 , (2.6)
where fD denotes the orthogonal projection (in the L 2 sense) of f onto the D-dimensional linear space generated by the set of functions {1l I j } 1≤j≤D . In this case, the risk is split into a stochastic term and a bias term which may be analyzed separately. This accounts for the fact that optimizing the squared L 2 -risk of histogram estimators has been a concern of many authors, in particular [START_REF] Scott | Averaged shifted histograms: Effective nonparametric density estimators in several dimensions[END_REF], [START_REF] Freedman | On the histogram as a density estimator: L 2 theory[END_REF], [START_REF] Daly | The construction of optimal histograms[END_REF], [START_REF] Wand | Data-based choice of histogram bin width[END_REF] and Birgé and Massart (1997).

Since the distribution of any selection procedure D(X 1 , . . . , X n ) only depends on the underlying distribution of the observations, it seems natural to evaluate its performances by a loss function which does not depend on the choice of the dominating measure. This is why some authors favour the systematic use of L 1 -loss for its nice invariance properties as explained by Devroye and Györfi (1985, p. 2) and the preface of [START_REF] Devroye | A Course in Density Estimation[END_REF].

Although it is less popular, may be because of its more complicated expression, we shall use here the squared Hellinger distance as our reference loss function to determine a suitable penalty, this choice being actually based on theoretical grounds only. First, as the L 1 -distance, it is a distance between probabilities, not only between densities. Then it is known that it is the natural distance to use in connection with maximum likelihood estimation and related procedures, as demonstrated many years ago by Le Cam (see, for instance, [START_REF] Le | Asymptotic Methods in Statistical Decision Theory[END_REF][START_REF] Le | Asymptotics in Statistics: Some Basic Concepts[END_REF]. Finally, the results of [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] that we use here are based on it.

Of course, the choice of a "nice" loss function is, in a large part, a question of personal taste. Hellinger distance has already been used as loss function in the context of regular histogram density estimation by [START_REF] Kanazawa | Hellinger distance and Akaike's information criterion for the histogram[END_REF] but other authors do prefer L p -losses and one should read for instance the arguments of Devroye mentioned above or those of [START_REF] Jones | On two recent papers of Y. Kanazawa[END_REF]. Therefore, although we shall base our choice of the procedure D on the Hellinger loss, we shall also use other loss functions, including squared L 2 , to evaluate its performances and compare it to other methods.

Hellinger risk

Before we consider the problem of choosing an optimal value of D we need an evaluation of the risk of regular histograms fD for a given value of D since the ratio (2.3) involves it. There is actually nothing special with regular histograms from this point of view and a general result in this direction is as follows. Given an histogram estimator fI of the form (2.7) below based on some arbitrary partition I, its risk is given by E f h 2 (f, fI ) . Asymptotic evaluations of this risk are given by [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] but we were unable to find a non-asymptotic bound for it in the literature. It can actually be proved (see our Appendix) that Theorem 1 Let f be some density with respect to some measure µ on X , X 1 , . . . , X n be an n-sample from the corresponding distribution and fI be the histogram estimator based on some partition

I = {I 1 , . . . , I D } of X , i.e. fI (X 1 , . . . , X n ) = 1 n D j=1 N j µ(I j ) 1l I j , with N j = n i=1 1l I j (X i ).
(2.7)

Setting p j = I j f dµ, we get E h 2 (f, fI ) ≤ h 2 (f, fI ) + D -1 2n with fI = D j=1 p j µ(I j ) 1l I j .
(2.8)

Moreover E h 2 (f, fI ) = h 2 (f, fI ) + D -1 8n [1 + o(1)],
(2.9)

when n(inf 1≤j≤D p j ) tends to infinity.

Remark: It should be noticed that fI minimizes both the L 2 -distance between f and the space H I of piecewise constant functions on I and the Kullback-Leibler information number K(f, g) between f and some element g of H I , but not the Hellinger distance h(f, H I ). In the case of a regular partition, fI = fD as in (2.6).

Penalized maximum likelihood estimators

The theoretical properties of penalized maximum likelihood estimators over spaces of piecewise constant densities, on which our work is based, have been studied by [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF]. We recall that a penalized maximum likelihood estimator derived from a penalty function D → pen(D) is the histogram estimator f D where D is a maximizer with respect to D of L n (D) -pen(D) with L n (D) given by (1.1). Roughly speaking, Castellan's results say (not going into details in order to avoid technicalities) that one should use penalties of the form

pen(D) = c 1 (D -1) 1 + c 2 L D 2 with c 1 > 1/2, L D > 0, (2.10)
where c 2 is a suitable positive constant and the numbers

L D satisfy D≥1 exp[-(D -1)L D ] = Σ < +∞. (2.11)
Let us observe that this family of penalties includes the classical Akaike's AIC criterion corresponding to pen(D) = D -1 (choose for instance c 1 = 3/4 and L D = L in a suitable way). Defining D(X 1 , . . . , X n ) as the maximizer of L n (D) -pen(D) for 1 ≤ D ≤ D = Γn/(log n) 2 , Castellan proves, under suitable assumptions (essentially that f is bounded away from 0), that

E f h 2 (f, f D) ≤ κ(c 1 ) inf 1≤D≤ D K(f, fD ) + n -1 pen(D) + n -1 κ ′ , (2.12)
where κ, κ ′ are positive constants, κ depending on c 1 , κ ′ on the parameters involved in the assumptions and in particular being increasing with respect to Σ. This bound and (2.11) suggest to choose some non-increasing sequence (L D ) D≥1 leading to some Σ of moderate size, which we shall assume from now on.

The asymptotic evaluations of Castellan also suggest to choose c 1 = 1 in order to minimize κ(c 1 ), at least when D opt goes to infinity. In this case, the penalty given by (2.10) can be viewed as a modified AIC criterion with an additional correction term which warrants its good behaviour when the number of observations and therefore the number of cells to be considered in the partition, are not large. Both criteria are equivalent when D tends to infinity.

It is also known (see for instance Birgé and Massart, 1998, Lemma 5) that, when log(f / fD ) is small, K(f, fD ) is approximately equal to 4h 2 (f, fD ). Therefore, under suitable assumptions on f , setting c 1 = 1, one can show, by the boundedness of the sequence (L D ) D≥1 , that

E f h 2 (f, f D) ≤ κ 1 inf 1≤D≤ D h 2 (f, fD ) + D -1 n + κ 2 n , (2.13)
where κ 2 is an increasing function of Σ. In view of Theorem 1, one can finally derive from (2.13) that, under suitable restrictions on f and for n large enough,

E f h 2 (f, f D) ≤ κ ′ 1 E f h 2 (f, fD opt ) + κ ′ 2 /n.
(2.14)

3 From theory to practice

Some heuristics

Although the asymptotic considerations suggest to choose c 1 = 1 (and this was actually confirmed by our simulations), the theoretical approach is not powerful enough to indicate precisely how one should choose the sequence (L D ) D≥1 in order to minimize the risk. It simply suggests that Σ should not be large in order to keep the remainder term κ 2 /n of moderate size when n is not very large. In order to derive a form of the penalty that leads to a low value of the risk, one needs to perform an optimization based on simulations and, at this stage, some heuristics will be useful. In particular we shall pretend that the asymptotic formula (2.9) is exact, and use the approximation

E f h 2 (f, fD ) ≈ h 2 (f, fD ) + (D -1)/(8n) which implies that E f h 2 (f, fD ) (8n) -1 for D ≥ 2.
Together with (2.14), this implies that

E f h 2 (f, f D) ≤ κ 3 E f h 2 (f, fD opt ) for D opt > 1. If D opt = 1, this bound still holds provided that 8nh 2 (f, 1l [0,1] ) ≥ 1 (3.1)
since f1 = 1l [0,1] , whatever f , which means that f is not too close to the uniform. This restriction confirms the arguments of Section 2.1.

If c 1 = 1, (2.10) can be written

pen(D) = D -1 + c 2 (D -1) 2 L D + L D .
Moreover, the constant c 2 is only known approximately and (2.11) requires that (D -1)L D tends to infinity with D. Since L D should not be large because it influences the risk as shown by (2.12), it seems natural to look for penalties of the form pen(D) = D -1 + g(D) where the function g tends to infinity but not too fast. We actually restricted our search to functions g(x) of the three following types:

αx β , 0 < β < 1; αx(1 + log x) -β , β > 0 and α(log x) β , β > 1, (3.2)
with α > 0, varying the values of both parameters. We also replaced the restriction 1 ≤ D ≤ D with D = Γn/(log n) 2 with some constant Γ > 0 of Castellan's Theorem by the simpler condition D = n/ log n which did not lead to any trouble in practise.

The operational procedure

We proceeded in two steps. The first one was an optimization step to choose a convenient form for the function g just mentioned; the second one was a comparison step to compare our new procedure with more classical ones. In both cases, we had to choose some specific densities to serve as references, i.e. for which we should evaluate the performances of the different estimators. The chosen densities are of piecewise polynomial form. To define them we used the trivial partition with a single element which leads to continuous densities, and some regular or irregular partitions with several elements. Both the partitions and the coefficients of the polynomials given by their linear expansion within the Legendre basis were drawn using a random device which ensured positivity. We also added to the resulting family some special piecewise constant densities which were known to be difficult to estimate and ended up with a set of 45 different densities, ranging from smooth to rather erratic, which are described in Figure 6 in the Appendix.

Many of these densities are far from smooth and this choice was made deliberately. Histograms are all purpose rough estimates which should cope with all kinds of densities. Testing their performances only with smooth densities like the normal or beta is not sufficient.

We then selected a large range of values for n, namely n = 25, 50, 100, 250, 500 and 1000 and this resulted in a set F of 264 pairs (f, n) after we excluded, in view of the previous arguments, six pairs which did not satisfy the requirement (3.1).

In both steps, we had to evaluate risks R n (f, f , ℓ) for various procedures f . There are typically no closed form formulas for such theoretical risks and we had to replace them by empirical risks based on simulations. We systematically used the same method: given the pair (f, n) we generated on the computer 1000 pseudo-random samples X j 1 , . . . , X j n , 1 ≤ j ≤ 1000 of size n and density f . We then performed all our computations replacing the theoretical distributions of losses of the procedures f at hand: P f [ℓ(f, f (X 1 , . . . , X n )) ≤ t] by their empirical counterparts

P n ℓ f, f (X 1 , . . . , X n ) ≤ t = 1 1000 1000 j=1 1l [0,t] ℓ f, f (X j 1 , . . . , X j n ) .
In particular we approximated the true risk R n (f, f , ℓ) by its empirical version

R n (f, f , ℓ) = 1 1000 1000 j=1 ℓ f, f (X j 1 , . . . , X j n )
and the upper 95% quantile of the distribution of ℓ(f, f (X 1 , . . . , X n )) by the corresponding upper 95% quantile Q (0.95) (n, f, f , ℓ) of the empirical distribution of the variables ℓ(f, f (X j 1 , . . . , X j n )). Note here that such computations required the evaluations of quantities of the form ℓ(f, f ), namely h 2 (f, f ) or f -f p p . Since both f (piecewise polynomial) and f (piecewise constant) were piecewise continuous, we could compute the losses by numerical integration separately on each of the intervals where both functions were continuous. The precise details of the procedures and the corresponding MATLAB functions can be found on the WEB site http://www.proba.jussieu.fr/~rozen/histograms.

The optimization

In this step, we wanted to compare the performances of the various penalized maximum likelihood estimators with penalties of the form pen(D) = D -1 + g(D) according to the possible values of g over the testing class F. As we previously mentioned, the performance of a selection procedure D(g) based on the penalty involving some function g can be evaluated by a comparison of its risk with the optimal risk corresponding to D = D opt .

