Keywords: Adaptation, Copula Density, Minimax Theory of Test, Goodness Test of Fit AMS Subject Classification -62G10, 62G20, 62G30

We consider the problem of testing hypotheses on the copula density from n bidimensional observations. We wish to test the null hypothesis characterized by a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the L 2 -norm from any density lying in the null hypothesis. The copula densities under consideration are supposed to belong to a range of Besov balls. According to the minimax approach, the testing problem is solved in an adaptive framework: it leads to a log log term loss in the minimax rate of testing in comparison with the non-adaptive case. A smoothness-free test statistic that achieves the minimax rate is proposed. The lower bound is also proved. Besides, the empirical performance of the test procedure is demonstrated with both simulated and real data.

A test of goodness-of-fit for the copula densities Ghislaine GAYRAUD * and Karine TRIBOULEY ♭

Introduction

Copulas became a very popular and attractive tool in the recent literature for modeling multivariate observations. The nice feature of copulas is that they capture the structure dependence among the components of a multivariate observation without requiring the study of the univariate margins. More precisely, Sklar's Theorem ensures that any d-varied distribution function H may be expressed as

H(x 1 , . . . , x d ) = C F 1 (x 1 ), . . . , F d (x d ) ,
where the F p 's are the margins and C is called the copula function. [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] states the existence and the uniqueness of C as soon as the random variables with joint law H are continuous.

Modeling the dependence is a great challenge in statistics, specially in finance or assurance where (for instance) the identification of the dependence structure between assets is essential. Many authors proposed parametrical families of copulas {C λ , λ ∈ Λ}, each of them being available to capture different dependence behavior. The elliptic family contains the Gaussian copulas and the Student copula which are often used in finance. For insurance purposes, heavy tails are needed and copulas coming from the archimedian family are used. Among others, the more common are the Gumbel copula, the Clayton copula or the Frank copula. In view to illustrate the different behaviours of the tails of several copula densities, some graphs corresponding to the models cited above are presented below. The parameters are chosen such a way that the associated Kendall's tau (i.e. the indicator of concordance/discordance) is identical in all illustrations. Since many parametric copula models are now available, the crucial choice for the practitioner is to identify the model which is well-adapted to data at hand. Many goodnessof-fit tests are proposed in the literature. [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF] give an excellent review and propose a detailed empirical study for different tests: we refer to this paper for any supplementary references. Roughly speaking, they study procedures based on empirical processes. Among others, they deal with rank-based versions of the Cramér-von-Mises and Kolmogorov-Smirnov statistics. They also consider test based on Kendall's transform. Basically, they restrict themselves to test statistics built from empirical distributions (empirical copula or transform of this latter). On a theoretical point of view, the asymptotic law under the null of the test statistic is stated in a number of papers (see by instance [START_REF] Deheuvels | La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d'indépendance[END_REF], [START_REF] Deheuvels | A kolmogorov-smirnov type test for independence and multivariate samples[END_REF] and [START_REF] Deheuvels | A nonparametric test of independence[END_REF]). It allows in particular to derive the critical value but generally the alternative is unspecified and the properties on the power are empirically given from simulations.

In our paper, it is supposed that the copula C admits a density copula c with respect to the Lebesgue measure. To our knowledge, [START_REF] Fermanian | Goodness-of-fit tests for copulas[END_REF] was the first author to propose a goodnessof-fit test based on nonparametric kernel estimations of the density copula. In the same spirit as the papers cited above, he derived the asymptotic law of the test statistic under the null. His results are valid for bandwidths greater than n -2/(8+d) which correspond to enough smooth copula densities.

Here, we focus on the minimax theory framework: we define the test problem as initiated by [START_REF] Ingster | On minimax nonparametric detection of a signal in gaussian white noise[END_REF]. One of the advantages of this point of view is to precisely define the alternative: it is then possible to quantify the risk associated with the test problem as the sum of the first type error and the second type of error. Since this risk measure provides a quality criterion, it is then possible to compare the test procedures. Indeed, the alternative H 1 (v n ) is defined from a positive quantity v n measuring the distance between the null and the latter. Obviously, the larger is this separating distance, the easier is the decision. The aim of the minimax theory is to determine the larger alternative for which the decision remains feasible. Solving the lower bound problem is equivalent to exhibit the faster separating rate v n such that the risk is bounded from below by a given positive constant α: this rate is called the minimax rate of testing. Next, the upper bound problem has to be solved exhibiting a test procedure whose risk is bounded from above by a given α, that is, the statistic test allows to distinguish the null from H 1 (v n ), where v n is the minimax rate.

In the white noise model or in the density model, the goodness-of-fit problem (stands as explained above) was solved for different regularity classes (Hölder or Sobolev or Besov) associated with various geometries: pointwise, quadratic and supremum norm. For fixed smoothness of the unknown density (minimax context), there is a rich literature summed-up in [START_REF] Ingster | Asymptotically minimax hypothesis testing for nonparametric alternatives, i, ii, iii[END_REF] and in [START_REF] Ingster | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF]. Optimal test procedures include orthogonal projections, kernel estimates or χ 2 procedures. Goodness-of-fit tests with alternatives of variable smoothness into some given interval (adaptive context) were introduced by [START_REF] Spokoiny | Adaptive hypothesis testing using wavelets[END_REF] for the L 2 -norm in the Gaussian white noise model and generalized by [START_REF] Spokoiny | Adaptive ans spacially adaptive testing of a nonparametric hypothesis[END_REF] to L p -norms. [START_REF] Ingster | Adaptive chi-square tests[END_REF] proved that a collection of χ 2 tests attains the adaptive rates of goodness-of-fit tests in L 2 -norm as well as for the density model.

For sake of simplicity, we restrict ourselves to bi-dimensional data but there is no theoretical obstacle to generalize our results to higher dimensions. Suppose that we observe n i.i.d. copies (X i , Y i ) i∈I where I = {1, . . . , n} of (X, Y ). The random vector (X, Y ) is drawn from the distribution function H expressed through the copula C. Moreover, it is assumed that C has a copula density c with respect to the Lebesgue measure and F and G stand for the cdf's of X and Y respectively. From (X i , Y i ) i∈I , we are interested in studying the goodness-of-fit problem when the null is a composite hypothesis H 0 : c ∈ C Λ for a general class C Λ of parametrical copula densities. Since the alternative is defined from the quadratic distance, we propose a goodness-of-fit test based on wavelet estimation of an integrated functional of the copula density. Indeed, [START_REF] Genest | Estimating copula densities through wavelets[END_REF] and [START_REF] Florent Autin | Thresholding methods to estimate the copula density[END_REF] show that the wavelet methods are an efficient tool to estimate the copula densities since these latter have very specifics behaviors. Unfortunately no direct observations (F (X i ), G(Y i )) for i ∈ I are available since F and G are unknown, the test statistic is then built with pseudoobservations ( F (X i ), G(Y i )) i∈I : as usual in the copula context, the quantities of interest are rank-based statistics. We provide an auto-driven test procedure and we produce its rate when the alternative contains a regular constraint: since the procedure is based on wavelet methods, the linked functional classes are the Besov classes B s,p,q . We give results for p ≥ 2 (dense case) and s ≥ 1/2. The constraint s ≥ 1/2 is due to the fact that pseudodata are used and then a minimal regularity is required in order to pay no attention to substitute the direct data with the ranked data. Observe that [START_REF] Kerkyacharian | Regression in random design and warped wavelets[END_REF] have the same constraint in the univariate regression model when the design is random with unknown distribution. Next, we prove that our procedure is minimax (and adaptive) optimal by exhibiting the minimax adaptive rate. This one looks like the minimax rate but an extra log log term appears: we prove that this loss is the price to paid for adaptivity. To our knowledge, the proof of the adaptive lower bound in the multivariate density model when the null is composite has never been clearly written.

Next, we allocate a part to empirical studies. Simulation allows us to show that, when the theoretical framework is respected, the power qualities of our test procedures are good. We choose to make simulations starting from the parametrical copula families presented at the beginning of the introduction and which are the more common for applications. We compare our simulation results with those of [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF]. Then, we study a very well known sample of real life data of [START_REF] Frees | Understanding relationships using copulas[END_REF] consisting of the indemnity payment (LOSS) and the allocated loss adjustment expense (ALAE) for 1500 general liability claims. The most popular model for the copula is a Gumbel copula model with parameter θ = 1.45 (which may be estimated by inverting the Kendall's tau) given in Figure 3. Among other results, it is empirically shown that the Gumbel and the Gaussian copula models are acceptable while Student, Clayton or Frank models are rejected. Figure 3 gives a wavelet estimator of the copula density of (LOSS, ALEA) by the method explained in [START_REF] Florent Autin | Thresholding methods to estimate the copula density[END_REF]. Visually, fitting the unknown copula with the Gumbel model seems indeed to be the most appropriated.

The paper is organized as follows. In Section 2, we first provide a general description of orthonormal wavelet bases, focusing on the mathematical properties that are essential to the construction of the statistics that we consider. In Section 3, we provide the inference procedures: first, we explain how to estimate the square L 2 -norm of the copula density as given in [START_REF] Florent Autin | Thresholding methods to estimate the copula density[END_REF]. Center: Gumbel copula density with parameter θ = 1.45. Right: Gaussien copula density with parameter ρ = 0.48. and next we derive the procedure of goodness-of-fit. The theoretical part is exposed in Section 4: first, we state very precisely the test problem under consideration; we define the criterion allowing to measure the quality of test procedures and define the separating minimax rate. In Section 5, the main results are stated: our test procedure is shown to be optimal in the sense defined in the previous section. Section 6 is devoted to practical results with both simulated and real data. We conclude these parts with a discussion in Section 7. The proof of the upper bound is given in Section 8 while the proof of the lower bound is given in Section 9. Finally, all technical or computational lemmas which are not essential to understand the main proofs, are postponed in appendices.