Ideally, one would like to minimize the ratio

M n (f, f D(g) , h 2 ) = R n (f, f D(g) , h 2 ) inf D≥1 R n (f, fD , h 2 ) , (3.3)
with respect to g for all pairs (f, n) ∈ F ′ . Of course the optimal strategy depends on the pair and we looked for some uniform bound for M n but it appeared that, roughly speaking, M n behaves as a decreasing function of D opt (f, n) and we actually tried to minimize approximately with respect to g sup

{(f,n) | D opt (f,n)=k} M n (f, f D(g) , h 2 ),
for all values of k simultaneously. Again, this is not a well-defined problem and some compromises were needed, but we finally concluded that the choice g(x) = (log x) 2.5 was the most satisfactory and ended up with the estimator f1 described in the introduction.

Some performances of our estimator

In order to evaluate the performances of f1 , we compared its risk with the oracle for all pairs (f, n) ∈ F and various loss functions. We actually considered, for all our comparisons, four typical loss functions, which are powers of either the Hellinger or some L p -distances. More precisely,

ℓ 0 (f, f ′ ) = h p 0 (f, f ′ ) and ℓ i (f, f ′ ) = f -f ′ p i p i for i = 1, 2, 3, with p 0 = p 2 = 2, p 1 = 1 and p 3 = 5.
In order to facilitate comparisons between the different loss functions and to balance the effect of the differences in the powers p i , we expressed our results in terms of a normalized version M ⋆ n of M n , setting, for any density f and estimator f ,

M ⋆ n (f, f , ℓ i ) = M n (f, f , ℓ i ) 1/p i .
The results of the comparisons are summarized in Table 1 below. For each n, we denote by F n the set of densities f such that (f, n) ∈ F. For n ≥ 100, F n contains all 45 densities we started with, but some (at most three when n = 25), which are too close to the uniform, had to be excluded for smaller values of n since they do not satisfy (3.1). Table 1 gives the values of

sup f ∈Fn M ⋆ n (f, f1 , ℓ i )
for the different values of n and i. We see that all values of M ⋆ n (f, f1 , h 2 ) are smaller than 1.5 and not much larger for the L 1 and L 2 losses, although our procedure was not optimized for those losses. Not surprisingly, the results for L 5 are worse for small values of n but improve substantially for n ≥ 500.

f ∈Fn M ⋆ n (f, f1 , ℓ i )
Apart from these worst-case results, it is also interesting to notice that some values of M ⋆ n (f, f1 , ℓ i ), when n is large and f is a "nice" density, are actually smaller than one, which means that, under favorable circumstances, our estimator can "beat" the oracle. This is not so surprising since the oracle has a fixed number D opt (f, n) of bins (the one which minimizes the risk, i.e. the average loss) independently of the sample, while our estimator tries to optimize the number of bins for each sample and can therefore adjust to the peculiarities of the sample. This means that it is the very notion of an oracle which is questionable as a reference.

Comparison with previous methods

Some historical remarks

The first methods used to decide about the number of bins were just rules of thumbs and date back to [START_REF] Sturges | The choice of a class interval[END_REF]. According to [START_REF] Wand | Data-based choice of histogram bin width[END_REF] such methods are still in use in many commercial softwares although they do not have any type of optimality property. Methods based on theoretical grounds appeared more recently and they can be roughly divided into three classes.

If the density to be estimated is smooth enough (has a continuous derivative, say), it is often possible, for a given loss function, to evaluate the optimal asymptotic value of the binwidth, the one which minimizes the risk, asymptotically. Such evaluations have been made by [START_REF] Scott | Averaged shifted histograms: Effective nonparametric density estimators in several dimensions[END_REF] and [START_REF] Freedman | On the histogram as a density estimator: L 2 theory[END_REF] for the squared L 2 -loss, by Devroye and Györfi (1985) for the L 1 -loss and by [START_REF] Kanazawa | Hellinger distance and Akaike's information criterion for the histogram[END_REF] for the squared Hellinger distance. Unfortunately, the optimal binwidth is asymptotically of the form cn -1/3 where c is a functional of the unknown density to be estimated and its derivative. Since an estimation of c involves complicated computations, most authors suggest a rule of thumbs to evaluate it, typically: pretend that the true density is normal. [START_REF] Wand | Data-based choice of histogram bin width[END_REF] proposes to estimate c by kernel methods.

Methods based on cross-validation have the advantage to avoid the estimation of an asymptotic functional and directly provide a binwidth from the data. An application to histograms and kernel estimators is given in [START_REF] Rudemo | Empirical choice of histograms and kernel density estimators[END_REF]. Theoretical comparisons between Kullback cross-validation and the AIC criterion are to be found in [START_REF] Hall | Akaike's information criterion and Kullback-Leibler loss for histogram density estimation[END_REF].

The third class of methods includes specific implementations for the case of regular histograms of general criteria used for choosing the number of parameters to put in a statistical model. The oldest method is the minimization of Akaike's AIC criterion (see [START_REF] Akaike | A new look at the statistical model identification[END_REF]). Akaike's method is merely a penalized maximum likelihood method with penalty pen(D) = D -1 in our case. In view of (1.2), our criterion is just a generalization of AIC criterion tuned for better performance with small samples. [START_REF] Taylor | Akaike's information criterion and the histogram[END_REF] derived the corresponding asymptotic optimal binwidth (under smoothness assumptions on the underlying density) which turns out to be the same as the asymptotically optimal binwidth for squared Hellinger risk, as derived by [START_REF] Kanazawa | Hellinger distance and Akaike's information criterion for the histogram[END_REF]. Related methods are those based on minimum description length and stochastic complexity due to Rissanen (see for instance [START_REF] Rissanen | Stochastic complexity and the MDL principle[END_REF]. Their specific implementation for histograms has been discussed in [START_REF] Hall | On stochastic complexity and nonparametric density estimation[END_REF].

Somewhat more exotic methods have been proposed by [START_REF] Daly | The construction of optimal histograms[END_REF] and [START_REF] He | Selecting the number of bins in a histogram: A decision theoretic approach[END_REF], the second one being based on Bayesian bootstrap.

The comparison study

In order to evaluate the performances of our method and to compare it to previous ones, we selected 14 different estimators f2 , . . . , f15 . The selection was based on the previous historical review and the precise description of the estimators is given in the Appendix. Let us just briefly mention that f2 and f3 are respectively L 2 and Kullback-Leibler cross-validation methods, f4 is the minimization of AIC, f5 and f6 are based on stochastic complexity and minimum description length respectively, f7 to f10 are estimators based on asymptotic evaluations of an optimal binwidth according to various criteria, f11 is Sturges'rule, f12 is due to Daly and f13 to He and Meeden. For completeness, we added two estimators f14 and f15 which do not look for the optimal regular partition. f14 is based on wavelet thresholding since such estimators are quite fashionable nowadays and also considered as very powerful. In order to have a fair comparison in terms of bias, we used the Haar wavelet basis in connection with a method from [START_REF] Herrick | Some new methods for wavelet density estimation[END_REF] which appeared to be the best among the different wavelet methods for density estimation we tried. Note that the resulting estimator, although piecewise constant, is not a regular histogram. The estimator f15 is due to Devroye and Lugosi and described in Section 10.3 of [START_REF] Devroye | Combinatorial Methods in Density Estimation[END_REF] but it selects only dyadic partitions.

We actually also studied the performances of modified versions of some of those estimates, as described by [START_REF] Rudemo | Empirical choice of histograms and kernel density estimators[END_REF], [START_REF] Hall | On stochastic complexity and nonparametric density estimation[END_REF], [START_REF] Wand | Data-based choice of histogram bin width[END_REF] and various thresholding strategies for Haar wavelets. Since the performances of the modified methods were similar to or worse than those of the original estimators, we do not include them here.

Analysis of the results

To compare the performances of the various estimators, we computed, for all 264 pairs (f, n) ∈ F, all estimators fk , 1 ≤ k ≤ 15 and the four selected loss functions, the values of M ⋆ n (f, fk , ℓ i ).We also used as a secondary index of performance the normalized ratio

Q ⋆ n (f, f , ℓ i ) = Q (0.95) (n, f, f , ℓ i )) inf D≥1 R n (f, fD , ℓ i ) 1/p i
of the empirical version of the 95% quantile of the distribution of f to the corresponding risk of the oracle. Dividing by the risk of the oracle does not influence the comparisons between the various estimators but substantially reduces the range of Q (0.95) (n, f, f , ℓ i ) when f varies in our test set which drastically improves the legibility of the results.

This simulation study resulted in a large set of data which had to be summarized. Therefore, for each n, ℓ and k we considered the set S(n, ℓ, k) of the

|F n | values of M ⋆ n (f, fk , ℓ) for f ∈ F n .
Our comparison of the estimates is based on the boxplots of the different sets S(n, ℓ, k). Here, the box provides the median and quartiles, the tails give the 10 and 90% quantiles and the additional points give the values which are outside this range. Figure 2 shows the boxplots corresponding to all methods for n = 25, 100 and 1000, squared Hellinger and L 2 losses. It is readily visible from these plots that estimators f6 to f15 are not satisfactory for n ≥ 100 as compared to the others. The complete set of results shows that their performances for other values of n and loss functions are not better. This is why we do not include them in the final figures to improve legibility. For the remaining 5 estimators, we provide hereafter in Figures 3 and4 the complete series of boxplots corresponding to all values of n and ℓ. Clearly, our method is the best from this point of view, especially for the smaller values of n. Apart from ours, the best method seems to be f5 , based on stochastic complexity. If the performances of f5 are roughly equivalent to those of f1 for large values of n (n ≥ 250), this is clearly not true for small n.

A few comments

As we previously noticed, some methods, corresponding to estimators fj with 6 ≤ j ≤ 15 were found to behave rather poorly on our test set, especially for large n. This is actually not surprising for Sturges'rule ( f11 ) which is a rule of thumbs, for f12 since the theoretical arguments supporting Daly's method are not very strong and for f13 since the decision theoretic arguments used by He and Meeden involve a very special loss function different from our criteria.

That all the methods ( f7 to f10 ) which define an asymptotically optimal binwidth from a smoothness assumption on the underlying density do not work well for estimating discontinuous densities is natural either. Since f15 only chooses dyadic partitions, this tend to result in an increased bias. Finally the method f14 , based on Haar waveled thresholding suffers from the same problem and moreover considers many more partitions than we do. It should rather be compared with selection procedures involving irregular partitions, since it is known (see for instance Birgé and Massart, 1997) that thresholding methods are equivalent to penalized methods for irregular dyadic partitions which do require heavier penalties.

Actually, all the methods that work reasonably well are either based on cross-validation or some complexity penalization arguments. It was therefore rather surprising for us to notice that the two estimators studied by [START_REF] Hall | On stochastic complexity and nonparametric density estimation[END_REF], which are asymptotically equivalent and have similar performances for moderate sample sizes according to the authors, appear to behave quite differently in our study, the estimator based on stochastic complexity being much better than the one based on minimum description length. This is probably due to the fact that the equivalence is really of an asymptotic nature and that the testing densities in Hall an Hannan are very smooth (normal and beta) while ours are not. Rewriting the three estimators fj with j = 1, 5, 6 as f Dj where Dj is the maximizer of L n (D) -π j (D), we compared the behaviours of π 1 , π 5 and π 6 for different simulated examples. Note that here π 1 (D) = pen(D) as defined by (1.2). The examples show that π 1 and π 5 are rather close while π 6 tends to be much smaller leading to larger values for D6 . An illustration of the phenomenon is shown in Figure 5 1 2 3 4 5 Since the risk R n of fI is given by

R n = E h 2 (f, fI ) = D j=1 E   1 2 I j f (x) - N j nµ(I j ) 2 dµ(x)   ,
(5.1) it suffices to bound each term in the sum. The generic term can be written, omitting the indices, setting l = µ(I), p = I f dµ and denoting by N a binomial B(n, p) random variable, as

R(I) = E   1 2 I f (x) - N nl 2 dµ(x)   = 1 2 I f (x) dµ(x) + E N n -E N n I f (x) l dµ(x) = p -E N n I f (x) l dµ(x).
Introducing f 1l I = l -1 p1l I and h 2 = h 2 (f 1l I , f 1l I ), we notice that

h 2 = 1 2 I f (x) - p l 2 dµ(x) = p - √ p I f (x) l dµ(x),
which implies that

R(I) = p -p -h 2 E N np = h 2 E N np + p 1 -E N np .
The conclusion then follows from the next lemma and (5.1).

Lemma 1 Let N be a binomial random variable with parameters n and p, 0 < p < 1, then

E N np > 1 - 1 -p 2np and E N np = 1 - 1 -p 8np 1 + O 1 np . Proof : Setting Z = N -np, we write E N/(np) = E 1 + Z/(np) . The first inequality follows from the fact that, for u ≥ -1, √ 1 + u ≥ 1 + u/2 -u 2 /2, E[Z] = 0 and Var(Z) = np(1 -p).
To get the asymptotic result, we use the more precise inequality

1 + u 2 - u 2 8 + u 3 16 - 5u 4 16 ≤ √ 1 + u ≤ 1 + u 2 - u 2 8 + u 3 16
together with the moments of order three and four of Z:

E Z 3 = np(1 -p)(1 -2p); E Z 4 = np(1 -p) 1 -6p + 6p 2 + 3np(1 -p) .

Our set of test estimators

Apart from f14 , each of the estimators fk , 1 ≤ k ≤ 15, that we consider in our study (see Section 4.2) is based on a specific selection method, Dk (X 1 , . . . , X n ) which derives the number of bins from the data, resulting in fk = f Dk with fD given by (2.1). For definiteness, we recall more precisely in this section the definitions of the various methods involved, adjusted to our particular situation of a support of length one. In the formulas below N j , as defined in (1.1), denotes the number of observations falling in the j-th bin.

The first 6 methods we considered are based on the maximization with respect to the number D of bins of some specific criterion. We recall that our estimator f1 is based on the minimization of D j=1 N j log N j + n log D -D -1 + (log D) 2.5 .

For L 2 and Kullback cross-validation rules D2 and D3 , the functions to maximize are given respectively (Rudemo, 1982 p. 69 and[START_REF] Hall | Akaike's information criterion and Kullback-Leibler loss for histogram density estimation[END_REF] The estimators f5 and f6 , respectively based on stochastic complexity and minimum description length considerations, involve the maximization [START_REF] Hall | On stochastic complexity and nonparametric density estimation[END_REF] Estimators f7 to f10 are all based on data driven evaluations lk , 7 ≤ k ≤ 10 of the binwidth. Since such evaluations do not lead to an integer number of bins when the support is [0, 1], we took for Dk the integer which was closest to l-1 k . For k = 7, 8, 9, the respective suggestions for lk by [START_REF] Taylor | Akaike's information criterion and the histogram[END_REF] or [START_REF] Kanazawa | Hellinger distance and Akaike's information criterion for the histogram[END_REF], Devroye and Györfi (1985) and [START_REF] Scott | Averaged shifted histograms: Effective nonparametric density estimators in several dimensions[END_REF] are 2.29σ 2/3 n -1/3 ; 2.72σn -1/3 ; and 3.49σn -1/3 , where σ2 denotes some estimator of the variance. We actually used for σ2 the unbiased version of the empirical variance. The previous binwidth estimates are actually based on the assumption that the shape of the underlying density is not far from a normal N (µ, σ 2 ) distribution. To avoid the use of such a rule of thumbs, Wand (1997, p. 62) suggests a more complicated evaluation for l10 and we used the one-stage rule that he denotes by h1 with M = 400 in his formula (4.1). For f11 , we merely used Sturges'rule with D11 the integer closest to 1 + log 2 n. where N j (k) denotes the number of observations falling in the j-th bin of the regular partition with k bins. We implemented for f13 the method given by [START_REF] He | Selecting the number of bins in a histogram: A decision theoretic approach[END_REF], without the restriction they impose that the number of bins should be chosen between 5 and 20 since such a restriction leads to poor results for small sample sizes. It was replaced by the less restrictive D > 1. We also computed f15 according to the method given in Chapter 10 of [START_REF] Devroye | Combinatorial Methods in Density Estimation[END_REF]. More precisely we used the histograms build by data splitting as described in their Section 10.3 with a maximal number of dyadic bins bounded by 2n (in order to avoid an algorithmic explosion) and a value of m set to the integer part of n/2. We actually also experimented smaller values of m but did not notice an improvement.

The last estimator f14 we used for the comparison is not a histogram but a piecewise constant function derived from an expansion within the Haar wavelet basis, the construction following the recommendations of [START_REF] Herrick | Some new methods for wavelet density estimation[END_REF] with the use of the normal approximation with p-value 0.01 and a finest resolution level set to log 2 U where U is the minimum of n 2 (to avoid an algorithmic explosion) and the inverse of the smallest distance between data.

Figure 1

 1 Figure 1: 6 examples, the dashed black line represents the true density and the thick grey line the estimator

Figure 2 :

 2 Figure 2: Hellinger (left) and L 2 (right) mean ratios for n = 25, 100, 1000.

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Hellinger, L 1 , L 2 and L 5 mean ratios for n = 25, 50, 100, 250, 500, 1000

N

  p. 452) by D(n + 1) j log(N j -1) + n log D, while AIC criterion D4[START_REF] Akaike | A new look at the statistical model identification[END_REF] corresponds to the maximization of D j=1 N j log N j + n log D -(D -1).

  -1/2) log(N j -1/2) -(n -D/2) log(n -D/2) + n log D -(D/2) log n.

  [START_REF] Daly | The construction of optimal histograms[END_REF] suggests to take D12 as the minimal value of D such that (

Table 1 :

 1 Maximum normalized mean ratio: sup

	i \ n	25	50 100 250 500 1000
	0	1.40 1.38 1.43 1.30 1.30 1.26
	1	1.48 1.54 1.49 1.34 1.33 1.26
	2	1.84 1.64 1.49 1.48 1.42 1.38
	3	2.94 2.89 2.85 2.55 1.62 1.53


Yves Rozenholc
email: yves.rozenholc@math.jussieu.fr

Nonparametric tests of change-point with tapered data

Keywords: Nonparametric, Tests, Change-Point, Periodogram, Tapered Data, Stationary Process Mathematics subject classifications (1991): 62G10, 62G20, 62M10, 62M15

In this work we build two families of nonparametric tests using tapered data for the off-line detection of change-points in the spectral characteristics of a stationary Gaussian process. This is done using the Kolmogorov-Smirnov's Statistics based on integrated tapered periodograms.

The convergence is obtained under the null hypothesis by means of a double indexed (frequency -time) process together with some extensions of Dirichlet and Fejer kernels. Consistency is proved using these statistics under the alternative.

Then using numerical simulations, we observe that the use of tapered data significantly improves the properties of the test, especially in the case of small samples.

Introduction

The problem of detecting a change-point in the properties of a process has been extensively studied, see for a general survey the books [START_REF] Basseville | Detection of abrupt changes in signals and dynamical systems[END_REF], [START_REF] Basseville | Detection of abrupt changes -Theory and applications[END_REF] and [START_REF] Brodsky | Nonparametric methods in change-point problems[END_REF] or more recently the works of [START_REF] Giraitis | The change-point problem for dependent observations[END_REF] or [START_REF] Lavielle | Optimal segmentation of random processes[END_REF].

The change point problem can be formulated in two different ways: a sequential problem called "online" and another one a posteriory called "off-line". We consider here this last case where one has to decide between homogeneity and change-point after observing a given set of random data.

Different kinds of changes can affect a stochastic process: changes in the mean, in the covariance structure, etc. In the present work, our aim is to detect changes in the spectrum of a strictly stationary time series, while assuming no change in the mean. As the distributions are a priori unknown, we restrict ourselves to a nonparametric framework and we build two families of nonparametric test-statistics for change-point detection using estimates of the spectral measure.

Several authors have investigated such problems of detecting change in the spectral distribution function in an off-line case. We can quote for instance [START_REF] Picard | Testing and estimating change-points in time series[END_REF] for Gaussian processes, [START_REF] Giraitis | A fuctionnal clt for nonparametric estimates of spectrum and the change point problem for a spectral function[END_REF], [START_REF] Giraitis | Testing and estimating in the change point problem of the spectral function[END_REF], [START_REF] Giraitis | The change-point problem for dependent observations[END_REF] for linear processes and [START_REF] Lavielle | Detection of changes in the spectrum of a multidimensionnal process[END_REF] for multidimensional Gaussian processes.

On the other hand, it is well known that, for estimating the spectral measure of a stationary process, the use of the periodogram requires a large number of data. To bypass this problem of sample sizes, Dahlhaus has shown, in a series of papers [START_REF] Dahlhaus | Spectral analysis with tapered data[END_REF] [14] [START_REF] Dahlhaus | Empirical spectral processes and their applications to time series analysis[END_REF] [START_REF] Dahlhaus | Nonparametric high resolution spectral estimation[END_REF] that the use of tapered data improves spectral estimation: The increase in the asymptotic variance is balanced by a reduction of the bias which leads to better results for small samples sizes. This is the old remedy to reduce leakage effects pointed out by [START_REF] Tukey | An introduction to the calculations of numerical spectrum analysis[END_REF] or more recently in the papers of [START_REF] Zhang | Reduction of the asymptotic bias of autoregressive and spectral estimators by tapering[END_REF], [START_REF] Sachs | Peak-intensive spectrum estimation[END_REF], and [START_REF] Janas | Consistency for non-linear functions of the periodogram of tapered data[END_REF].

2 Statement of the results

Notations and assumptions

Let G denote the set of stationary real centered Gaussian processes with absolutely continuous spectral measure with density f w.r.t. the normalized Lebesgue measure on the torus T = [-π, +π[, dλ(α) = dα/2π, which satisfy • (A-i) F (λ) = λ 0 f (α)dλ(α) is an increasing function on [0, π],

• (A-ii) G 2 = π 0 f 2 (α)dλ(α) < +∞. We test the following hypotheses H 0 : "(X j ) j=1...T is the restriction to {1, . . . , T } of a process belonging to G". against H 1 : "there exists k 0 , 1 < k 0 < T such that X j = X 1 j if j ≤ k 0 and X j = X 2 j if j > k 0 where X 1 and X 2 are both in G and have two unknown different spectral measures F 1 and F 2 (F i denotes the spectral measure of X i , i = 1, 2)".