Wavelet Setting

Wavelet expansion

In the univariate case, we consider a wavelet basis of L 2 ([0, 1]) (see [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF]). Let φ be the scaling function and let ψ be the same notation for the associated wavelet function and its usual modifications near the frontiers 0 and 1. They are chosen compactly supported on [0, L], L > 0. Let j in IN, k 1 in Z Z and for any univariate function Φ, set Φ j,k 1 (•) = 2 j/2 Φ(2 j •-k 1 ). In the sequel, we use wavelet expansions for bivariate functions and we keep the same notation as for the univariate case. Then, a bivariate wavelet basis is built as follows:

φ j,k (x, y) = φ j,k 1 (x)φ j,k 2 (y), ψ (1)
j,k (x, y) = φ j,k 1 (x)ψ j,k 2 (y), ψ (2) j,k (x, y) = ψ j,k 1 (x)φ j,k 2 (y), ψ (3)
j,k (x, y) = ψ j,k 1 (x)ψ j,k 2 (y),
where the subscript k = (k 1 , k 2 ) indicates the number of components of the functions φ j,k and ψ j,k . For a given j ∈ IN, the set

{φ j,k , ψ ǫ ℓ,k ′ , ℓ ≥ j, (k, k ′ ) ∈ Z Z 2 × Z Z 2 , ǫ = 1, 2, 3}
is an orthonormal basis of L 2 ([0, 1] 2 ) and the expansion of any real bivariate function Φ in L 2 ([0, 1] 2 ) is given by:

Φ(x, y) = k∈Z Z 2 A j,k φ j,k (x, y) + ∞ ℓ=j k∈Z Z 2 ǫ=1,2,3 B ǫ ℓ,k ψ ǫ ℓ,k (x, y),
where the scaling coefficients and the wavelet coefficients are

∀j ∈ IN, ∀k ∈ Z Z 2 , A j,k = [0,1] 2 Φφ j,k , B ǫ j,k = [0,1] 2 Φψ ǫ j,k .
The Parseval Equality immediately leads to the expansion of the square L 2 -norm of the function Φ:

Φ 2 = T j + B j , (1)
where the trend and the detail terms are respectively:

T j = k∈Z Z 2 (A j,k ) 2 and B j = ∞ ℓ=j k∈Z Z 2 3 ǫ=1 (B ǫ ℓ,k ) 2 . ( 2
)
Notice that, since the support of Φ is [0, 1] 2 , the sum over the indices k is finite: there are no more than (2 j + L) 2 terms in the sum (recall that L is the length of the support of φ).

In order to simplify the notations, the bounds of variation of k and ǫ in expansion of any Φ, are omitted in the sequel.

Besov Bodies and Besov spaces

Dealing with wavelet expansions, it is natural to consider Besov bodies as functional spaces since they are characterized in term of wavelet coefficients as follows.

Definition 1. For any s > 0, p ≥ 1 and any radius M > 0, a d-varied function Φ belongs to the ball b s,p,∞ (M ) of the Besov body b s,p,∞ if and only if its sequence of wavelet coefficients B ǫ j,k satisfies

∀j ∈ IN, k∈Z Z 2 3 ǫ=1 |B ǫ j,k | p < M 2 -j(s+d/2-d/p)p .
The Besov body b s,p,∞ coincides with the more standard Besov space B s,p,∞ when there exists an integer N strictly larger than s and such that the q-th moment of the wavelet ψ vanishes for any q = 0, . . . , N -1. It is possible to build univariate wavelets whose support is included in [0, 2N -1] satisfying this property for any choice of N (see the Daubechies wavelets).

In the sequel, we need to bound the detail term B j defined in [START_REF] Butucea | Nonparametric homogeneity tests[END_REF]. We use the following inequality

∀j ∈ IN, B j ≤ ∞ ℓ=j   k∈Z 2 3 ǫ=1 B ǫ ℓ,k p   2/p K 2 2j 1-2/p ,
where K is a positive constant depending on the supports of Φ and ψ. Assuming that the function Φ belongs to b s,p,∞ (M ) with s, p and M as in Definition 1, the following inequality holds

∀j ∈ IN, B j ≤ K 2 -2js , (3)
where K is a positive constant depending on the supports of Φ, ψ and on the radius M . When Φ is a copula density, K = M 2/p 3(L + 1) 2 1-2/p .

Statistical Procedures

Assuming that the copula density c belongs to L 2 ([0, 1] 2 ), we first explain the procedure to estimate the square

L 2 -norm of c θ = c 2 := [0,1] 2 c 2 ,
which is used to define the alternative of the goodness-of-fit test. The statistical methods depend on parameters (the level j for the estimation procedure and j and the critical value t j for the test procedure) which are discussed and determined in an optimal way in Section 5.

It is fundamental to Notice that, for any bivariate function Φ, one has

IE c [Φ(U, V )] = IE h [Φ(F (X), G(Y ))] , (4)
where h stands for the joint density of (X, Y ). This means in particular that the wavelet coefficients {c j,k , c ǫ ℓ,k , ℓ ≥ j, k ∈ Z Z 2 , ǫ = 1, 2, 3} of the copula density c on the wavelet basis {φ j,k , ψ ǫ ℓ,k , ℓ ≥ j, k ∈ Z Z 2 , ǫ = 1, 2, 3} are equal to the coefficients of the joint density h on the warped wavelet family

{φ j,k (F (•), G(•)), ψ ǫ ℓ,k (F (•), G(•)), ℓ ≥ j, k ∈ Z Z 2 , ǫ = 1, 2, 3}.
The statistical procedures are based on the wavelet expansion of the copula density c, for which the wavelet coefficients have to be estimated.

Procedures to estimate θ

Let J be a subset of IN and consider a given j in J. Motivated by the wavelet expansion (1), we propose to estimate θ with an estimator of the trend T j omitting the detail term B j . Using the orthonormality property of the wavelet basis, it leads to estimate the square of the coefficients of the copula density on the scaling function. As usual, a U -statistic associated with the empirical coefficients is used in order to remove the bias terms. Due to (4), we first consider the following family of statistics { T j , j ∈ J} defined by

T j = k θ j,k ,
where θ j,k is the following U -statistic

θ j,k = 1 n(n -1) n i 1 ,i 2 =1 i 1 =i 2 φ j,k (F (X i 1 ), G(Y i 1 )) φ j,k (F (X i 2 ), G(Y i 2 )) .
Since no direct observation (F (X i ), G(Y i )) is usually available, it is replaced in θ j,k by the pseudo observation ( F (X i ), G(Y i )), where F , G denote some estimator of the margins. To preserve the independence given by the observations, we split the initial sample

(X i , Y i ) i∈I into disjoint samples (X i , Y i ) i∈I 1 and (X i , Y i ) i∈I 2 with I 2 ∪ I 1 = I, I 2 ∩ I 1 = ∅,
and whose size is n 1 and n 2 respectively. The sub-sample with indices in I 1 is used to estimate the marginal distributions and the second one with indices in I 2 is devoted to the computation of the U -statistic. We consider the usual empirical distribution functions:

F (x) = 1 n 1 i∈I 1 1I {X i ≤x} and G(y) = 1 n 1 i∈I 1 1I {Y i ≤y} .
It leads to the family { T j , j ∈ J} of estimators of θ

T j = k θ j,k , with θ j,k = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 φ j,k R i 1 n 1 , S i 1 n 1 φ j,k R i 2 n 1 , S i 2 n 1 ,
where R p = n 1 F (X p ) and S p = n 1 G(Y p ), p ∈ I 1 , could be viewed as estimates of the rank statistics of X p and Y p respectively.

Test Procedures

In this part, we consider a family of known bivariate copula densities

C Λ = {c λ , λ ∈ Λ} indexed by a parameter λ varying in a given set Λ ⊂ IR d Λ , d Λ ∈ IN * . From the observations (X i , Y i ) i∈I
, our aim is to test the goodness-of-fit between any c λ and a copula density c, which is enough distant in the L 2 -norm, from the parametric family C Λ . Acting as in paragraph 3.1, we estimate the square L 2 -norm between c and a fixed element c λ lying in the family C Λ by

T j (λ) = k θ j,k (λ), (5)
for

θ j,k (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 φ j,k R i 1 n 1 , S i 1 n 1 -c j,k (λ) × φ j,k R i 2 n 1 , S i 2 n 1 -c j,k (λ) ,
where {c j,k (λ), k ∈ Z Z 2 , j ∈ IN} denote the known scaling coefficients of the target copula density c λ . Notice that, if direct observations (F (X i ), G(Y i )) i∈I would be available, the appropriate test statistic T j (λ) would be

T j (λ) = k θ j,k (λ),
where

θ j,k (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k (λ)) × (φ j,k (F (X i 2 ), G(Y i 2 )) -c j,k (λ)) .
Now we are ready to build the test procedures. Let us give a set of indices J and a set of critical values {t j , j ∈ J} and define {D Λ j , j ∈ J}, the family of test statistics

D Λ j = 1I inf λ∈Λ T j (λ) > t j ,
allowing to test if c belongs to the parametric family C Λ = {c λ , λ ∈ Λ}. Notice that Λ = {λ 0 } leads to the single null hypothesis H 0 : c = c λ 0 . We are also interested in building auto-driven procedures by considering all the tests in the family

D Λ = max j∈J D Λ j = 1I max j∈J ( inf λ∈Λ T j (λ) -t j ) > 0 . (6)
The sequence of parameters t j of the method are determined in an optimal way in Section 5. We explain in Section 4 what "optimal way" means in giving a presentation of the minimax theory for our framework.

Minimax Theory

We adopt the minimax point of view to solve the problem of hypothesis testing, initiated by [START_REF] Ingster | On minimax nonparametric detection of a signal in gaussian white noise[END_REF] in Gaussian white noise. A review of results obtained in problems of minimax hypothesis testing is available in [START_REF] Ingster | Asymptotically minimax hypothesis testing for nonparametric alternatives, i, ii, iii[END_REF] and [START_REF] Ingster | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF]. Let us describe this approach.

Minimax hypothesis testing Problem

As in the previous section, we consider C Λ = {c λ , λ ∈ Λ} a given functional class of copula densities. For any given τ = (s, p, M ), with s > 0, p ≥ 1, M > 0, the following statistical problem of hypothesis testing is considered,

H 0 : c = c λ ∈ C Λ against H 1 : c ∈ Γ(v n (τ )), (7)
with

Γ(v n (τ )) = b s,p,∞ (M ) ∩ c : inf c λ ∈C Λ c -c λ ≥ v n (τ ) ,
where b s,p,∞ (M ) is the ball of radius M of the Besov body b s,p,∞ defined in Definition 1 and v n (τ ) is a sequence of positive numbers, depending on τ and decreasing to zero as n goes to infinity. Recall that g denotes the L 2 -norm of any function g in L 2 ([0, 1] 2 ). Observe that the functional class Γ(v n (τ )), which determines the alternative H 1 , is characterized by three parameters: the regularity class b s,p,∞ where the copula density is supposed to belong, the L 2 -norm which is the geometrical tool measuring the distance between both hypotheses, and the sequence v n (τ ).

According to the principle of the minimaxity, the regularity space and the loss function are chosen by the statistician. Notice that the parameter τ could be known or unknown. Obviously, our aim is to consider tests which are able to detect alternatives defined with sequences v n (τ ) as small as possible. It can be shown ( [START_REF] Ingster | Asymptotically minimax hypothesis testing for nonparametric alternatives, i, ii, iii[END_REF]) that v n (τ ) cannot be chosen in an arbitrary way: indeed, if v n (τ ) is too small, then H 0 and H 1 cannot be distinguished with a given error α ∈ (0, 1). Therefore, solving hypothesis testing problems via the minimax approach consists in determining the smallest sequence v n (τ ) for which such a test is still possible and to indicate the corresponding test functions. The smallest sequence v n (τ ) is called the minimax rate of testing. Let D n be a test statistic i.e. an arbitrary function with possible values 0, 1, measurable with respect to (X i , Y i ) i∈I and such that we accept H 0 if D n = 0 and we reject it if D n = 1.