Let h be a non negative C 1 -function defined on [0, 1]: the taper. For k = 0, 1, . . . , T and s ∈ [0, 1], let us define

H k p (α) = k j=1 h p ( j T )e -ijα , H k p = H k p (0), H 0 p = 0, H p (s) = s 0 h p (u)du, H p = H p (1). ( 1
)
Consider now the tapered periodograms I k (α) (respectively ǏT -k (α) built on the first k data (resp. on the last Tk data), as in [START_REF] Dahlhaus | Spectral analysis with tapered data[END_REF]

d k (α) = k j=1 X j h( j T )e -ijα , I k (α) = |d k (α)| 2 H k 2 , ǏT -k (α) = |d T (α) -d k (α)| 2 H T 2 -H k 2 (2)
2 with I 0 (α) = ǏT (α) = 0. The associated estimates of the spectral distribution function are respectively

F k (λ) = λ 0 I k (α)dλ(α), FT -k (λ) = λ 0 ǏT -k (α)dλ(α). (3)
The asymptotic properties of F T have been studied, in the non tapered case (h identically equal to 1), in [START_REF] Ibragimov | On estimation of the spectral function of a stationnary gaussian process[END_REF], [START_REF] Malevitch | The asymptotic behaviour of an estimate for the spectral function of a stationnary gaussian process[END_REF], and in the tapered case by [START_REF] Dahlhaus | Spectral analysis with tapered data[END_REF] [14] [START_REF] Dahlhaus | Nonparametric high resolution spectral estimation[END_REF]. In particular, Dahlhaus has first pointed out the interest of data tapering by numerical simulations results and then, through his theory of "high-resolution", [START_REF] Dahlhaus | Nonparametric high resolution spectral estimation[END_REF] proved it. For a spectral density having high peaks the estimates based on tapered data behave better (especially around the peaks). Indeed, for small T , this is because their bias is reduced. For large T , they are uniformly more efficient with respect to the supremum of the integrated mean square error; this supremum is taken on a class of densities which contain the ARMA processes, whose roots belong to the disk with radius 1 -1/T .

Denote by [a] the integer part of a and define

Y 1 T (λ, s) = √ T H [sT ] 2 H T 2 F [sT ] (λ) -F T (λ) , (4)
Y 2 T (λ, s) = √ T H [sT ] 2 H T 2 H T 2 -H [sT ] 2 H T 2 F [sT ] (λ) -FT -[sT ] (λ) . (5)
The associated tests statistics are

S i T (h) = sup λ∈[0,π] sup s∈[0,1] Y i T (λ, s) , i = 1, 2 (6)
and the critical regions are, using the notations of (A-ii) and ( 1)

R j T = S j T (h) > c H 2 G 2 H 4 , j = 1, 2. ( 7
)
The properties of the statistic S 2 T (h) under the null hypothesis H 0 in the non tapered case (i.e. h ≡ 1) have been studied by [START_REF] Picard | Testing and estimating change-points in time series[END_REF]. In the non tapered case, [START_REF] Giraitis | Testing and estimating in the change point problem of the spectral function[END_REF] have studied tests for linear (non necessarily Gaussian) processes using analogous test-statistics.

Main Results

Our first aim is to obtain the asymptotic distributions of the above statistics in order to get the asymptotic level of our test.

Let us stress upon the fact that the critical regions are not free. The unknown parameter G 2 defined in (Aii) remains in the test. Hence, this value must be known or substituted by a consistent estimate G 2 in order to perform the test. For example G 2 can be replaced by Theorem 1 Assume that (X j ) j∈N belongs to G. Then, as T → +∞, under the null hypothesis H 0 , the two sequences (Y 1 T ) and (Y 2 T ) converge in distribution in D to the centered Gaussian process

G 2 = 1 2 |u|< √ T   1 T T -|u| j=1 X j X j+u   2 . ( 8
) Let D = D([0, π] × [0, 1], R)
1 H 2 Z(λ, s) - H 2 (s) H 2 Z(λ, 1) (λ,s)∈[0,π]×[0,1] (9)
where H 2 and H 2 (.) are defined in [START_REF] Basseville | Detection of abrupt changes in signals and dynamical systems[END_REF] and where Z(., .) is the continuous Gaussian process with mean zero and covariance function

E Z(λ, s)Z(λ ′ , s ′ ) = s∧s ′ 0 h 4 (u)du. λ∧λ ′ 0 f 2 (α)dλ(α). ( 10
)
The proof of Theorem 1 is given in Section 4. Now, we can study the levels of the tests associated with the critical regions R j T defined in [START_REF] Brodsky | Nonparametric methods in change-point problems[END_REF].

Corollary 1.1 Assume that (X j ) j∈N belongs to G. Then, as T → +∞, lim

T →+∞ P H0 (R j T ) = P( sup u∈[0,1] sup s∈[0,1] B(u, s) - H 2 [H -1 4 (s.H 4 )] H 2 B(u, 1) > c)
where H -1 4 (.) denotes the inverse (or pseudo-inverse) function of H 4 (.) and B is the standard bidimensional Brownian motion on R 2 centered with covariance function

E B(u, s)B(u ′ , s ′ ) = (u ∧ u ′ ).(s ∧ s ′ ).
The proof in given in Section 6. Therefore, for T large enough, we obtain approximations for the levels which only depend on the taper h. To obtain this result, we introduce, as in [START_REF] Picard | Testing and estimating change-points in time series[END_REF] [START_REF] Giraitis | Testing and estimating in the change point problem of the spectral function[END_REF], the process Z T with values in D defined by

Z T (λ, s) = H [sT ] 2 √ T (F [sT ] (λ) -F (λ)). (11)
Then, the following holds for Y 1 T and Y 2 T ,

Y 1 T (λ, s) = T H T 2 Z T (λ, s) - H [sT ] 2 H T 2 Z T (λ, 1) (12)
Y 2 T (λ, s) = Y 1 T (λ, s) + T H [sT ] 2 (H T 2 ) 2 Ω T (λ, s) (13)
where

Ω T (λ, s) = 2 √ T Re λ 0   [sT ] j=1 h( j T )X j e -ijα T j=[sT ]+1 h( j T )X j e ijα   dλ(α). (14)
All the results quoted above rely on the next theorem which states, under the null hypothesis H 0 , the weak convergence of the process Z T (., .). Theorem 2 Assume that (X j ) j∈N belongs to G. Then (Z T ) T ∈N converges in distribution to the centered Gaussian process Z of C([0, π] × [0, 1], R) with covariance function given in [START_REF] Csensov | Limit theorems for some classe of random functions[END_REF].

The proof of Theorem 2, is given in Section 4. Some preliminary results are useful for obtaining the limiting covariances together with the tightness of the sequence Z T (., .).

Preliminary results

This section is devoted to studying the kernels that appear in the proof of the functional central limit theorem.

Let us define, for p and j arbitrary integers and for α ∈ [-π, π], the functions

L p j L p j (α) =      j π 2 , if |α| ≤ 2 j , π(p -p ) log p (e p j|α|/2) |α| , if 2 j ≤ |α| ≤ π.
For α ∈ R, denote by L p j the 2π-periodic extension of L p j .

Property 1 The function L p j (α) has the following properties 1. For all p and j, α → L p j (α) is continuous, odd, and, for α in [0, π], non increasing. It is an increasing function w.r.t. j and p.

2. For all p, q, n > 1 and j, there is a constant K, which does not depend on j such that (a).

T L p j (α)dλ(α) ≤ K(log j) p+1 , (b)
.

T [L p T (α)] n dλ(α) ≤ KT n-1 , (15) (c).
T

L p j (γ + α)L q j (β -α)dλ(α) ≤ KL p+q+1 j (β + γ). ( 16
)
The proof is omitted here, since this result is similar to the one stated in [12, lemma 1 and 2]. The only difference lies in the definition of the L p T . A detailed proof can be found in [START_REF] Rozenholc | Détection de ruptures non paramétriques à l'aide de données rabotées[END_REF].

Lemma 3.1 Let |α| ≤ π, and consider two integers

k 1 , k 2 , k 1 < k 2 , set k2 j=k1+1 h( j T )e -ijα ≤ C h L 0 k2-k1 (α).
Proof Now, let us consider the Dirichlet kernels ∆ k (α) = H k 0 (α) (see [START_REF] Basseville | Detection of abrupt changes in signals and dynamical systems[END_REF]). By an Abel transformation, we obtain

k2 j=k1+1 h( j T )e -ijα =   h( k 2 T )∆ k2-k1 (α) - k2-1 j=k1+1 h( j + 1 T ) -h( j T ) ∆ j-k1 (α)   e -ik1α .
So,

k2 j=k1+1 h( j T )e -ijα ≤   h( k 2 T ) + k2-1 j=k1+1 h( j + 1 T ) -h( j T )   sup j=1,...,k2-k1 |∆ j (α)|.
As h is C 1 , the term between brackets is bounded by a constant C h depending only on h. As the functions L p j (α) are increasing with j, we just have to prove that, for |α| ≤ π

|∆ j (α)| = sin(jα/2) sin(α/2) ≤ L 0 j (α).
Since sin x ≤ x, for x ∈ [0, π] and 2 π x ≤ sin x for x ∈ [0, π/2], then one easy checks that

• for |α| ≤ 2/j, |∆ j (α)| ≤ j |α| 2 2 π |α| 2 = πj 2 = L 0 j (α), (17)
• for 2/j ≤ |α| ≤ π,

|∆ j (α)| ≤ 1 2 π |α| 2 = π |α| = L 0 j (α). ( 18
)
This completes the proof of Lemma 3.1 Q.E.D.

Let us now introduce some kernels based on the taper h, which are approximate identities for convolution. Set s = min(s 1 , . . . , s m ) and define on T m-1 the function Φ s1,...,sm T by Φ s1,...,sm

T (γ 1 , . . . , γ m-1 ) =        H [s 1 T ] 1 (γ1)...H [s m-1 T ] 1 (γm-1)H [sm T ] 1 (- m-1 j=1 γj ) H [sT ] m , if H [sT ] m = 0, 1, otherwise. ( 19
)
Lemma 3.2 Let m ∈ N, s 1 , . . . , s m in [0, 1] and h be C 1 such that H m (s) = 0. Then, the sequence of functions (Φ s1,...,sm T ) ∞ T =1 is an approximate identity for convolution.

First (see [START_REF] Rudin | Functionnal analysis[END_REF]Chaper 6]), recall that a family (Φ T ) T ∈N of Lebesgue integrable functions on T m-1 with values in C is called an approximate identity for convolution if, for γ = (γ 1 , . . . , γ m-1 ) and ||γ|| = sup i=1,...,m-1 |γ i |, (i) sup

T ∈N T m-1 |Φ T (γ)| dλ(γ) < ∞, (ii) lim T →∞ T m-1 Φ T (γ)dλ(γ) = 1, (iii) lim T →∞ T m-1 \{γ,||γ||<δ}
|Φ T (γ)| dλ(γ) = 0, for all δ > 0.

Then, for all bounded continuous complex functions g on T m-1 , lim

T →∞ ||Φ T * g -g|| ∞ = 0.
Proof This result was shown by [START_REF] Dahlhaus | Spectral analysis with tapered data[END_REF]Lemma 3] in the case s 1 = . . . = s m = 1. We follow the sketch of his proof in our case,that is to say that (Φ s1,...,sm T ) ∞ T =1 satisfies the three assertions (i)-(iii). As the functions L 0 k (α) increase with k, by applying Lemma 3.1 to k 1 = 0 and k 2 = [sT ], we get

H [sT ] 1 (α) ≤ C h L 0 [sT ] (α) ≤ C h L 0 T (α).
Therefore,

|Φ s1,...,sm T (γ 1 , . . . , γ m-1 )| ≤ (C h ) m H [sT ] m L 0 T (γ 1 ) . . . L 0 T (γ m-1 )L 0 T (- m-1 j=1 γ j ). (20)
Assertion (i) is obtained noting that the limit of the sequence (

H [sT ] m T ) is H m (s) = 0

by assumption, and using inequality (16).