Definition 2. Assuming τ to be known, the sequence v n (τ ) is the minimax rate of testing H 0 versus H 1 if relations ( 8) and ( 9) are fulfilled:

• for any given α 1 ∈ (0, 1), there exists a > 0 such that

lim n→+∞ inf Dn sup c λ ∈C Λ IP λ (D n = 1) + sup c∈Γ(a vn(τ )) IP c (D n = 0) ≥ α 1 , (8)
where the infimum is taken over any test statistic D n ,

• there exists a sequence of test statistics (D ⋆ n ) n for which for any given α 2 in (0, 1), it exists A > 0 such that

lim n→+∞ sup c λ ∈C Λ IP λ (D ⋆ n = 1) + sup c∈Γ(A vn(τ )) IP c (D ⋆ n = 0) ≤ α 2 , (9)
where IP c , respectively IP λ denotes the distribution function associated with the copula density c, respectively with c λ .

Adaptation

Nevertheless, since the copula function itself is unknown, the a priori knowledge on τ could appear unrealistic. Therefore, the purpose of this paper is to solve the previous problem of test in an adaptive framework i.e. in supposing that τ = (s, p, M ) is unknown but varying in a known set S. Comparing the adaptive case with the non-adaptive case, it has been proved in different frameworks that a loss of efficiency in the rate of testing is unavoidable (see for instance [START_REF] Spokoiny | Adaptive hypothesis testing using wavelets[END_REF], [START_REF] Gayraud | Adaptive minimax testing in the discrete regression scheme[END_REF]). This loss is expressed as t n , a positive constant or a sequence of positive numbers increasing to infinity with n (as slow as possible), which appears in the rate of testing v nt -1 n (τ ). Similarly to the minimax rate of testing, we define the adaptive minimax rate of testing as follows.

Definition 3. The sequence v nt -1 n (τ ) is the adaptive minimax rate of testing if relations [START_REF] Gayraud | Adaptive minimax testing in the discrete regression scheme[END_REF] and [START_REF] Genest | Comment on "Understanding relationships using copulas[END_REF] are satisfied

• for any given α 1 ∈ (0, 1), there exists a > 0 such that

lim n→+∞ inf Dn   sup c λ ∈C Λ IP λ (D n = 1) + sup τ ∈S sup c∈Γ(a v nt -1 n (τ )) IP c (D n = 0)   ≥ α 1 , ( 10
)
where the infimum is taken over any test statistic D n ,

• there exists a sequence of universal test statistics D ⋆ n (free of τ ) such that, for any given α 2 in (0, 1), there exists A > 0 such that

lim n→+∞   sup c λ ∈C Λ IP λ (D ⋆ n = 1) + sup τ ∈S sup c∈Γ(A v nt -1 n (τ )) IP c (D ⋆ n = 0)   ≤ α 2 ( 11
)
where t n is either a positive constant or a sequence of positive numbers increasing to infinity with n as slow as possible.

Notice that relations [START_REF] Gayraud | Adaptive minimax testing in the discrete regression scheme[END_REF] and ( 11) (instead of relations ( 8) and ( 9)) mean that the minimax rate of testing v n (τ ) is contaminated by the term t n in the adaptive setting. Observe that the same phenomenon is observed in the estimation problem where an extra logarithm term tn = log(n) has often (but not always) to be paid for the adaptation.

Main results

In this section, we focus on test problems for which the parametric family C Λ is included in some b s Λ ,p Λ ,∞ (M Λ ) where s Λ > 0, p Λ ≥ 1 and M Λ > 0 are known.

Our theoretical results concern the minimax resolution of the problem of hypothesis testing defined in [START_REF] Deheuvels | A nonparametric test of independence[END_REF] in an adaptive framework. Theorem 1 states the result of the lower bound (see relation [START_REF] Gayraud | Adaptive minimax testing in the discrete regression scheme[END_REF]). Then, Theorem 2 exhibits the rate achieved by the test procedure proposed in Section 3 (see relation [START_REF] Genest | Comment on "Understanding relationships using copulas[END_REF]). Comparing the rate of our procedure with the fastest rate given in Theorem 1 leads to Theorem 3 establishing the optimality of our procedure. First, let us state the assumption which gives a control of the complexity of C Λ .

• A0: the set Λ is compact in IR d Λ and sup (x,y)∈[0,1] 2 |c λ (x, y) -c λ ′ (x, y)| ≤ Q λ -λ ′ ν IR d Λ , ∀λ, λ ′ ∈ Λ,
where ν is a positive real, Q is a positive constant and • IR d Λ denotes the Euclidean norm in IR d Λ .

Lower Bound

As it is usual for composite null hypotheses, the result of the lower bound requires the existence of a particular density c λ 0 ∈ C Λ (see assumption AInf below) in order to construct a randomized class of functions which must be included in the alternatives.

• AInf: there exists a parameter λ 0 in Λ such that

∀(u, v) ∈ [0, 1] 2 , c λ 0 (u, v) > m, with m > 0.
Theorem 1. Suppose that S defined by

S = {τ = (s, p, M ), s ≥ 1/2, p ≥ 2, M > 0 : s -2/p ≤ s Λ -2/p Λ , M Λ ≤ M } ( 12
)
is nontrivial (see [START_REF] Spokoiny | Adaptive hypothesis testing using wavelets[END_REF]), which means that there exist p ≥ 2, M > 0 and 0 < s min < s max such that ∀s ∈ [s min , s max ], (s, p, M ) ∈ S and assume that A0 and AInf hold. Set

v nt -1 n (τ ) = (nt -1 n ) -2s/(4s+2) with t n = log(log(n)).
Then, it exists a positive constant a such that

lim n→+∞   inf Dn {sup λ∈Λ IP λ (D n = 1) + sup τ ∈S sup c∈Γ(a v nt -1 n (τ )) IP c (D n = 0)}   = 1, ( 13
)
where the infimum is taken over any test function D n .

Upper Bound

Theorem 2 deals with relation [START_REF] Genest | Comment on "Understanding relationships using copulas[END_REF] which holds for the test statistic D Λ defined by relation [START_REF] Deheuvels | A kolmogorov-smirnov type test for independence and multivariate samples[END_REF] as soon as the parameters of the methods are chosen as follows. The set J = {⌊j 0 ⌋, . . . , ⌊j ∞ ⌋} is determined by

2 j 0 = log(n 2 ) log(n 1 ), 2 j∞ = n 2 log(n 2 ) 1/2 ∧ n 1 log(n 1 ) 1/2-1/2q , ( 14
)
where q is the order of differentiability of the scaling function φ. The critical values satisfy

∀j ∈ J, t j = 3µ 2 j n 2 log log(n 2 ), ( 15
)
where µ is a positive constant such that µ > 2K g K 1 , and K g and K 1 are positive constants depending on φ ∞ , c ∞ , c λ ∞ and the length of the support of φ (see Lemma 3).

Theorem 2. Let us choose n 1 = π n and n 2 = (1π)n for some π in (0, 1). Assume that the scaling function φ is continuously q-differentiable for

q ≥   1 - log n 2 log(n 2 ) log n 1 log(n 1 )   -1 .
Moreover assume that any density c under the alternatives or any c λ under the null are uniformly bounded. Then, the test statistic D Λ defined by ( 6) is such that

lim n 1 ∧n 2 →+∞ sup c λ ∈C Λ IP λ (D Λ = 1) = 0. ( 16
)
Assume that A0 holds, then there exists a positive constant A such that

lim n 1 ∧n 2 →+∞ sup τ ∈S sup c∈Γ(Av nt -1 n (τ )) IP c (D Λ = 0) = 0, ( 17
)
where v nt -1 n (τ ) = (n 2 t -1 n 2 ) -2s/(4s+2) and t n 2 = log(log(n 2 )).
Relation [START_REF] Genest | Comment on "Understanding relationships using copulas[END_REF] of the upper bound holds since both relations ( 16) and ( 17) are satisfied. Notice also that relation [START_REF] Ingster | On minimax nonparametric detection of a signal in gaussian white noise[END_REF] indicates that the test statistic D Λ is asymptotically of any level in (0, 1).

Optimality

As a corollary of Theorem 1 and Theorem 2, we obtain Theorem 3. Under the assumptions of Theorem 1 and Theorem 2, our test procedure defined by Relation ( 6) is adaptive optimal over the range of parameters τ ∈ S where S is defined by equation [START_REF] Genest | Estimating copula densities through wavelets[END_REF].

Practical results

The purpose of this section is to provide several examples to investigate the performances of the test procedure presented in Section 3. This part is not exactly an illustration of the theoretical part since it does not focus on the separating rate between the alternative and the null hypothesis, but it is devoted to the study of our test procedure from a risk point of view. Note also that we do not use exactly the theoretical procedure described in the previous section. As usual for practical purpose, we replace theoretical quantities by more adapted quantities obtained with resampling methods. In the first part, we fix the test level α = 5% and we study the empirical power function. In the second part, we present an application to some economical series.

Methodology

On the contrary to the estimation problem, a smooth wavelet is not needed. The test statistic is then computed with the Haar wavelet since it has a small support and then it leads to a fast computation time. The critical value of the test is determined with bootstrap methods: the standard deviation of the test statistic is computed thanks to N boot = 20 resampling. The size of the simulated samples is n = 2048 which is reasonable for bi-dimensional problems in an asymptotic context. For the real life data example, the number of data is around n = 4000. For the simulation part, the empirical level of the test is derived from N M C = 500 replications for each test problem.

Simulations

The setup of our simulations is closely related to the work of [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF], except that they consider small samples (of size 150) since their test procedures are based on the empirical copula distribution (and thus generate parametrical rates). To explore various degrees of dependance, three values of Kendal's tau are considered, namely τ = 0.25, 0.50, 0.75 for the following copula families: Clayton, Gumbel, Frank, Normal and Student with four degrees of freedom (df). Calculations are made with the MatLab Sofware. The results of the simulations are presented in Table 1. For an easier reading, the estimated standard errors of the empirical powers are presented in italics. Furthermore, for each testing problem we highlighted the estimated errors of the first type (estimators of α = 0.05) using bold characters. In brackets, we give the results obtained by [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF] with their test procedures, denoted CvM and built on rank-based versions of the familiar Cramér-von Mises statistics. It would be also possible, if one is interested in, to compare with the different test procedures (based on the empirical copula distribution) proposed also by [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF].