Set h s = h.1 [0,s] and let us check (ii). One has

T m-1 Φ s1,...,sm T (γ)dλ(γ) = 1
H [sT ] m T j1=1 . . . T jm=1 h sm ( j m T ) m-1 l=1 h s l ( j l T ) T e i(j l -jm)γ l dλ(γ l ) = 1 H [sT ] m T j=1 h s1 ( j T ) . . . h sm ( j T ) = 1 H [sT ] m T j=1 h s ( j T ) m = 1 H [sT ] m [sT ] j=1 h( j T ) m = 1.
Fix δ > 0 and T 0 such that δ ≥ π T0 . Now, using that {||γ|| ≥ δ} = ∪ m-1 j0=1 {|γ j0 | ≥ δ} and noticing that

|γ j0 | ≥ δ ≥ 2 T implies L 0 T (γ j0 ) = 2 γj 0 ≤ 2 δ , inequality (20) leads to T m-1 \{γ,||γ||<δ} |Φ T (γ 1 , . . . , γ m-1 )| m-1 j=1 dλ(γ j ) ≤ π(C h ) m δH [sT ] m m-1 j0=1 T m-2 L 0 T (- m-1 j=1 γ j ) 1 ≤ j ≤ m -1 j = j 0 L 0 T (γ j )dλ(γ 1 ) . . . dλ(γ m-1 ).
Then, inequalities [START_REF] Dahlhaus | Nonparametric high resolution spectral estimation[END_REF] give that this last term is an O( ln m-1 T T

). We obtain assertion (iii) using again

the assumption

H m (s) = 0 to control H [sT ]
m . This completes the proof of Lemma 3.2 Q.E.D.

Functional central limit theorem

To obtain a functional limit theorem for the process Z T defined in [START_REF] Dacunha-Castelle | Probabilités et statistiques -Tome 2[END_REF], we follow a classical scheme. We first investigate the limits of the finite dimensional distributions of Z T and then we prove the tightness of Z T .

Finite dimensional distributions

We first prove that Z T is asymptotically uniformly centered, then exhibit its limiting covariance function and finally study its finite dimensional distributions.

Proposition 1 Suppose (X j ) belongs to G. Then, as T → ∞ sup s∈[0,1],λ∈[0,π] |E Z T (λ, s)| = o(1).
Proof We extend to tapered data inequality (1.5) of [25, p. 369]. If f is a function defined on the torus T, denote by f (n) its n th Fourier coefficient. The expectation of Z T satisfies, using definitions (1), ( 2) and ( 11)

E Z T (λ, s) = H [sT ] 2 √ T λ 0 E I [sT ] (α) -f (α) dλ(α) = H [sT ] 2 √ T +∞ n=-∞ ϕ(n) E I [sT ] (n) -f (n) , with ϕ = 1 [0,λ] and ϕ(n) = (e -inλ -1)/n. Now, let K k (α) = 1 k H k 1 (α) 2 be the tapered generalization of the Fejer kernel. It follows that E I k (α) = K k * f (α), so that E I k (n) = K k (n). f (n).
Hence, the Fourier coefficients of K k are null for n > k, since they verify

K k (n) = 1 H k 2 inf(k,(k-n)) j=sup(1,(1-n)) h( j T )h( j + n T ).
The expectation of Z T splits into two terms

E Z T (λ, s) = H [sT ] 2 √ T   [sT ] n=-[sT ] ϕ(n) f (n) K [sT ] (n) -1 - |n|>[sT ] ϕ(n) f (n)   = H [sT ] 2 √ T A [sT ] -B [sT ] (21)
with

A [sT ] = [sT ] n=-[sT ] ϕ(n) f (n) K [sT ] (n) -1 and B [sT ] = |n|>[sT ] ϕ(n) f (n).
Using |n ϕ(n)| ≤ 2 and the Schwarz inequality, we get

B 2 [sT ] ≤ 4 |n|>[sT ] 1 |n| 2 |n|>[sT ] f (n) 2 . Since |n|>[sT ] 1 |n| 2 ≤ 2[sT ] -1 , we get H [sT ] 2 √ T B [sT ] ≤ ||h|| 2   [sT ] T |n|>[sT ] | f (n)| 2   1/2 . ( 22
)
Fix ε > 0, the behavior of B [sT ] varies according to the position of s w.r.t. ε.

• If 0 ≤ s < ε, using [sT ] T ≤ ε and the Parseval equality yields

H [sT ] 2 √ T B [sT ] ≤ ||h|| 2 f 2 ε 1/2 . • If ε ≤ s ≤ 1, as [sT ] T ≤ 1, H [sT ] 2 √ T B [sT ] ≤ ||h|| 2 |n|>[εT ] | f (n)| 2 1/2 . Taking ε = 1/ √ T , we get that sup s∈[0,1] H [sT ] 2 √ T B [sT ] ≤ ||h|| 2 sup   1 √ T , |n|> √ T | f (n)| 2   1/2 = o(1). ( 23
)
To study

A [sT ] , first consider ∆ [sT ] (n) -1 = 1 H [sT ] 2   inf([sT ],([sT ]-n)) j=sup(1,(1-n)) h( j T ) h( j + n T ) -h( j T ) - j (n) h 2 ( j T )   (24)
where

(n) j means [sT ] j=[sT ]-n+1 if n > 0 and
-n j=1 if n < 0. This sum has at most n terms, so

j (n) h 2 ( j T ) ≤ |n| h 2 .
Expanding h in Taylor series, leads to

H [sT ] 2 ∆ [sT ] (n) -1 ≤ [sT ] h |n| T h ′ + |n| h 2 ≤ C h |n| . (25)
Plugging this in A [sT ] (see 21), we get

| H [sT ] 2 √ T A [sT ] | ≤ C h √ T |n|≤[sT ] |n ϕ(n) f (n)| ≤ C h 2 √ T |n|≤[sT ] | f (n)| ≤ C h 2 √ T |n|≤T | f (n)|. ( 26
)
As Proposition 2 If (X j ) is in G, then, for all s 1 and s 2 in [0, 1] and all λ 1 and λ 2 in [0, π],

f is in L 2 , |n|≤T | f (n)| = o( √ T ),
lim T →∞ cov[Z T (λ 1 , s 1 ), Z T (λ 2 , s 2 )] = λ1∧λ2 0 f 2 (α)dλ(α) s1∧s2 0 h 4 (u)du.
Proof As Z T (., .) is uniformly centered, it is enough to prove this result for the centered process Z T defined by

Z T (λ, s) = Z T (λ, s) -E Z T (λ, s) = 1 √ T λ 0 |d [sT ] (α)| 2 -E |d [sT ] (α)| 2 dλ(α) (27)
where d k (α) is defined in [START_REF] Basseville | Detection of abrupt changes -Theory and applications[END_REF]. Clearly,

cov( Z T (λ 1 , s 1 ), Z T (λ 2 , s 2 )) = λ1 0 λ2 0 G(α 1 , α 2 , s 1 , s 2 )dλ(α 1 )dλ(α 2 ) ( 28
)
where

G(α 1 , α 2 , s 1 , s 2 ) = E d [s1T ] (α 1 ) 2 d [s2T ] (α 2 ) 2 -E d [s1T ] (α 1 ) 2 E d [s2T ] (α 2 ) 2 . Let us study the term G(α 1 , α 2 , s 1 , s 2 ). Since d k (.) is a Gaussian process, setting ξ 1 = d [s1T ] (α 1 ), ξ 2 = d [s1T ] (-α 1 ), ξ 3 = d [s2T ] (α 2 ) and ξ 4 = d [s2T ] (-α 2 ), vector (ξ 1 , ξ 2 , ξ 3 , ξ 4
) satisfies the Gaussian identity

E ξ 1 ξ 2 ξ 3 ξ 4 = E ξ 1 ξ 2 E ξ 3 ξ 4 + E ξ 1 ξ 3 E ξ 2 ξ 4 + E ξ 1 ξ 4 E ξ 2 ξ 3 . (29)
Therefore,

E ξ 1 ξ 3 E ξ 2 ξ 4 =   [s1T ] j1=1 h( j 1 T )e -ij1α1 [s2T ] j3=1 h( j 3 T )e ij3α2 T e -i(j1-j3)β1 f (β 1 )dλ(β 1 )     [s1T ] j2=1 h( j 2 T )e -iij2α1 [s2T ] j4=1 h( j 4 T )e ij4α2 T e -i(j2-j4)β2 f (β 2 )dλ(β 2 )   = T 2 dλ(β 1 )dλ(β 2 )f (β 1 )f (β 2 ) H [s1T ] 1 (α 1 + β 1 )H [s2T ] 1 (-α 2 -β 1 )H [s1T ] 1 (-α 1 + β 2 )H [s2T ] 1 (α 2 -β 2 )
where H k p (α) is defined in [START_REF] Basseville | Detection of abrupt changes in signals and dynamical systems[END_REF]. Substituting α 2 in -α 2 , we obtain the expression of E ξ 1 ξ 4 E ξ 2 ξ 3 . Now for γ = (γ 1 , γ 2 , γ 3 ), let us define

g(γ) = T 1 λ1 (α)1 λ2 (γ 1 + γ 2 -α)f (γ 1 -α)f (γ 3 + α)dλ(α) + T 1 λ1 (α)1 λ2 (α -γ 1 -γ 2 )f (γ 1 -α)f (γ 3 + α)dλ(α) (30)
and

Φ s1,s2,s3,s4 T (γ) = H [s1T ] 1 (γ 1 )H [s2T ] 1 (γ 2 )H [s3T ] 1 (γ 3 )H [s4T ] 1 (- 3 j=1 γ j ) H [s1T ]∧[s2T ] 4 . ( 31
)
With these notations the covariances, defined in [START_REF] Lavielle | Optimal segmentation of random processes[END_REF], may be expressed as

cov Z T (ϕ 1 , s 1 ), Z T (ϕ 2 , s 2 ) = H [s1T ]∧[s2T ] 4 T Φ s1,s1,s2,s2
T * g(0, 0, 0). [START_REF] Pollard | Convergence of stochastic processes[END_REF] Noting that g(0, 0, 0) = λ1∧λ2 0 f 2 (α)dλ(α), and that

H [s1T ]∧[s2T ] 4
/T tends to s 1 ∧ s 2 as T → ∞, we achieve the proof using Lemma 3.2.

Q.E.D.

Proposition 3 Assume that (X j ) is in G. Then, for all λ 1 , . . . , λ l in [0, π], all s 1 , . . . , s m in [0, 1], and all real numbers a j1,j2 , j 1 = 1, . . . , l, j 2 = 1, . . . , m, the random variable

l j1=1 m j2=1 a j1,j2 Z T (λ j1 , s j2 )/Var   l j1=1 m j2=1 a j1,j2 Z T (λ j1 , s j2 )   D -→ N (0, 1).
Proof The asymptotic normality of the finite dimensional distributions is obtained using the same approach as Ibragimov in [25, Th. 3.1 pp.379-80]. In the quadratic decomposition proposed by Ibragimov

l j1=1 m j2=1 a j1,j2 Z T (λ j1 , s j2 ) = (B T XT , XT ) -E (B T XT , XT )
where XT is the column vector t (X 1 , . . . , X T ), only the T × T matrix B T changes

B T =    1 √ T l j1=1 m j2=1 a j1,j2 λj 1 0 h( k 1 T )h( k 2 T )e i(k2-k1)α 1 1≤k1,k2≤[sj 2 T ] dλ(α)    1≤k1,k2≤T
So we only need to generalize the upper-bound for ||B T ||. We have

||B T || = 1 √ T sup ||x||=1 j1,j2 a j1,j2 T k1,k2=1 x k1 x k2 λj 1 0 h( k 1 T )h( k 2 T )e i(k2-k1)α dλ(α) ≤ ml √ T sup j1,j2 |a j1,j2 | sup ||x||=1 T T k=1 x k h( k T )e -ikα 2 dλ(α).
This is the L 2 -norm of the function having for Fourrier coefficients the finite sequence (x 1 h( 1 T ), . . ., x T h( T T )). Hence, using the Parseval equality and the Schwarz inequality

T T k=1 x k h( k T )e -ikα 2 dλ(α) = x 2 1 h 2 ( 1 T ) + . . . + x 2 T h 2 ( T T ) ≤ ||h|| 2 ∞ ||x|| 2 ≤ ||h|| 2 ∞ .
Thus,

||B T || ≤ ml √ T sup j1,j2 |a j1,j2 | ||h|| 2 ∞ . (33)
So

||B T || is always a O(T -1/2
) and we can apply the sketch of proof proposed by Ibragimov.

Q.E.D.

Tightness

To prove the tightness of the process Z T (., .) we use as in [START_REF] Picard | Testing and estimating change-points in time series[END_REF] the Csensov tightness criterion (see [START_REF] Csensov | Limit theorems for some classe of random functions[END_REF]) for continuous processes:

A family of process {Y T , T > 0} of C([0, π] × [0, 1], R) is tight if 1. The family {Y T (0, 0), T > 0} is tight in R, 2. The family {Y T (0, .), T > 0} is tight in C([0, 1], R), 3. The family {Y T (., 0), T > 0} is tight in C([0, π], R),
4. There exist constants C > 0, γ 1 > 1, γ 2 > 0 and a modulus of continuity ω defined on [0, π] such that, for all

B = [λ 1 , λ 2 ] × [s 1 , s 2 ] included in [0, π] × [0, 1], E Y T (B) γ1 ≤ C[(s 2 -s 1 )ω(λ 2 -λ 1 )] γ2
where

Y T (B) = Y T (λ 2 , s 2 ) -Y T (λ 2 , s 1 ) + Y T (λ 1 , s 1 ) -Y T (λ 1 , s 2 ).
As the process Z T (., .) is not continuous with respect to its second coordinate, we introduce the process L T (., .) with continuous sample path defined by

L T (λ, s) = 1 √ T λ 0 l T (α, s)dλ(α) (34)
where l T (., .) is the polygonal line which joins the points ((α, k/T ), z T (α, k/T )) where

z T (α, s) = [sT ] j=1 X j h( j T )e -ijα 2 -E [sT ] j=1 X j h( j T )e -ijα 2 = d [sT ] (α) 2 -E d [sT ] (α) 2 .
We first prove that the continuous process L T (., .) satisfies Csensov tightness criterion (Proposition 4), then to obtain the tightness of Z T (see [11, Proposition 4 Assume that (X j ) is in G, then the family of processes (L T ) is tight.

Proof Since here, we have L T (0, 0) = L T (0, .) = L T (., 0) = 0 the first three conditions of Csensov criterion are obvious. Define for

B = [λ 1 , λ 2 ] × [s 1 , s 2 ] L T (B) = 1 √ T λ2 λ1 [l T (α, s 2 ) -l T (α, s 1 )] dλ(α). ( 35
)
Our aim is to control

E L T (B) 2 = 1 T E λ2 λ1 λ2 λ1 [l T (α, s 2 ) -l T (α, s 1 )] [l T (-β, s 2 ) -l T (-β, s 1 )] dλ(α)dλ(β). (36)
The technical part of this proof is to control E L T (B)

2 when 0 ≤ s 2 -s 1 ≤ 1 T .
The general case is a direct consequence of this particular one. For this and before considering the general case, let us denote by

∆ n (α) = n j=1 e -ijα the Dirichlet kernel and ϕ n (λ) = λ 0 ∆ n (α)dλ(α). ( 37
)
[39, Lemma 8.2 p.57] assure that the functions ϕ n are continuous on the torus and are uniformly bounded in n and λ so one can find a uniform modulus of continuity ω(.) for the family (ϕ n ) n∈N i.e. which satisfies sup 0<|µ-λ|<δ |ϕ n (µ)ϕ n (λ)| ≤ ω(δ) for all integer n, we prove the following lemma Lemma 4.1 Let ω(.) be an uniform modulus of continuity for all the ϕ n defined by [START_REF] Sachs | Peak-intensive spectrum estimation[END_REF], then for

B = [λ 1 , λ 2 ] × [s 1 , s 2 ] such that [s 1 T ] ≤ s 1 T < s 2 T ≤ [s 1 T ] + 1, the following inequality holds E L T (B) 2 ≤ C 1 (s 2 -s 1 ) (λ 2 -λ 1 ) ∨ ω 2 (λ 2 -λ 1 ) .
Proof We note k = [s 1 T ] and Y j = h(j/T )X j the tapered data. Then, using that 0 ≤ s 2s 1 ≤ 1/T , it follows

l T (α, s 2 ) -l T (α, s 1 ) = T (s 2 -s 1 ) z T (α, k + 1 T ) -z T (α, k T ) (38)
and

E L T (B) 2 ≤ (s 2 -s 1 ) λ2 λ1 λ2 λ1 cov z T (α, k + 1 T ) -z T (α, k T ), z T (β, k + 1 T ) -z T (β, k T ) dλ(α)dλ(β). (39)
Now, noting that

z T (α, k + 1 T )-z T (α, k T ) = Y 2 k+1 -E Y 2 k+1 +2Y k+1 Re k j=1 Y j e -i(k+1-j)α -2E   Y k+1 Re k j=1 Y j e -i(k+1-j)α   ,
we obtain

E L T (B) 2 ≤ (s 2 -s 1 )(λ 2 -λ 1 ) 2 var Y 2 k+1 + Re (R 1 ) + R 2 , with (40)
R 1 = 4(s 2 -s 1 )(λ 2 -λ 1 ) λ2 λ1 cov Y k+1 k j=1 Y j e -i(k+1-j)α , Y 2 k+1 dλ(α), R 2 = 4(s 2 -s 1 ) λ2 λ1 λ2 λ1 cov Y k+1 k j=1 Y j cos[(k + 1 -j)α], Y k+1 k j=1 Y j cos[(k + 1 -j)β] dλ(α)dλ(β).
Since (X j ) is a Gaussian stationary process, var X 2 j is constant, and

(s 2 -s 1 )(λ 2 -λ 1 ) 2 var Y 2 k+1 ≤ πvar X 2 0 ||h|| 2 ∞ (s 2 -s 1 )(λ 2 -λ 1 ). (41)
The two last right terms of (40) are controlled by the two following lemma Lemma 4.2 :

|R 1 | ≤ 8 √ πE X 2 0 ||h|| 4 ∞ ||f || 2 (s 2 -s 1 )(λ 2 -λ 1 )
Proof Using the identity for Gaussian vector

cov   Y k+1 k j=1 Y j e -i(k+1-j)α , Y 2 k+1   = 2E Y 2 k+1 k j=1 e -i(k+1-j)α E Y j Y k+1 = 2E X 2 0 h( k + 1 T ) 3 T f (β) k j=1 h( j T )e -i(k+1-j)(α+β) dλ(β) = 2E X 2 0 h( k + 1 T ) 3 H k 1 * f (α) (42) with H k 1 (α) = e -i(k+1)α H k 1 (α). Lemma 4.3 Let H k 1 (α) = k j=1 h(j/T )e -ij(k+1-j)α , then for all f in L 2 (T), H k 1 * f 2 ≤ ||h|| ∞ ||f || 2 .
Proof Let f (m) be the Fourier coefficients of f , then

T H k 1 (α)f (α + β)dλ(α) = T k j=1 h( j T )e i(j-k-1)α m∈Z f (m)e im(α+β) dλ(α) = m∈Z f (m)e imβ T k j=1 h( j T )e i(j+m-k-1)α dλ(α) = k j=1 f (k + 1 -j)h( j T )e i(k+1-j)β .
By Parseval equality, it follows

H k 1 * f 2 2 = k j=1 h 2 ( j T ) f (k + 1 -j) 2 ≤ ||h|| 2 ∞ ||f || 2 2 .
Q.E.D. Now, applying Schwarz inequality and Lemma 4.3 to (42), it follows

|R 1 | ≤ 8 √ πE X 2 0 ||h|| 3 ∞ (s 2 -s 1 )(λ 2 -λ 1 ) H k 1 * f 2 ≤ 8 √ πE X 2 0 ||h|| 4 ∞ ||f || 2 (s 2 -s 1 )(λ 2 -λ 1 ). Q.E.D. Lemma 4.4 : |R 2 | ≤ 4 ||h|| 2 ∞ (||f || 2 2 + (E X 2 0 ) 2 )(s 2 -s 1 ) (λ 2 -λ 1 ) ∨ ω 2 (λ 2 -λ 1 )
Proof Using again the identity for Gaussian vector

R 2 = 4(s 2 -s 1 ) λ2 λ1 λ2 λ1 cov   Y k+1 k j=1 Y j cos[(k + 1 -j)α], Y k+1 k j=1 Y j cos[(k + 1 -j)β]   dλ(α)dλ(β) = S 1 + S 2 , with (43)
S 1 = 4(s 2 -s 1 )   λ2 λ1 E   Y k+1 k j=1 Y j cos[(k + 1 -j)α]   dλ(α)   2 , S 2 = 4(s 2 -s 1 )E Y 2 k+1 E   λ2 λ1 k j=1 Y j cos[(k + 1 -j)α]dλ(α)   2 .
The bound for S 1 is close the one of R 1 developed in Lemma 4.2. By Schwarz inequality and lemma 4.3 it follows

|S 1 | ≤ 4(s 2 -s 1 ) λ2 λ1 E   Y k+1 k j=1 Y j e -i(k+1-j)α   dλ(α) 2 ≤ 4(s 2 -s 1 ) λ2 λ1 H k 1 * f (α)dλ(α) 2 ≤ 4(s 2 -s 1 )(λ 2 -λ 1 ) ||h|| 2 ∞ ||f || 2 2 . ( 44
)
To control S 2 in (43), let us first consider ω a uniform modulus of continuity for all the integrated Dirichlet kernels λ 0 ∆ n (α)dλ(α) where ∆ n = H k 0 (see 1), see for example [39, lemma Lemma 8.2 p.57] for the existence of such modulus.