Let us now summarize the conclusions made from the simulation results.

• Our test is degenerated: we almost always accept H 0 (when H 0 is true) while the procedure of Genest et al. [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF] produces an excellent estimation of the prescribed level α. It is a characteristic of the adaptive minimax procedures.

• For small level of dependence τ = 0.25, our procedure is very competitive and produces (almost) always a better empirical power than the CvM test. The results are spectacular when the fit c λ 0 is a Student(4).

• When a large Kendal's tau is considered, our procedure fails when the data are issued from a Clayton copula density. The procedure is not available to recognize a structure of dependence modeled with a Clayton.

• The improvment of our results with respect to the CvM test is decreasing with the Kendal's tau. The CvM test becomes better when the tau is increasing whereas for us it is the opposite.

In conclusion, we recommend the use of our test procedures when the Kendal's tau is not too large since it seems to outperform the existing procedures based on the copula distribution. This situation corresponds to our theoretical setup related to the functional spaces in which the unknown copula density is supposed to live. Unfortunately, the practical results do not give hope for using this procedure when the copula densities present high peaks (as it is the case for the Clayton copula density with a large tau).

Real data

We present now an application to real data of our test procedure. The level of each test (with simple null hypothesis or multivariate null hypothesis) is α = 5%. To obtain the empirical level, N = 50 replications of our procedure computed with the half of the available data (chosen randomly) is used. Table 2 gives the empirical probability to reject the null hypothesis and the final decision. "Yes" means that we accept that the structure of dependence belongs to the considered family and "No" that we reject the fitting.

We consider the data of [START_REF] Frees | Understanding relationships using copulas[END_REF], which were also analyzed by [START_REF] Genest | Comment on "Understanding relationships using copulas[END_REF], [START_REF] Klugman | Fitting bivariate loss distributions with copulas[END_REF], [START_REF] Chen | Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection[END_REF] and [START_REF] Genest | Goodness-of-fit procedures for copula models based on the integral probability transformation[END_REF], among others. The data consist of the indemnity payment (LOSS) and the allocated loss adjustment expense (ALAE) for 1466 general liability claims.

We consider the following test problems:

H 0 : c ∈ C Λ
where the parametrical family C Λ is described in Table 2. Since the Kendall's tau computed with the sample is τ = 0.31, we choose an adapted grid of parameters for each parametrical family of copula densities. Next, assuming that the density copula of the data belongs to a fixed parametric family, we estimate the parameter λ

• by λ in inverting the Kendall's tau (third part of Table 2 where

H 0 : c = c b λ ).
• by λ in minimizing the average square error (ASE) computed thanks to the benchmark given in Figure 3 (fourth part of Table 2 where H 0 : c = c e λ ). For information, we give the relative ASE computed with c e λ into brackets.

The various authors who analyzed this data set concluded that the Gumbel copula provides an adequate representation of the underlying dependence structure. The Gumbel parametric family of extreme-value copulas captures the fact that almost all large indemnity payments generate important adjustment expenses (e.g., investigation and legal costs) while the effort invested in the treatment of a small claim is more variable. Accordingly, the copula exhibits positive but asymmetric dependence. Confirming this result, the adaptive method of estimation proposed by [START_REF] Florent Autin | Thresholding methods to estimate the copula density[END_REF] provides a benchmark (see Figure 3) for the copula density associated with the data.

Discussion

The paper is mainly devoted to construct an optimal procedure for solving a general nonparametric problem of test: both hypotheses are composite, very general parametric family could be considered under the null. Our procedure is proved asymptotically to be adaptive minimax and the minimax separating rate is exhibited over a range of Besov balls.

Thanks to the simulations and a application to real data, our procedure seems to be competitive on the power point of view even if the setting of test under consideration is, in the simulation study, clearly parametric.

It is worthwhile to point out that the copula model requires more regularity (than the usual density model) since the approximation due to the rank-based statistics needs to be accurate enough (see Lemma 4).

One must notice that only copulas densities belonging to dense Besov spaces (i.e. defined with a parameter p larger than 2) are under consideration in this paper although several copula densities with a strong dependence structure belong to sparse Besov spaces (i.e. defined with a parameter p smaller than 2). As it is illustrated in the simulation study, our test procedure fails for the Clayton copula density with large parameters. This density is suspected to belong to a sparse Besov ball. The study of sparse Besov balls would require the determination of a new test strategy which would lead to another minimax rate of testing: these objectives are beyond those of the present paper and will be explored in a further work since the set of copulas densities contains a number of sparse functions. For sparse Besov balls and in the white noise model for testing the existence of the signal, [START_REF] Lepskii | Minimax nonparametric hypothesis testing: the case of inhomogeneous alternative[END_REF] proved that the minimax testing rate in the sparse and the dense cases is different. They also proved that it is possible to built an adaptive minimax (non linear) procedure of test for the sparse case.

A very close problem is the sample comparison test (problem with two samples). It could be interesting to test if the structure of dependence between a couple of variables V 1 = (X, Y ) is the same as for another couple V 2 = (Z, T ). This problem of tests could be stated as follows:

H 0 : c V 1 = c V 2 against H 1 : (c V 1 , c V 2 ) ∈ Γ(v n (τ )), with Γ(v n ) = {c V 1 ∈ b s 1 ,p 1 ,∞ (M 1 )} ∩ {c V 2 ∈ b s 2 ,p 2 ,∞ (M 2 )} ∩ {(c V 1 , c V 2 ) : c V 1 -c V 2 ≥ v n .}
where v n is the separating rate of both hypotheses. In an analogous way as in Section 3, the rule for the comparison test would be

D = 1I max j∈J ( k θ j,k -t j ) > 0 with θ j,k = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 φ j,k R X i 1 n 1 , R Y i 1 n 1 -φ j,k R Z i 1 n 1 , R T i 1 n 1 × φ j,k R X i 2 n 1 , R Y i 2 n 1 -φ j,k R Z i 2 n 1 , R T i 2 n 1 ,
where R X , R Y , R Z , R T are the rank statistics associated with X, Y, Z, T . Using the same tools as in [START_REF] Butucea | Nonparametric homogeneity tests[END_REF], in which the homogeneity in law of the both samples is studied, it is possible to prove that this test is adaptive optimal and that the minimax rate of testing is

v n = n log(log(n 2 )) -2(s 1 ∧s 2 )/(4(s 1 ∧s 2 )+2)
.

Obviously, all these test procedures could be used in the multivariate framework (d > 2), but as usual in the nonparametric context, it will provide slower minimax rates of testing.

Proof of Theorem 2

Recall that for any given λ ∈ Λ, IP λ (respectively IP c ) denote the distribution associated with density c λ , respectively with c. In the same spirit, denote also IE λ and Var λ (respectively IE c and Var c ) the expectation and the variance with respect to IP λ , respectively to IP c . When no index appears in IE or in IP it means that the underlying distribution is either IP c or IP λ .

Expansion of the statistics of interest

Fix a level j in J. For the test problem, the statistic of interest T j (λ) (for λ ∈ Λ) defined in ( 5) is an estimate of

T j (λ) = k θ j,k (λ) = k (c j,k -c j,k (λ)) 2 ,
which is the quantity that we need to detect under the alternative. It would be useful to expand the statistic T j (λ) as follows

T j (λ) = 2T ⋄ j (λ) + T j ♥ + T ♠ j + 2T ♣ j (λ) + T j (λ) (18) = 2 k θ ⋄ j,k (λ) + k θ ♥ j,k + k θ ♠ j,k + 2 k θ ♣ j,k (λ) + k θ j,k (λ),
where

θ ♥ j,k = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k ) ×(φ j,k (F (X i 2 ), G(Y i 2 )) -c j,k ) θ ♠ j,k = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 φ j,k R i 1 n 1 , S i 1 n 1 -φ j,k (F (X i 1 ), G(Y i 1 )) × φ j,k R i 2 n 1 , S i 2 n 1 -φ j,k (F (X i 2 ), G(Y i 2 )) θ ♣ j,k (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 φ j,k R i 1 n 1 , S i 1 n 1 -φ j,k (F (X i 1 ), G(Y i 1 )) × (φ j,k (F (X i 2 ), G(Y i 2 )) -c j,k (λ)) θ ⋄ j,k (λ) = 1 n 2 i 1 ∈I 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k ) (c j,k -c j,k (λ)).
The sequence {c j,k } j,k denotes the unknown scaling coefficients of the unknown copula density c. Recall that

T j (λ) = k θ j,k (λ), with θ j,k (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k (λ)) × (φ j,k (F (X i 2 ), G(Y i 2 )) -c j,k (λ)) .
The following lemma gives some evaluation for the first moments of each statistic of interest.

Lemma 1. Let q be a positive integer and assume that φ is continuously q-differentiable.

Let j be a level smaller than j ∞ defined in [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF]. Then, it exists some positive constant κ which may depend on φ, c ∞ , c λ ∞ and M such that

IE T j (λ) = T j (λ) and Var T j (λ) ≤ κ 2 j n 2 2 + 2 j n 2 T j (λ) IE|T ♠ j | ≤ κ log(n 1 ) n 1 IE c |T ♣ j (λ)| ≤ κ log(n 1 ) n 1 T j (λ) 1/2 and IE λ (T ♣ j (λ)) 2 ≤ κ 2 j log(n 1 ) n 2 n 1 .
Using the Bernstein Inequality, we establish the following bound for the deviation of the statistic T ⋄ j (λ) under the alternative. The proof is postponed to Appendix B.

Lemma 2. For any level j, for all x > 0

IP c |T ⋄ j (λ)| ≥ x ≤ exp -K( n 2 2 x 2 n 2 T j (λ) + n 2 x2 j T j (λ) 1/2 ) ,
where K is a positive constant depending on L, φ ∞ and c ∞ .

Using a result from [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF], we establish the following bound for the deviation of the U -statistics T j (λ) and T ♥ j . The proof is postponed to Appendix C.

Lemma 3. For any level j, as soon as x ≥ 2 j n -1 2 log(log(n 2 )), for all µ > 0,

IP λ | T j (λ)| > µx + IP c |T ♥ j | > µx ≤ K g (log(n 2 )) -δ
for any positive δ ≤ µ 2 (K g K 1 ) -1 , where K g is an universal positive constant given in [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF] and K 1 is a positive constant depending on L, φ ∞ and either c λ ∞ or c ∞ depending on the underlying distribution i.e. either IP λ or IP c .

Proof of Relation (16) (First type error)

Let us fix λ ∈ Λ and set

p λ = IP λ max j∈J inf λ ′ ∈Λ T j (λ ′ ) -t j > 0 .
Notice that under the null T ⋄ j (λ) = T j (λ) = 0 and T j ♥ = T j (λ).

Using expansion [START_REF] Ingster | Adaptive chi-square tests[END_REF], we get

p λ ≤ j∈J IP λ inf λ ′ ∈Λ T j (λ ′ ) > t j ≤ j∈J IP λ T j (λ) > t j ≤ j∈J IP λ | T j (λ)| > t j 3 + IP λ |T ♠ j | > t j 3 + IP λ |T ♣ j (λ)| > t j 3
Due to Lemma 1 and using Markov Inequality, we obtain

p λ ≤ j∈J IP λ | T j (λ)| > t j 3 + j∈J IE λ |T ♠ j | (t j /3) + IE λ (T ♣ j (λ)) 2 (t j /3) 2 ≤ j∈J IP λ | T j (λ)| > t j 3 +K j∈J (t j /3) -1 log(n 1 ) n 1 + (t j /3) -2 2 j log(n 1 ) n 1 n 2 .
Notice that T j (λ) is centered under IP λ , then applying Lemma 3, where t j is t j = 3µ 2 j n -1 2 log(log(n 2 )), the constant µ is defined in [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF] and since card(J) ≤ log(n 2 ), one obtains

p λ ≤ K g card(J) (log(n 2 )) -δ + Kcard(J)2 -j 0 n 2 log(n 1 ) n 1 log log(n 2 ) +Kcard(J)2 -j 0 log(n 1 )n 2 2 n 2 n 1 log log(n 2 ) ≤ K g (log(n 2 )) 1-δ + K2 -j 0 log(n 1 ) log(n 2 ) log log(n 2 ) ,
where the last inequality holds since δ satisfies δ ≤ µ 2 (2K g K 1 ) -1 (see Lemma 3). Since µ is such that µ > 2K g K 1 , relation ( 16) is proved if one takes δ = µ 2 (2K g K 1 ) -1 .

Proof of Relation (17) (Second type error)

Let us fix τ ∈ S and c ∈ Γ(Av nt -1 n (τ )) and set

p c = IP c max j∈J inf λ∈Λ T j (λ) -t j ≤ 0 .
Using the expansion (18), we get, for any

j ⋆ ∈ J p c ≤ IP c inf λ 2T ⋄ j ⋆ (λ) + T j (λ) + T ♥ j ⋆ + T ♠ j ⋆ + 2T ♣ j ⋆ (λ) ≤ t j ⋆ ≤ IP c inf λ 2T ⋄ j ⋆ (λ) + T j ⋆ (λ) ≤ 2t j ⋆ +IP c T ♥ j ⋆ + T ♠ j ⋆ + 2 inf λ T ♣ j ⋆ (λ) ≥ t j ⋆ ≤ IP c inf λ 2T ⋄ j ⋆ (λ) + T j ⋆ (λ) ≤ 2t j ⋆ + IP c T ♥ j ⋆ ≥ t j ⋆ /3 +IP c T ♠ j ⋆ ≥ t j ⋆ /3 + IP c inf λ T ♣ j ⋆ (λ) ≥ t j ⋆ /6 = p c1 (j ⋆ ) + p c2 (j ⋆ ) + p c3 (j ⋆ ) + p c4 (j ⋆ ). ( 19
)
Let us explain how j ⋆ is chosen. From the wavelet expansion (1) and Lemma 1, one has

IE c T j ⋆ (λ) = T j ⋆ (λ) = (c -c λ ) 2 -B j ⋆ (λ),
where T j ⋆ , B j ⋆ are defined in (2) and t ⋆ j is the critical value given in [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF]. Since c is in

Γ(Av nt -1 n (τ )) and c λ lies in b s Λ ,p Λ ,∞ (M Λ ) ⊂ b s,p,∞ (M ), the function (c-c λ ) is in b s,p,∞ (M ). We can choose j ⋆ such that 2 j ⋆ = K 3µ n 2 log log(n 2 ) 1/(2s+1)
, which is possible due to our choice of j ∞ and because s ≥ 1/2; the constant K appears in [START_REF] Chen | Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection[END_REF]. It implies that

B j ⋆ ≤ t j ⋆ since B j ⋆ ≤ K2 -2j ⋆ s (see Inequality (3)). Next, since c ∈ Γ(Av nt -1 n (τ )), one has (c -c λ ′ ) 2 ≥ A 2 (v nt -1 n (τ )) 2 for all λ ′ ∈ Λ. Focusing on rates v nt -1 n (τ ) combined with positive constant A which satisfy 4t j ⋆ ≤ (Av nt -1 n (τ )) 2 , one obtains t j ⋆ IE c T j ⋆ (λ) = t j ⋆ T j ⋆ (λ) ≤ 1/3. ( 20
)
Coming back to the evaluation of the probability terms (see relation ( 19)), we first consider p c1 (j ⋆ ). Consider an η-net Λ η on the set Λ that is for any λ in Λ, denote λ the closest (in the Euclidean sense) element in Λ η to λ (closer than η). Due to assumption A0, let us prove that for any j ∈ J, T ⋄ j ( λ) + T j ( λ) is close to T ⋄ j (λ) + T j (λ):

|T ⋄ j ( λ) -T ⋄ j (λ)| = k   1 n 2 i 1 ∈I 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k ) c j,k ( λ) -c j,k (λ)   ≤ k c j,k ( λ) -c j,k (λ) 2 1/2 ×   k   1 n 2 i 1 ∈I 2 (φ j,k (F (X i 1 ), G(Y i 1 )) -c j,k )   2   1/2 ≤ Qη ν 2 2j (2 φ ∞ + c ∞ 2 -4j ) 1/2 .
In the same way, one has,

|T j (λ) -T j ( λ)| ≤ κQη ν 2max( c , c λ ) + Qη ν .
Choosing η = n -b with bν > 1, then by [START_REF] Kerkyacharian | Regression in random design and warped wavelets[END_REF] and applying Lemma 2, we get

p c1 (j ⋆ ) ≤ λ∈Λ n -b IP c 2T ⋄ j ⋆ (λ) + T j ⋆ (λ) ≤ 2t j ⋆ ≤ λ∈Λ n -b IP c T ⋄ j ⋆ (λ) ≤ -T j ⋆ (λ)/6 ≤ λ∈Λ n -b exp -K n 2 2 2j ⋆ T j ⋆ (λ) ∧ n 2 2 j ⋆ T j ⋆ (λ) 1/2 ≤ λ∈Λ n -b exp -K 2 n 2 t j ⋆ ∧ n 2 2 j ⋆ t 1/2 j ⋆ ≤ ( D(Λ) n -b ) d Λ exp -K 3 2 j ⋆ (log log(n 2 )) 1/2 ∧ n 2 2 j ⋆ /2 1/2 (log log(n 2 )) 1/4 (21)
where D(Λ) is the diameter of Λ and K, K 2 and K 3 are positive constants. Both terms behind the minus sign in the exponential of the right hand side of the last inequality tend to infinity with a power of n since s > 1/2. This implies that p c1 (j ⋆ ) goes to zero as n goes to infinity. Now, it remains to verify that p c2 (j ⋆ ), p c3 (j ⋆ ) and p c4 (j ⋆ ) are going to zero as n 1 ∧ n 2 goes to infinity. Using again the bound [START_REF] Kerkyacharian | Regression in random design and warped wavelets[END_REF], Lemma 3 for some positive δ, Lemma 1 and the definition of the critical value [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF], one gets

p c2 (j ⋆ ) + p c3 (j ⋆ ) + p c4 (j ⋆ ) ≤ IP c T ♥ j ⋆ ≥ t j ⋆ /3 + +IP c T ♠ j ⋆ ≥ t j ⋆ /3 + IP c T ♣ j ⋆ (λ) ≥ t j ⋆ /6 ≤ K g (log(n 2 )) -δ + 9 IE c |T ♠ j ⋆ (λ)| 2 t 2 j ⋆ + 6 IE c |T ♣ j ⋆ | t j ⋆ ≤ K g (log(n 2 )) -δ + 9κ 2 j ⋆ log(n 1 ) n 1 n 2 t 2 j ⋆ + 6κ log(n 1 ) n 1 t j ⋆ ≤ K g (log(n 2 )) -δ + 6κ 2 -j ⋆ n 2 n 1 log(n 1 ) log log(n 2 ) , (22)
which tends to zero with our choice of j ⋆ and where κ is the positive constant appearing in Lemma 1. Inequalities ( 21) and ( 22) entail that the right hand side of ( 19) is less than any α ∈ (0, 1) as n is large enough. To finish the proof, observe that the choice of v nt -1 n (τ ) is driven by the fact that it corresponds to the smallest sequence such that 4t ⋆ j ≤ (Av nt -1 n (τ )) 2 , which leads to

v nt -1 n (τ ) ≥ 2 j ⋆ µ log(log n 2 ) n 2 1/2 ≥ n 2 log log(n 2 )
2s/(4s+2)

.

Proof of Theorem 1

Without loss of generality, we suppose that the support of the scaling function φ and its associated wavelet function ψ is [0, 1]. Moreover recall that 1 0 ψ ǫ = 0. Let us give some a > 0 which must be small enough.

Discretisation of S

For any given τ = (s, p, M ) ∈ S, denote by j(τ ) the level

2 j(τ ) = (nt -1 n ) 2/(4s+2)
and define s j the solution of the equation j = j(s j , p, M ) for any resolution level j ∈ J = {j s max , . . . , j s min } ⊂ {j 0 , . . . , j ∞ } with j s max = ⌊j(s max , p, M )⌋ and j s min = ⌊j(s min , p, M )⌋.

Consider now the set S n = {τ j = (s j , p, M ), j ∈ J } which appears as a discretisation version of a subset of S whose cardinality is of order O(log(n)).