Lemma 4.5 |S 2 | ≤ 4 ||f || 2 (V (h)) 2 E Y 2 0 (s 2 -s 1 )ω 2 (λ 2 -λ 1 )
Proof Let us denote by Y j = Y k+1-j , we have to control the following term

V = E   λ2 λ1 k j=1 Y j cos(jα)dλ(α)   2 ≤ E λ2 λ1 k j=1 Y j e -ijα dλ(α) 2 . ( 45
)
Considering this last inequality, it follows

V ≤ E   λ2 λ1 k j1=1 Y j e -ij1α dλ(α) λ2 λ1 k j2=1 Y j e ij2β dλ(β)   = λ2 λ1 λ2 λ1 k j1=1 k j2=1 E ( Y j1 Y j2 )e -ij1α e ij2β dλ(α)dλ(β) = λ2 λ1 λ2 λ1 k j1=1 k j2=1   h( k + 1 -j 1 T )h( k + 1 -j 2 T ) T f (γ)e -i(j2-j1)γ dλ(γ)   e -ij1α e ij2β dλ(α)dλ(β) 13 = T   λ2 λ1 k j1=1 h( k + 1 -j 1 T )e -ij1(α-γ) dλ(α)     λ2 λ1 k j2=1 h( k + 1 -j 2 T )e ij2(β-γ) dλ(β)   f (γ)dλ(γ) = T λ2 λ1 k j=1 h( k + 1 -j T )e -ij(α-γ) dλ(α) 2 f (γ)dλ(γ).
Let us denote λ i = λ iγ. By an Abel transformation (e -ijα = ∆ j (α) -∆ j-1 (α)), we have

λ2 λ1 k j=1 h( k + 1 -j T )e -ij(α-γ) dλ(α) = h( 1 T ) λ2 λ1 ∆ k (α)dλ(α)+ k-1 j=1 h( k + 1 -j T ) -h( k -j T ) λ2 λ1 ∆ j (α)dλ(α) it follows that λ2 λ1 k j=1 h( k + 1 -j T )e -ij(α-γ) dλ(α) ≤ V (h).ω(λ 2 -λ 1 )
and using Schwarz inequality

V = E   λ2 λ1 k j=1 Y j cos(jα)dλ(α)   2 ≤ √ 2π ||f || 2 (V (h)) 2 ω 2 (λ 2 -λ 1 ).
Reporting this last inequality in the definition of S 2 (see ( 43)), it follows that

|S 2 | ≤ 4 ||f || 2 ||h|| 2 ∞ (V (h)) 2 E X 2 0 (s 2 -s 1 )ω 2 (λ 2 -λ 1 ).
Q.E.D. Now, using (44) and Lemma 4.5, we obtain for R 2 defined in (43)

|R 2 | ≤ C 2 (s 2 -s 1 ) ω 2 (λ 2 -λ 1 ) ∨ (λ 2 -λ 1 )
so the Lemma 4.4 is proved Q.E.D. Finally using (41), Lemma 4.2 and Lemma 4.4, we prove that when [s

1 T ] ≤ s 1 T < s 2 T ≤ [s 1 T ] + 1, the following inequality holds E L T (B) 2 ≤ C 1 (s 2 -s 1 ) (λ 2 -λ 1 ) ∨ ω 2 (λ 2 -λ 1 ) ,
this ends the proof of Lemma 4.1

Q.E.D. To close the proof of the tightness, we have to extend the result of Lemma 4.1 to the general case where |s 2s 1 | ≥ 1/T . It is the object of the following Lemma. Lemma 4.6 Let ω(.) an uniform modulus of continuity for all ϕ n (see [START_REF] Sachs | Peak-intensive spectrum estimation[END_REF]), then for

B = [λ 1 , λ 2 ] × [s 1 , s 2 ] such that |s 2 -s 1 | ≥ 1/T the following inequality holds E L T (B) 2 ≤ C 2 (s 2 -s 1 ) (λ 2 -λ 1 ) ∨ ω 2 (λ 2 -λ 1 ) . Proof We consider t i = [s1T ]+i T , for i ≥ 0 and we note p = [s 2 T ] -[s 1 T ]. So we have t 0 ≤ s 1 < t 1 < . . . < t p ≤ s 2 < t p+1 .
Clearly,

L T (B) = L T ([λ 1 , λ 2 ] × [s 1 , t 1 ]) + p-1 j=1 L T ([λ 1 , λ 2 ] × [t j , t j+1 ]) + L T ([λ 1 , λ 2 ] × [t p , s 2 ]) (46)
and by Schwarz inequality it follows, using that 1/T ≤ s 2s 1 and [

s 2 T ] -[s 1 T ] -1 ≤ (s 2 -s 1 )T E L T (B) 2 ≤ C 1 (p + 1) 1 T ω 2 (λ 2 -λ 1 ) ∨ (λ 2 -λ 1 ) ≤ C 1 [s 2 T ] -[s 1 T ] + 1 T ω 2 (λ 2 -λ 1 ) ∨ (λ 2 -λ 1 ) ≤ 3C 1 (s 2 -s 1 ) ω 2 (λ 2 -λ 1 ) ∨ (λ 2 -λ 1 ) .
Q.E.D. As for δ > 0, ω(δ) = ω 2 (δ) ∨ δ clearly define a modulus of continuity, the fourth condition of Csensov criterion holds for E | L T (B)| 2 with γ 1 = 2 and γ 2 = 1 so the continuous process (L T ) T ∈N is tight in C([0, π] × [0, 1], the proof of Proposition 4 is complete.

Q.E.D.

Proposition 5

The two families of processes (L T ) T and (Z T ) T are contiguous (see [?, Chap 3.1 pp. [START_REF] Giraitis | A fuctionnal clt for nonparametric estimates of spectrum and the change point problem for a spectral function[END_REF][START_REF] Giraitis | Testing and estimating in the change point problem of the spectral function[END_REF][START_REF] Giraitis | The change-point problem for dependent observations[END_REF][START_REF] Hannan | Testing for a jump in the spectral function[END_REF][START_REF] Hinkley | Inference about the change point in a sequence of random variables[END_REF][START_REF] Hinkley | Inference about the change point from a cumulative sum tests[END_REF] or [?, Chap. 6 pp.85-91] for definition of the contiguity).

Proof Since Proposition 1 holds we prove that (L T ) and the centered process ( Z T ) (see ( 27)) are contiguous. For this we prove the following more general result:

P sup λ∈[0,π] sup s∈[0,1] L T (λ, s) -Z T (λ, s) > ǫ → 0 as T → +∞.
Applying the fourth condition of Csensov criterion to the process (L T ) with

B = [0, λ] × [ k T , k+1 T ], it follows for k = 1, . . . , T E L T ([0, λ] × [ k T , k + 1 T ]) 2 ≤ C ω(π) T . ( 47
) As λ → L T ([0, λ] × [ k T , k+1 T ]) is continuous on [0, π], it exists λ k such that sup λ∈[0,π] L T ([0, λ] × [ k T , k + 1 T ]) = L T ([0, λ k ] × [ k T , k + 1 T ]) .
As k takes only a finite number of values, there exists k 0 such that sup k=1,...,T

L T ([0, λ k ] × [ k T , k + 1 T ]) = sup k=1,...,T sup λ∈[0,π] L T ([0, λ] × [ k T , k + 1 T ]) = L T ([0, λ k0 ] × [ k 0 T , k 0 + 1 T ]) .
follows from this last equality and from (47) that

E sup k=1,...,T sup λ∈[0,π] L T ([0, λ] × [ k T , k + 1 T ]) 2 = E L T ([0, λ k0 ] × [ k 0 T , k 0 + 1 T ]) 2 ≤ C ω(π) T . (48)
Now, as Z T (λ, s) = L T (λ, [sT ]/T ), it follows, using that l T is piecewiese affine

L T (λ, s) -Z T (λ, s) = L T (λ, s) -L T (λ, [sT ] T ) = T (s - [sT ] T ) L T ([0, λ] × [ [sT ] T , [sT ] + 1 T ]).
Applying (48), its follows

E sup λ∈[0,π] sup s∈[0,1] L T (λ, s) -Z T (λ, s) 2 ≤ C ω(π) T .
Using Tchebicev inequality, we deduce that

P sup λ∈[0,π] sup s∈[0,1] L T (λ, s) -Z T (λ, s) > ǫ ≤ C ω(π) T ǫ 2 → 0 as T → +∞.
So (L T ) and (Z T ) are contiguous. Q.E.D.

Asymptotic distribution of the test statistics under the null hypothesis

We prove in this section our main theorem that is Theorem 1. The convergence of the statistic constructed on (Y T 1 ) is a direct consequence of relation ( 12) and Theorem 2: as the limiting process

Z is in C([0, π] × [0, 1], R),

30]

) and to remark that the sequence (

H T 2
T ) T converges to H 2 which is assumed to be finite and non zero.

We also have to prove that the two processes (Y T j ) T , j = 1, 2 have the same limiting distribution. For this we prove the convergence in distribution to 0 under the null hypothesis of the sequence of processes (Ω T ) T >0 defined by ( 13) and [START_REF] Dahlhaus | Small sample effects in time series analysis : a new asymptotic theory and a new estimate[END_REF].

Theorem 3 If X . is a process in G then Ω T → 0 as T → ∞.
Proof Let us first investigate the distributions of (Ω T (., .)) T Lemma 5.1 If X . is a process in G then process (Ω T (., .)) T converges in probability to 0.

Proof The proof splits into to steps

Step 1 Process (Ω T (., .)) T is uniformly asymptotically unbiased. Remark that

Ω T (λ, s) = Z T (λ, 1) -Z T (λ, s) -Z * T (λ, 1 -s) (49)
where

Z * T (λ, 1 -s) = ȞT -[sT ] 2 √ T FT -[sT ] (λ) -F (λ) is the "reverse-time" process of Z T constructed on h( T T )X T , . . . , h( [sT ]+1 T )X [sT ]+1 .
Clearly, the process Z * T has the same properties as Z T . In particular, Z * T satisfies Proposition 1. Therefore, the same is true for Ω T and we have sup

(λ,s)∈[0,π]×[0,1] |E Ω T (λ, s)| = o(1).
Step 2 For all (λ, s), the variance of Ω T (λ, s) tends to 0 as T → ∞. Following the same lines of proof as in Proposition 2 and considering that Z * T (λ, 1s) is defined on the variables X T , . . . , X [sT ]+1 and using the taper h on the set { T T ,. . . , [sT ]+1 T } which converge to [s, 1], we prove that lim

T →∞ cov(Z * T (λ, 1 -s), Z * T (λ, 1 -s ′ )) = λ 0 f 2 (α)dλ(α) [s,1]∧[s ′ ,1] h 4 (u)du = λ 0 f 2 (α)dλ(α) 1 s∨s ′ h 4 (u)du.
Now, as Z T (λ, 1) = Z * T (λ, 1), using (49) to compute var Ω T (λ, s) gives lim

T →∞ var Ω T (λ, s) = 2 lim T →∞ cov(Z T (λ, s), Z * T (λ, 1 -s)).
We just need to prove that cov(Z T (λ, s), Z * T (λ, 1s)) → 0. Following the same lines as in Lemma 2, it follows that cov(Z T (λ, s), Z * T (λ, 1s)) = Ψ s T * g(0, 0, 0)

with g defined by [START_REF] Pages | Détections de changements brusques des caractéristiques spectrales d'un signal numérique[END_REF]. For all γ = (γ 1 , γ 2 , γ 3 )

Ψ s T (γ) = 1 T H [sT ] 1 (γ 1 )H [sT ] 1 (γ 2 ) ȞT -[sT ] 1 (γ 3 ) ȞT -[sT ] 1 (-γ 1 -γ 2 -γ 3 ) with ȞT -k 1 (α) = H T 1 (α) -H k 1 (α). Since Ψ s T = H [sT ] 4 T (Φ s,s,1,1 T -Φ s,s,1,s T -Φ s,s,s,1 T + Φ s,s,s,s T
) (see [START_REF] Picard | Testing and estimating change-points in time series[END_REF] for the definition of Φ s1,s2,s3,s4 T ) using that Φ s,s,1,1 T , Φ s,s,1,s T , Φ s,s,s,1 T and Φ s,s,s,s T are approximate identities (see Lemma 3.2) their product of convolution with g at point (0, 0, 0) tends to g(0, 0, 0) as T tends to ∞. It follows that the right term of (50) tends to 0.

Applying Chebishev inequality, Lemma 5.1 is obtained.

Q.E.D.

Lemma 5.2 If X . is a process in G then the sequence of processes (Ω T ) T ∈N is tight.