Prior and parametric family included in the alternatives

For any s j ∈ S n , define a prior π j which is concentrated on the class of the random functions

c j (u, v) = c λ 0 (u, v) + k 3 ǫ=1 δ k u j (n)ψ ǫ j,k (u, v),
where c λ 0 is defined in assumption AInf and

P (δ k = 1) = P (δ k = -1) = 1/2 and u j (n) = C 1 M (nt -1 n ) - 2(s j +1) 4s j +2
for C 1 such that 3M 2 C 2 1 = 2a 2 . Let j be any index in J. Since ψ = 0 and when a is small enough (to guarantee that c j ≥ 0), c j is a density. Easy calculations imply that

c j -c λ 0 2 = M 2 C 2 1 (v nt -1 n (τ j )) 2 > a 2 (v nt -1 n ) 2 .
Moreover, if a is small enough, we have 3 C p 1 < 1 and

2 j(s j +1-2/p)p k ǫ | c j ψ ǫ jk | p = 2 j(s j +1-2/p)p k ǫ |u j (n)| p = 3C p 1 M p ≤ M p , implying that c j ∈ b s j ,p,∞ (M ). Denote by A j,n (a) the set of densities A j,n (a) = {c ∈ b s j ,p,∞ (M ) : inf λ∈Λ c -c λ 2 > a 2 (v nt -1 n (τ j )) 2 }
. and consider the variation between both distributions IP λ 0 and IP Π V ar(IP λ 0 , IP Π ) = 1 2

dIP Π dIP λ 0 -1 dIP λ 0 , where dIP Π dIP λ 0 = 1 N n j∈ J dIP j dIP λ 0 = 1 N n j∈ J IE (n) π j [ c j c λ 0 ],
and N n = card ( J). Assuming that the following assertion holds

lim n→∞ inf j∈ J π j (c ∈ A j,n (a)) = 1, (23)
we deduce that the left hand side (LHS) of relation ( 13) without the limit is bounded from below by

LHS ≥ IP λ 0 (D n = 1) + sup τ j ∈Sn sup c∈A j,n (a) IP c (D n = 0) ≥ 1 -V ar(IP λ 0 , IP Π )(1 + o n (1)),
as n large enough. Since the supports of the functions c j and c j ′ are disjoint for j = j ′ , one has

1 -V ar(IP λ 0 , IP Π ) ≥ 1 - 1 2 1 N 2 n j∈ J IE λ 0   n i=1 c j (U i , V i ) c λ 0 (U i , V i ) dπ j (c j ) 2 -1   ≥ 1 -o n (1) provided that lim n→∞ 1 N 2 n j∈ J IE λ 0   n i=1 c j (U i , V i ) c λ 0 (U i , V i ) dπ j (c j ) 2   = 0. (24)
Relation ( 13) is thus proved if ( 23) and ( 24) are satisfied. The remaining proofs are given in the sequel.

Proof of Relation (23)

Let Λ ′ be a subsect of Λ. We have

π j inf λ∈Λ c j -c λ 2 ≤ a 2 (v nt -1 n (τ j )) 2 ≤ π j inf λ∈Λ/Λ ′ c j -c λ 2 ≤ a 2 (v nt -1 n (τ j )) 2 + π j inf λ∈Λ ′ c j -c λ 2 ≤ a 2 (v nt -1 n (τ j )) 2 (25)
Consider the particular subset Λ ′ defined by

Λ ′ = {λ ∈ Λ : c λ 0 -c λ 2 ≤ 6C 2 1 M 2 (v nt -1 n (τ j )) 2 }. Notice that λ ∈ Λ/Λ ′ =⇒ c λ -c j 2 ≥ a 2 (v nt -1 n (τ j ))
2 due to the choice of C 1 . It implies that the first term in the right hand side of ( 25) is null and then, it remains to prove that

lim n→∞ π j inf λ∈Λ ′ c j -c λ 2 ≤ a 2 (v nt -1 n (τ j )) 2 = 0. ( 26
)
Since λ is in Λ ′ , we get

c λ -c j 2 2 = c λ 0 -c λ 2 + k ǫ u j (n) 2 + 2 k ǫ δ k u j (n)B j,k,λ,λ 0 ≥ 3C 2 1 M 2 (v nt -1 n (τ j )) 2 + 2 k δ k u j (n) ǫ B j,k,λ,λ 0 , where B j,k,λ,λ 0 = ψ ǫ j,k (c λ 0 -c λ ).
Therefore assertion ( 26) is equivalent to

lim n→∞ π j inf λ∈Λ ′ 2 k δ k u j (n)B j,k,λ,λ 0 ≤ -a 2 (v nt -1 n (τ j )) 2 = 0. or lim n→∞ π j sup λ∈Λ ′ 2 k (-δ k )u j (n)B j,k,λ,λ 0 ≥ a 2 (v nt -1 n (τ j )) 2 = 0. ( 27
)
We can construct in the Euclidean metric an η-net Λ ′ η on the subset Λ ′ . For any λ in Λ ′ , denote λ the closest element in Λ ′ η to λ in the Euclidean sense. Then for any λ ∈ Λ ′ , we have by assumption A0:

| k δ k u j (n)(B j,k,λ,λ 0 -B j,k, λ,λ 0 )| ≤ u j (n) k |B j,k,λ,λ 0 -B j,k, λ,λ 0 | ≤ u j (n) k Qη ν 2 -j ψ ǫ ∞ ≤ 2 j(s+1) 2 2j Qη ν 2 -j ψ ǫ ∞ ≤ κ2 js η ν ,
where κ is a positive constant depending on Q, C 1 , M and ψ ǫ ∞ . Chosing η = n -b , with bν > smax 2smax+1 , the proof of relation ( 27) is then reduced to the proof of

lim n→∞ Card(Λ ′ n -b )π j 2 k (-δ k )u j (n)B j,k, λ,λ 0 ≥ a 2 (v nt -1 n (τ j )) 2 = 0 lim n→∞ (T n b ) d Λ π j 2 k (-δ k )u j (n)B j,k, λ,λ 0 ≥ a 2 (v nt -1 n (τ j )) 2 = 0, ( 28
)
where Diam is the diameter of Λ. Finally, relation ( 26) is proved applying Bernstein inequality in the right hand side of relation (28). Indeed Bernstein inequality is applied to

π j 2 k (-δ k )u j (n)B j,k, λ,λ 0 ≥ a 2 (v nt -1 n (τ j )) 2 ,
with the i.i.d. centered random variables Z k = -δ k B j,k, λ,λ 0 . In particular, Notice that 2 , where K 1 and K 2 are positive constants. Notice also that it leads to an exponential bound of order exp(-2 j ).

|Z k | < K 1 v nt -1 n (τ j ), k Var(Z k ) ≤ K 2 (v nt -1 n (τ j ))

Proof of Relation (24)

Set

l n,π = n i=1 c j (U i , V i ) c λ 0 (U i , V i ) dπ j (c j ).
Due to the fact that the functions ψ ǫ j,k have disjoint support, it is possible to rewrite c j as follows

c j = c λ 0 k (1 + δ k D j,k ) for D j,k = u j (n) ǫ ψ ǫ j,k c λ 0 .
Then,

l n,π = k n i=1 (1 + δ k D j,k (U i , V i ))dπ j (δ k ) = k 1 2 n i=1 (1 + D j,k (U i , V i )) + n i=1 (1 -D j,k (U i , V i )) , and
l 2 n,π = k 1 4 2 n i=1 1 + D 2 j,k (U i , V i ) + 2 n i=1 1 -D 2 j,k (U i , V i ) +H D j,k (U i , V i ), D bt j,k (U t , V t ) t∈{1,...,i-1,i+1,...,n}
, where b t is either 0 or 2. Due to the independence of the data and acting as in [START_REF] Pouet | Tests minimax non-paramétriques : hypothèse nulle composite et constantes exactes[END_REF], it can be shown that

IE λ 0 H D j,k (U i , V i ), D bt j,k (U t , V t ) t∈{1,...,i-1,i+1,...,n} = 0.
Therefore,

IE λ 0 [l 2 n,π (U i , V i )] ≤ k 1 + IE λ 0 D 2 j,k (U i , V i ) n + 1 -IE λ 0 D 2 j,k (U i , V i ) n ≤ k cosh nIE λ 0 D 2 j,k (U i , V i ) .
Using the inequality log(cosh(u)) ≤ Ku 2 where K is a fixed constant and since c λ 0 is bounded from below by m, one obtains

1 N 2 n j∈ J exp(log(IE λ 0 l n,π ) 2 ) ≤ 1 N 2 n j∈ J exp Kn 2 k IE λ 0 D 2 j,k (U i , V i ) 2 ≤ 1 N 2 n j∈ J exp 3 2 K m 2 n 2 2 2j u j (n) 4 ≤ log(n) κ log(n)(1 + o n (1))
,

where κ = K(3C 2 1 M 2 ) 2 m -2 = 4Ka 4 m -2
. Choosing a small enough and κ < 1, Relation (24) is then proved.

10 Appendix A: Proof of Lemma 1

In this part, κ denotes any positive constant which may depend on φ, M and on c , c λ .

Notations and Preliminaries

Let us define or recall some notations that will be used below. For any k ∈ Z Z 2 , set

ξ k (X i , Y i ) = φ j,k F (X i ), G(Y i ) -φ j,k (F (X i ), G(Y i )) ω λ j,k (X i , Y i ) = φ j,k (F (X i ), G(Y i )) -c j,k (λ) ω ∞ j,k (X i , Y i ) = φ j,k (F (X i ), G(Y i )) -c j,k ,
where i is in I 2 . First, the localization property of the scaling function implies that only few ξ k (X i , Y i ) will be used since the others are zero. Indeed, one has the following result Lemma 4. For any k ∈ Z Z 2 , let us denote

N j = card {i ∈ I 2 ; ξ k (X i , Y i ) = 0} .
Let δ > 0. For any level j such that

2 j ≤ 2 3 √ δ + 1 n 2 log(n 2 ) 1/2 , one has P(N j > 2(2L + 3)n 2 2 -j )) ≤ K(n -δ 1 + n -δ 2 ).
We refer to [START_REF] Genest | Estimating copula densities through wavelets[END_REF] for the proof of this lemma since a similar result is established with an estimate F built on the whole sample: it guarantees in particular that F (X (i:n) ) = i/n, where X (i:n) denotes the i-th (among n) order statistic. In our case, the situation is different since F (X (i:n 2 ) ) is based on the observations lying in the subsample whose indices are in I 1 whereas it is calculated in an observation lying in the subsample whose indices are in I 2 ; nevertheless, applying the Dvoretsky-Kiefer-Wolfovitz Inequality, the following deviation inequality holds. For any ǫ > 0, P b F = P F (X (i:n 2 ) ) -i n 2 ≥ 2ǫ is bounded from above by

P b F ≤ P F (X (i:n 2 ) ) -F (X (i:n 2 ) ) ≥ ǫ + P F (X (i:n 2 ) ) -F (X (i:n 2 ) ) ≥ ǫ ≤ P F -F ∞ ≥ ǫ + P F -F ∞ ≥ ǫ ≤ K n -δ 1 + n -δ 2 ,
as soon as we take ǫ = δ log(n 1 )/(2n 1 ) ∨ δ log(n 2 )/(2n 2 ). Here F represents the empirical margin computed with the subsample whose indices in I 1 and F , the empirical margin computed with the subsample whose indices in I 2 .