Proof Using that process (Ω T (., .)) T is uniformly asymptotically unbiased, it is enough to prove that

(Ω T -E Ω T ) is contiguous to the C-tighted process L ′ T = 1 √ T λ 0 l ′ T (α, s)dλ(α)
where l ′ T (α, s) is the polygonal line which joins the points ((α, k/T ), z ′ T (α, k/T )) with

z ′ T (α, s) =   k j1=1 Y j e -ij1α T j2=k+1 Y j e ij2α   -E   k j1=1 Y j e -ij1α T j2=k+1 Y j e ij2α   ,
and with Y j = h(j/T )X j and k = [sT ]. Equation [START_REF] Alley | Rupture dans une série chronologique. application à la détection de rupture dans les champs aléatoires stationnaires[END_REF] to [START_REF] Zygmund | Trigonometric series. Volumes I and II combined[END_REF] are still valid, replacing z T , l T , L T , L T by

z ′ T , l ′ T , L ′ T , L ′ T , so that z ′ T (α, k + 1 T ) -z ′ T (α, k T ) =   k+1 j1=1 Y j e -ij1α T j2=k+2 Y j e ij2α - k j1=1 Y j e -ij1α T j2=k+1 Y j e ij2α   -E   k+1 j1=1 Y j e -ij1α T j2=k+2 Y j e ij2α - k j1=1 Y j e -ij1α T j2=k+1 Y j e ij2α   =   Y k+1 e -i(k+1)α T j2=k+2 Y j e ij2α - k j1=1 Y j e -ij1α Y k+1 e i(k+1)α   -E   Y k+1 e -i(k+1)α T j2=k+2 Y j e ij2α - k j1=1 Y j e -ij1α Y k+1 e i(k+1)α   = Y k+1 (d k (α) -ďT -k-1 (α)) -E Y k+1 (d k (α) -ďT -k-1 (α)) with d k (α) defined in (2) and ďT -k (α) = d T (α)-d k (α). It follows that, for [s 1 T ] ≤ s 1 T < s 2 T ≤ [s 1 T ]+1, E | L ′ T (B)| 2 ≤ (s 2 -s 1 ) λ2 λ1 λ2 λ1 cov Y k+1 (d k (α) -ďT -k-1 (α)), Y k+1 (d k (β) -ďT -k-1 (β)) dλ(α)dλ(β).
The righthand term is similar to the quantity R 2 appearing in (40), therefore, it is controlled in a same way (see Lemma 4.4). Such control leads to the following inequality

E | L ′ T (B)| 2 ≤ C (s 2 -s 1 ) (λ 2 -λ 1 ) ∨ ω 2 (λ 2 -λ 1 )
which holds for [s 1 T ] ≤ s 1 T < s 2 T ≤ [s 1 T ] + 1. Now, as ( 46)-(48) hold for L ′ T (B), Lemma 4.6 and Proposition 5 can be extended to process L ′ T (B) and Ω T . Therefore (Ω T ) T >0 is contiguous to a C-tighted process so is tighted (see [3]).

Q.E.D.

This proves Theorem 3 together with the asymptotic equivalence of the sequence (Y 1 T ) T >0 and (Y 2 T ) T >0 . So the two families of test statistics S 1 T (h) and S 2 T (h) are also asymptotically equivalent Q.E.D.

Critical region and applications

First, our purpose here, is to obtain the asymptotic form of the reject region. Then, we prove the consistency of our tests when G 2 is known or not. Finally, we present some numerical simulation results. We observe on these simulations that tapering improves detection.

Critical region

We establish here the following result Corollary 2 When T is large, the level of the test associated with the critical regions R j T , j = 1, 2 can be approximated using the limits lim

T →+∞ P H0 (R j T ) = P( sup u∈[0,1] sup s∈[0,1] B(u, s) - H 2 [H -1 4 (s.H 4 )] H 2 B(u, 1) > c)
where H -1 4 (.) is the inverse or pseudo-inverse function of H 4 (.) and where B is the Gaussian process of C([0, 1] 2 , R) with mean zero and covariance function

E B(u, s)B(u ′ , s ′ ) = u ∧ u ′ .s ∧ s ′
Proof Since the two statistics S j T (h), j = 1, 2 have the same limit under H 0 we only prove Corollary1.1 for S 1 T (h). Let us consider the time change

u = G 2 (λ) G 2 u ′ = G 2 (λ ′ ) G 2 v = H 4 (s) H 4 v ′ = H 4 (s ′ ) H 4 with G 2 (λ) = λ 0 f 2 (
α)dλ(α) and H 4 (.) defined in [START_REF] Basseville | Detection of abrupt changes in signals and dynamical systems[END_REF]. Denote by Z the process defined on

[0, 1] × [0, 1] by Z(u, v) = Z(λ, s). It is easy to verify sup λ∈[0,π] sup s∈[0,1] 1 H 2 Z(λ, s) - H 2 (s) H 2 Z(λ, 1) = 1 H 2 sup u∈[0,1] sup v∈[0,1] Z(u, v) - H 2 (H -1 4 (H 4 .v)) H 2 Z(u, 1) .
The process Z/ √ G 2 H 4 has the same covariance function that the process B defined in Corollary 1.1. So, it is enough to prove Corollary 1.1 Q.E.D.

Consistency and practical use

If G 2 is known, it is clear that under H 1 the statistic S 2 T (h) converges in probability to +∞. So our test is consistent. To get such a result for S 1 T (h), we need more assumptions on the two processes before and after the change-point, due to the presence of the crossed term Ω T (see [START_REF] Dahlhaus | Empirical spectral processes and their applications to time series analysis[END_REF] and (14)), for example.

Theorem 4 Under H 1 , if the processes X 1 and X 2 are independent, the statistic S 1 T (h) converges in probability to +∞.

Proof We prove that S 1 T (h) has the same limit as S 2 T (h). Using (13), we only have to prove that the process Ω T is bounded in probability. For this, we remark first that, as the processes X 1 and X 2 being independent, Ω T is unbiased. Let Φ s0 T = Φ s0,s0,s0,s0 T defined in [START_REF] Picard | Testing and estimating change-points in time series[END_REF] and

Φ s0 T (γ) = K [s0T ] 1 (γ 1 )K [s0T ] 1 (γ 2 )K [s0T ] 1 (γ 3 )K [s0T ] 1 (- 3 j=1 γ j ) H T 2 -H [s0T ] 2 with K [sT ] 1 (α) = T j=[sT ]+1
h( j T )e -ijα . Following the proof of Proposition 2 we obtain

var 2 Ω T (λ 0 , s 0 ) ≤ H [s0T ] 2 T T 3 Φ s0 T (γ)dλ(γ) T f 1 (γ 1 -α)f 1 (γ 2 + α)dλ(α) + H T 2 -H [s0T ] 2 T T 3 Φ s0 T (γ)dλ(γ) T f 2 (γ 1 -α)f 2 (γ 2 + α)dλ(α) (51)
where γ = (γ 1 , γ 2 , γ 3 ) with dλ(γ) = dλ(γ) 1 dλ(γ) 2 dλ(γ) 3 , f 1 and f 2 denote the spectral density before and after change-point. The sequence of kernels ( Φ s0 T ) T is an approximated identity for convolution (see Lemma 3.2). It is easy to see that the sequence ( Φ s0 T ) T defined also an approximated identity for convolution. So the right term in (51) converges and the covariance is bounded. Applying the Bienayme-Chebyshev inequality, we get the announced result.

Q.E.D.

Consider now the case where G 2 is unknown. We also have under H 1 to control the estimated G 2 defined in (8) which appears in the critical regions (see [START_REF] Brodsky | Nonparametric methods in change-point problems[END_REF]. In fact, we can show that it converges under H 1 to a finite value which is 1 2

s 2 0 f 1 2 2 + (1 -s 0 ) 2 f 2 2 2 1/2 .
So the test can be extended to the case G 2 unknown.

Numerical results

To present our numerical results we consider the polynomial taper

h ρ,n (x) =    4 n .( x ρ ) n .(1 -x ρ ) n x ∈ [0, ρ 2 ), 1 x ∈ [ ρ 2 , 1 -ρ 2 ], h ρ,n (1 -x) x ∈ (1 -ρ 2 , 1],
with n ≥ 1 and 0 ≤ ρ ≤ 1. The parameter ρ controls how much tapered the data are: as ρ goes to 0, h ρ,n tends to 1 which corresponds to the untapered case. The parameter n controls the smoothness of the tapering: the larger n is, the smoother the tapering. Before presenting some simulation results, we have to say a few words about the tabulation of the limiting distribution.

Since the limit distribution of our test is not independent from h, we need to tabulate it for each h. It may be of interest to bypass this problem by looking for a family of tapers (h T ) T indexed by T such that h T converges to 1 when T goes to +∞. Doing this we obtain the following result Theorem 5 If (h ρ T ,n T ) T is a sequence of tapers such that • (ρ T ) T is a sequence of integer which tends to 0 when T goes to +∞,

• lim inf T →∞ T.ρ T > 0,

• (n T ) T is a bounded sequence such that n T ≥ 1, the level of the test associated to the critical regions R j T , j = 1, 2 can be approximated using the following limits lim Proof The proof may be found in totality in the thesis [33, Th 10.0.12 pp. 90-94].

Q.E.D. In fact for small T , it is more interesting to adapt our taper h ρ,n to the expecting number of false alarms. The following tables present for T = 50 and T = 150 the percentage of false alarms detected with each taper for 1000 trajectories of i.i.d. N (0, 1)

• T = 50 On the first table, we see that the h ρ,n adapted taper is, for T = 50, obtained for ρ between 0.85 and 0.90. For T = 150, the second table shows that we have to take a value of ρ between 0.60 and 0.65. As the value of n does not change these results significantly we have only worked with n = 4.

α
For these two adapted tapers, we have tested a change-point in a process X t constructed as follows

• X 0 , . . . , X [s0T ] are i.i.d. Gaussian N (0, 1).

• X [s0T ]+1 , . . . , X T are the trace of an AR(1) with root 0.3.

The following table gives the percentage of alarms detected when s = 0.1 and s = 0.5 and s = 0.9 with or without taper These last two tables clearly show the effect of tapering when T is small (T = 50). In this case, we have, on average, twice more alarms detected with the taper. The case T = 150 shows that, even if we always have a better score with the taper, this taper effect is less when T grows up.

To conclude this numerical part we represent the field Y 2 T (., .) with or without taper. We clearly see on the following two figures what the taper effect is: smoothness and concentration.

1 TY 2 T

 12 be the space of real functions on [0, π] × [0, 1] which are right-continuous with left-hand limits, endowed with the Skorohod topology. Let C = C([0, π] × [0, 1], R) be the space of continuous real functions on [0, π] × [0, 1], with the uniform convergence topology. Clearly processes Y and are D-valued. The following holds

  so the expression in the right hand-side of (26) goes to 0 as T → +∞ uniformly in s ∈ [0, 1]. Joining this and (23) completes the proof of Proposition 1 Q.E.D.

  Déf. 7.3.25-b p.219]) we prove the contiguity (see [?, Chap 3.1 pp.19-24] or [?, Chap. 6 pp.85-91]]) of Z T and L T (Proposition 5).

  we just need to apply the classical functional theorems in C (see [3, Th. 4.4 p 27 & Th. 5.1 p.

1 ]

 1 |W (u, s)| > c)where W (u, s) = B(u, s)s B(u, 1), and where B is defined in corollary 1.1.

fig 1 :

 1 fig 1 : Y 2 T with tapering fig 2 : Y 2 T without tapering