Study of T j (λ)

Rewrite θ j,k (λ) in T j (λ) = k θ j,k (λ) as follows

θ j,k (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 ω λ j,k (X i 1 , Y i 1 )ω λ j,k (X i 2 , Y i 2 ).
For all i ∈ I 2 , one has IE(ω λ j,k (X i , Y i )) = c j,kc j,k (λ), which implies that

IE( T j (λ)) = k θ j,k (λ) = T j (λ).
Moreover for p = k, one obtains

IE( θ j,k (λ) θ j,p (λ)) = 1 (n2(n2 -1)) 2 X i 1 =i 2 =i 3 =i 4 IE h ω λ j,k (Xi 1 , Yi 1 ) i IE h ω λ j,p (Xi 3 , Yi 3 ) i IE h ω λ j,k (Xi 2 , Yi 2 ) i IE h ω λ j,p (Xi 4 , Yi 4 ) i +4 1 (n2(n2 -1)) 2 X i 1 =i 2 =i 3 IE h ω λ j,k (Xi 1 , Yi 1 ) i IE h ω λ j,p (Xi 3 , Yi 3 ) i IE h ω λ j,k (Xi 2 , Yi 2 )ω λ j,p (Xi 2 , Yi 2 ) i +2 1 (n2(n2 -1)) 2 X i 1 =i 2 IE h ω λ j,k (Xi 1 , Yi 1 )ω λ j,p (Xi 1 , Yi 1 ) i IE h ω λ j,k (Xi 2 , Yi 2 )ω λ j,p (Xi 2 , Yi 2 ) i ≤ θ j,k (λ)θj,p(λ) + 4 n2 (c j,k -c j,k (λ)) (cj,p -cj,p(λ)) »Z " φ j,k - Z φ j,k c λ « " φj,p - Z φj,pc λ « c - + 2 n2(n2 -1) »Z " φ j,k - Z φ j,k c λ « " φj,p - Z φj,pc λ « c -2 ,
which implies that

Var( c Tj (λ)) = IE X k d θ j,k (λ) ! 2 -IE X k d θ j,k (λ) ! 2 ≤ 4 n2 X k,p (c j,k -c j,k (λ)) (cj,p -cj,p(λ)) »Z " φ j,k - Z φ j,k c λ « " φj,p - Z φj,pc λ « c - + 2 n2(n2 -1) X k,p »Z " φ j,k - Z φ j,k c λ « " φj,p - Z φj,pc λ « c -2 .
Applying the Hölder Inequality and the consequence of the Parseval Equality, we get

X kp »Z " φ j,k - Z φ j,k c λ « " φj,p - Z φj,pc λ « c -2 ≤ 2 2 0 @ X k,p »Z φ j,k φj,pc -2 + 2 X k,p »Z φ j,k c λ Z φj,pc -2 + X k »Z φ j,k c λ -2 ! 2 1 A ≤ 2 2 X k Z φ 2 j,k c ! 2 + 2 Z c 2 Z c 2 λ + "Z c 2 λ « 2 ! ≤ κ 2 2j .
We conclude that

Var( T j (λ)) ≤ κ    4 n 2   k,p θ j,k (λ)θ j,p (λ)   1/2 2 j + 2 2j n 2 (n 2 -1)    ≤ κ 2 j n 2 T j (λ) + 2 2j n 2 2 ,
which is the announced result for T j (λ).

10.1.2 Study of T ♠ j and T ♣ j (λ) Let us denote

A i 1 = [ξ k (X i 1 , Y i 1 )] , D i 1 = k,p (IE [ξ k (X i 1 , Y i 1 )ξ p (X i 1 , Y i 1 )]) 2 B i 1 ,i 2 = k [ξ k (X i 1 , Y i 1 )ξ k (X i 2 , Y i 2 ] , C i 1 ,i 2 = ξ k (X i 1 , Y i 1 )ξ p (X i 2 , Y i 2 ).
We need the following results which are stated in the lemma below Lemma 5. Assume that the scaling function is q-differentiable. For any level j ≤ j ∞ , there exists some positive constant κ depending on φ, its derivatives and on c ∞ (which might be c λ ∞ for some λ ∈ Λ) such that for any distinct indices i 1 , i 2 , one obtains

IE|A i 1 | ≤ κ log(n 1 ) n 1 1/2 (29) IE|B i 1 ,i 2 | ≤ κ2 2j log(n 1 ) n 1 , IE|C i 1 ,i 2 | ≤ κ log(n 1 ) n 1 (30) |D i 1 | ≤ 2 6j log(n 1 ) n 1 2 .
We prove relation (29) in the next section, relations (30) are proven in [START_REF] Genest | Estimating copula densities through wavelets[END_REF]. We have

IET ♠ j = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 IE[B i 1 ,i 2 ].
Using Lemma 4 and Lemma 5, it follows

IE|T ♠ j | ≤ 1 n 2 (n 2 -1) (n 2 2 -j ) 2 2 2j log(n 1 ) n 1 ≤ log(n 1 ) n 1 .
Moreover, we get

T ♣ j (λ) = 1 n 2 (n 2 -1) i 1 ,i 2 ∈I 2 i 1 =i 2 k ξ k (X i 1 , Y i 1 )ω λ j,k (X i 2 , Y i 2 ) .
By Hölder Inequality and from lemmas 4 and 5, one obtains

IE|T ♣ j (λ)| ≤ 1 n 2 (n 2 -1) n 2 i 1 ,i 2 ∈I 2 i 1 =i 2 k (IE(A i )) 2 k (IEω λ j,k (X i 2 , Y i 2 )) 2 1/2 .
Remembering that IEω λ j,k (X i 2 , Y i 2 ) = (c j,kc j,k (λ)) for any index i 2 , we get

IE|T ♣ j (λ)| ≤ 1 n 2 (n 2 -1) (n 2 2 -j )n 2 2 2j log(n 1 ) n 1 T j (λ) 1/2 ≤ K log(n 1 ) n 1 T j (λ) 1/2
.

Let us study the moments of T ♣ j (λ) under IP λ . Since IE λ ω λ j,k (X i , Y i ) = 0 for any k and i, we obviously have IE λ T ♣ j (λ) = 0 and

IE λ (T ♣ j (λ)) 2 = 1 n 2 (n 2 -1) 2 i 1 =i 2 T i 1 ,i 2 + 1 n 2 (n 2 -1) 2 i 1 =i 2 =i 3 S i 1 ,i 2 ,i3 , where T i 1 ,i 2 = k,p IE λ [ξ k (X i 1 , Y i 1 )ξ p (X i 1 , Y i 1 )] IE λ ω λ j,k (X i 2 , Y i 2 )ω λ j,p (X i 2 , Y i 2 ) , S i 1 ,i 2 ,i 3 = k,p IE λ [ξ k (X i 1 , Y i 1 )ξ p (X i 2 , Y i 2 )] IE λ ω λ j,k (X i 3 , Y i 3 )ω λ j,p (X i 3 , Y i 3 ) .
By Hölder Inequality, we have

T i 1 ,i 2 = k,p IE λ [ξ k (X i 1 , Y i 1 )ξ p (X i 1 , Y i 1 )] IE λ ω λ j,k (X i 2 , Y i 2 )ω λ j,p (X i 2 , Y i 2 ) ≤ D 1/2 i 1   k,p IE λ ω λ j,k (X i 2 , Y i 2 )ω λ j,p (X i 2 , Y i 2 ) 2   1/2
With Parseval Equality, we get

k,p IE λ ω λ j,k (X i 2 , Y i 2 )ω λ j,p (X i 2 , Y i 2 ) 2 ≤ k,p φ j,k φ j,p c λ 2 ≤ K k φ 2 j,k c 2 λ ≤ K 2 2j ,
which combining with Lemma 5, implies that

T i 1 ,i 2 ≤ K 2 6j log(n 1 ) n 1 2 1/2 2 2j 1/2 ≤ 2 4j log(n 1 ) n 1 .
In the same way,

S i 1 ,i 2 ,i 3 ≤ K   k,p (IE λ C i 1 ,i 2 ) 2   1/2   k,p IE λ ω λ j,k (X i 2 , Y i 2 )ω λ j,p (X i 2 , Y i 2 ) 2   1/2 ≤ 2 2j 1/2 2 4j log(n 1 ) n 1 2 1/2 ≤ 2 3j log(n 1 ) n 1 .
From Lemma 4, one has

IE λ (T ♣ j (λ)) 2 ≤ K 1 n 2 2 (n 2 -1) 2 (n 2 2 -j )n 2 2 4j log(n 1 ) n 1 +K 1 n 2 2 (n 2 -1) 2 (n 2 2 -j ) 2 n 2 2 3j log(n 1 ) n 1 ≤ K 2 j log(n 1 ) n 2 n 1 .

Proof of Lemma 5

The following expansion is crucial because it allows to reduce the study to univariate variables.

ξ k (X i , Y i ) = ξ k 1 (X i )ξ k 2 (Y i ) (31) +ξ k 1 (X i )φ jk 2 (G(Y i )) + ξ k 2 (Y i )φ jk 1 (F (X i )) ,
where the univariate statistics ξ k 1 (X i ) and ξ k 2 (Y i ) are defined as follows

ξ k 1 (X i ) = φ j,k 1 F (X i ) n 1 -φ j,k 1 (F (X i )) ξ k 2 (Y i ) = φ j,k 2 G(Y i ) n 1 -φ j,k 1 (G(Y i )).
Assuming that φ is continuously q-differentiable, we get

ξ k 1 (X i ) = ẑk 1 (X i ) + ŵk 1 (X i ), where ẑk 1 (X i ) = q-1 ℓ=1 2 jℓ ℓ! ( F (X i ) -F (X i )) ℓ φ (ℓ) j,k 1 (F (X i )) and ŵk 1 (X i ) = 2 qj F (X i ) b F (X i ) φ (q) j,k 1 (t) (F (X i ) -t) q-1 dt.
A direct application of the Dvoretsky, Kiefer and Wolfovitz Inequality leads to the following bound

IP( F -F ∞ > ǫ) ≤ K exp(-2n 1 ǫ 2 ) ≤ Kn -δ 1 ,
as soon as ǫ = 0.5 δ log(n 1 )/n 1 . In the sequel, we take such an ǫ with δ large enough. Since j ≤ j ∞ where j ∞ is defined in [START_REF] Genest | Goodness-of-fit tests for copulas: A review and a power study[END_REF], observe that 2 j ǫ ≤ 1 and then we get

|ẑ k 1 (X i )| ≤ K 2 j ǫ max ℓ=1,...q-1 |φ (ℓ) j,k 1 (F (X i ))|(1 + o P (1)) | ŵk 1 (X i )| ≤ K 2 (q+1/2)j ǫ q (1 + o P (1))
which leads to the following bound

|ξ k 1 (Xi)| ≤ K " 2 (q+1/2)j ǫ q + 2 j ǫ max ℓ=1,...q-1 |φ (ℓ) j,k 1 (F (Xi))| « (1 + oP (1)).
The same kind of result obviously holds for ξ k 2 (Y i ). In the sequel, we need the following evaluations (which also hold for any derivatives of φ). Using expansion (31), we get

ξ k (X i , Y i ) = S 1 + S 2 ,
where

S 1 = ξ k 1 (X i )ξ k 2 (Y i ), S 2 = ξ k 1 (X i )φ j,k 2 (G(Y i )) + ξ k 2 (Y i )φ j,k 1 (F (X i )) .
Using (32), we get IE|S 1 | ≤ K 2 (2q+1)j ǫ 2q + 2 (q+1)j ǫ q+1 + 2 j ǫ 2 , IE|S 2 | ≤ K 2 qj ǫ q + ǫ .

If 2 j ≤ (n 1 / log(n 1 )) Applying Bernstein Inequality to the Z ′ i s leads to prove Lemma 2.

12 Appendix C: Proof of Lemma 3

Ω(Z i 1 , Z i 2 )| > x   ≤ K g exp - 1 K g min x 2 C 2 , x D , x B 2/3 , x A 1/2
, where

A = Ω(•, •) ∞ , B 2 = ñ IE[Ω 2 (Z 1 , •)] ∞ + IE[Ω 2 (•, Z 2 )] ∞ , C 2 = ñ2 IE[(Ω(Z 1 , Z 2 )) 2 ]
and

D = ñ sup Ω 1 ,Ω 2 {IE[Ω(Z 1 , Z 2 )Ω 1 (Z 1 )Ω 2 (Z 2 )] : IE[Ω 2 1 (Z 1 )] ≤ 1; IE[Ω 2 2 (Z 2 )] ≤ 1}.
We apply this proposition for Z i = (F (X i ), G(Y i )), ñ = n 2 and the kernel

Ω c (Z i 1 , Z i 2 ) = k {φ j,k (Z i 1 ) -IE c[φ j,k (Z i 1 )]} × {φ j,k (Z i 2 ) -IE c[φ j,k (Z i 2 )]} ,
which is considered under the distribution IP c where c is either c λ or c. The quantities A, B, C and D are evaluated in the following lemma which is proved in the next section.

Lemma 6. There exists some positive constant K 1 larger than either

12L 2 φ 2 ∞ ∨ (2 c ∞ ) ∨ 2L 2 φ 2 ∞ ∨ 4 c ∞ ( c ∞ + 3L 4 φ 2 ∞ )) such that A ≤ K 1 2 2j , B 2 ≤ K 1 n 2 2 2j , C 2 ≤ K 1 n 2 2 2 2j , D ≤ K 1 n 2
, where c is either c λ or c.

Again define c as c λ or c, then applying both the result of [START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF] and Lemma 6, for any level j and any x ≥ 2 j ((n 2 -1)n 2 ) -1/2 log(log(n 2 )), it immediately follows that which ends the proof of Lemma 3. Notice that

k,p IE c[φ j,k (U i 1 , V i 1 )φ j,p (U i 1 , V i 1 )] ≤ 2 2j , k (IE c[φ j,k (U, V )]) 2 = k c2 j,k ≤ c 2 ≤ M.
We get

A = k (φ j,k (u 1 , v 1 ) -IE c[φ j,k (U, V )]) (φ j,k (u 2 , v 2 ) -IE c[φ j,k (U, V )]) ∞ ≤ k φ j,k (u 1 , v 1 )φ j,k (u 2 , v 2 ) ∞ + 2 k φ j,k (u 1 , v 1 )IE c[φ j,k (U, V )] ∞ + k (IE c[φ j,k (U, V )]) 2 ∞ ≤ L 2 2 2j φ 2 ∞ + 2L 2 φ ∞ c 2 2 j + c 2 2 ≤ K2 2j ,
where K ≥ 2L 2 φ 2 ∞ and

B 2 = 2n 2 k,p IE c [(φ j,k (U i 1 , V i 1 ) -IE c[φ j,k (U, V )]) (φ j,p (U i 1 , V i 1 ) -IE c[φ j,p (U, V )])] × (φ j,k (u 2 , v 2 ) -IE c[φ j,k (U, V )]) (φ j,p (u 2 , v 2 ) -IE c[φ j,p (U, V )]) ∞ ≤ 2n 2 k,p
φ j,k φ j,p cφ j,k c φ j,p c (φ j,k (u 2 , v 2 ) -IE c[φ j,k (U, V )])

× (φ j,p (u 2 , v 2 ) -IE c[φ j,p (U, V )]) ∞ ≤ 2n 2 (2 c ∞ )   k,p φ j,k (u 2 , v 2 )φ j,p (u 2 , z 2 ) ∞ + 2 k,p φ j,k (u 2 , v 2 )IE c[φ j,k (U, V )] ∞ + k,p IE c[φ j,k (U, V )]IE c[φ j,p (U, V )] ∞   ≤ (4n 2 c ∞ ) 2 2j L 4 2 φ 2 ∞ + 2L 2 2 j φ ∞ + 2 2j c ∞ ≤ K n 2 2 2j
where K ≥ 4 c ∞ ( c ∞ + 3L 4 φ 2 ∞ ). Moreover, Applying again the inequality of Hölder to k (c 1 (k)) 2 (the same occurs for c 2 (k)), one gets

C 2 = n 2 2 k,p IE c [(φ j,k (U i 1 , V i 1 ) -IE c[φ j,k (U, V )]) (φ j,p (U i 1 , V i 1 ) -IE c[φ j,p (U, V )])] ×IE c [(φ j,k (U i 2 , V i 2 ) -IE c[φ j,k (U, V )]) (φ j,p (U i 2 , V i 2 ) -IE c[φ j,p (U, V )])] = n 2
k (c 1 (k)) 2 ≤ k ( (φ j,k -IE cφ j,k (U, V ))Ω 1,c c1I [ k 1 2 j , 2N-1+k 1 2 j ]×[ k 2 2 j , 2N-1+k 2 2 j ] ) 2 ≤ k (φ j,k -IE cφ j,k (U, V )) 2 c × (Ω 1,c ) 2 1I [ k 1 2 j , 2L-1+k 1 2 j ]×[ k 2 2 j , 2L-1+k 2 2 j ] c ≤ c ∞ (Ω 1,c ) 2 c k 1I [ k 1 2 j , 2L-1+k 1 2 j ]×[ k 2 2 j , 2L-1+k 2 2 j ] ≤ 12 φ 2 ∞ L 2 ,
since IE c(Ω 1,c (U )) 2 ) ≤ 1. It follows that D ≤ K n 2 , where K > 12L 2 φ 2 ∞ .

Figure 1 :

 1 Figure 1: Kendall's tau= 0.25. Left: Bi-dimensional Gaussian copula density with parameter ρ = 0.4. Right: Bi-dimensional Student copula density with parameter (ρ, ν) = (0.4, 1).

Figure 2 :

 2 Figure 2: Kendall's tau= 0.25. Left: Bi-dimensional Frank copula density with parameter θ = 2.5. Center: Bi-dimensional Gumbel copula density with parameter θ = 1.33. Right: Bidimensional Clayton copula density with parameter θ = 0.66.

Figure 3 :

 3 Figure3: Left: Thresholded wavelet estimator for the copula density of (LOSS, ALEA) as given in[START_REF] Florent Autin | Thresholding methods to estimate the copula density[END_REF]. Center: Gumbel copula density with parameter θ = 1.45. Right: Gaussien copula density with parameter ρ = 0.48.

11 Appendix B : Proof of Lemma 2 2 i∈I 2 2 1/ 2 ≤ 2 ≤ 2 ×

 2222222 Let us denote T ⋄ j (λ) = n -1 Z i whereZ i = k (φ jk (F (X i ), G(Y i ))c jk ) (c jkc jk (λ)) , IE c Z i = 0, |Z i | ≤ k (φ jk (F (X i ), G(Y i ))c jk ) 2 k (c jkc jk (λ)) K 1 2 j 2 T j (λ) 1/K 2 j T j (λ) 1/2 ,andVar c (Z i ) ≤ k,p IE (φ jk (F (X i ), G(Y i ))c jk ) (φ jp (F (X i ), G(Y i ))c jp ) × |(c jkc jk (λ)) (c jpc jp (λ))| ≤ k,p IEφ 2 jk (F (X i ), G(Y i ))IEφ 2 jp (F (X i ), G(Y i )) 1/|(c jkc jk (λ)) (c jpc jp (λ))| ≤ c ∞ k (c jkc jk (λ)) 2 = c ∞ T j (λ).

1 ,i 2 ∈I 2 i 1 =i 2 Ωc (Zi 1 , Zi 2 )

 1212 Kg exp (-δ log(log(n2))) .

12. 2

 2 Proof of Lemma 6Let us denote (U, V ) = (F (X), G(Y )) any pair of random variables whose marginal distribution are both uniform on [0, 1]. Denote c the copula density which is c λ or c; in the same spirit, the coefficients cj,k stand for c j,k (λ) or c j,k . Recall thatc j,k (λ) = IE λ [φ j,k (F (X), G(Y ))] = c λ (u, v)φ j,k (u, v)dudv. c j,k = IE [φ j,k (F (X), G(Y ))] = c(u, v)φ j,k (u, v)dudv.

( 2 = n 2 2 2 ≤ n 2 2where K ≥ 2 c 2 ∞(c 1 (k)) 2 k(c 2

 22222122 IE c [φ j,k (U i 1 , V i 1 )φ j,p (U i 1 , V i 1 )] -IE c [φ j,k (U, V )] IE c [φ j,p (U, V )]) k,p φ j,k φ j,p cφ j,k c φ j,p c . Denote u Ω 1 ,Ω 2 = IE c[Ω c(Z 1 , Z 2 )Ω 1,c (Z 1 )Ω 2,c (Z 2 )] and for i = 1, 2, put c i (k) = (φ j,k -IEcφ j,k (U, V ))Ω i,c c.By Hölder Inequality, we getu Ω 1 ,Ω 2 = k (φ j,k -IE cφ j,k (U, V ))Ω 1,c c (φ j,k -IE cφ j,k (U, V ))Ω 2,c c ≤ k (k)) 2 .

  1/2-1/2q , we obtain IE|ξ k (X i , Y i )| ≤ ǫ which ends the proof.

  12.1 U-StatisticLet us first recall the result of[START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF]. Proposition 1. (Theorem 3.3 p. 21[START_REF] Giné | Exponential and moment inequalities for u-statistics[END_REF]) It exists an universal positive constant K g < ∞ such that, if Ω is a bounded canonical kernel of two variables for the i.i.d. Z i 1 , Z i 2 , i 1 , i 2 ∈ {1, . . . , ñ}, where ñ ∈ IN, for any x > 0, we have

		
	IP	 |	i 1 ,i 2

Copula

True copula τ = 0.25 τ = 0.50 τ = 0.75 under H 0 Gumbel Clayton 1.00 0.0000 (0.72) 1.00 0.0000 (0.99) 1.00 0.0000 (