Salim Bouzebda
email: salim.bouzebda@upmc.fr

Amor Keziou
email: amor.keziou@upmc.fr

A new test procedure of independence in Copula models via χ 2 -divergence

Keywords: dependence function, Gumbel copula, pseudo-likelihood, asymptotic theory, multivariate rank statistics, semiparametric estimation. Mathematics Subject Classification :62F03, 62F10, 62F12, 62H12, 62H15

We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of χ 2 -divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.

Introduction and motivations

Parametric models for copulas have been intensively investigated during the last decades.

Copulas have become of great interest in applied statistics, because of the fact that they constitute a flexible and robust way to model dependence between the marginals of random vectors. The reader may refer to the following books for excellent expositions of the basics of copula theory : [START_REF] Nelsen | An introduction to copulas[END_REF] and [START_REF] Joe | Multivariate models and dependence concepts[END_REF]. In this framework, semiparametric inference methods, based on pseudo-likelihood, have been applied to copulas by a number of authors (see, e.g., [START_REF] Shih | Inferences on the association parameter in copula models for bivariate survival data[END_REF], [START_REF] Wang | On assessing the association for bivariate current status data[END_REF], [START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF] and the references therein). Throughout the available literature, investigations on the asymptotic properties of parametric estimators, as well as the relevant test statistics, have privileged the case where the parameter is an interior point of the admissible domain. However, for most parametric copula models of interest, the boundaries of the admissible parameter spaces 1 include some important parameter values, typically among which, that corresponding to the independence of margins. This paper concentrates on this specific problem. We aim, namely, to investigate parametric inference procedures, in the case where the parameter belongs to the boundary of the admissible domain. In particular, the usual limit laws both for parametric copula estimators and test statistics become invalid under these limiting cases, and, in particular, under marginal independence. Motivated by this observation, we will introduce a new semiparametric inference procedure based on χ 2 -divergence and duality technique. We will show that the proposed estimator remains asymptotically normal, even under the marginal independence assumption. This will allow us to introduce test statistic of independence, his study will be made, both under the null and alternative hypotheses.

It is well known since the work of [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] that the joint behavior of a bivariate vector (X 1 , X 2 ) with d.f. F(x 1 , x 2 ) := P(X 1 ≤ x 1 , X 2 ≤ x 2 ), and continuous marginal d.f.'s F i (x i ) := P (X i ≤ x i ), i = 1, 2, is characterized by the copula (or dependence function) C(•, •) associated with F(•, •). The copula function is defined, for all (u 1 , u 2 ) ∈ (0, 1) 2 , through the identity

C(u 1 , u 2 ) := P {F 1 (X 1 ) ≤ u 1 , F 2 (X 2 ) ≤ u 2 } .
Many useful multivariate models for dependence between X 1 and X 2 turn out to be generated by parametric families of copulas of the form {C θ ; θ ∈ Θ}, typically indexed by a vector valued parameter θ ∈ Θ ⊆ R p (see, e.g., Kimeldorf and Sampson (1975a), [START_REF] Kimeldorf | Uniform representations of bivariate distributions[END_REF], and [START_REF] Joe | Parametric families of multivariate distributions with given margins[END_REF]). The nonparametric approach to copula estimation has been initiated by [START_REF] Deheuvels | Propriétés d'existence et propriétés topologiques des fonctions de dépendance avec applications à la convergence des types pour des lois multivariées[END_REF], who introduced and investigated the empirical copula process. In addition, [START_REF] Deheuvels | Nonparametric test of independence[END_REF][START_REF] Deheuvels | A Kolmogorov-Smirnov type test for independence and multivariate samples[END_REF] described the limiting behavior of this empirical process (see, also Fermanian et al. (2004a) and the references therein). In this paper, we consider semiparametric copula models with unknown marginals.

In order to estimate the unknown true value of the parameter θ ∈ Θ, which we denote, throughout the sequel, by θ T ∈ Θ, some semiparametric estimation procedures, based on the maximization, on the parameter space Θ, of properly chosen pseudo-likelihood criterion, have been proposed by [START_REF] Oakes | Multivariate survival distributions[END_REF], and studied by [START_REF] Genest | A semiparametric estimation procedure of dependence parameters in multivariate families of distributions[END_REF], [START_REF] Shih | Inferences on the association parameter in copula models for bivariate survival data[END_REF], [START_REF] Wang | On assessing the association for bivariate current status data[END_REF] and [START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF] among others. In each of these papers, some asymptotic normality properties are established for √ n θ -θ T , where θ = θn denotes a properly chosen estimator of θ T . This is achieved, provided that θ T lies in the interior, denoted by Θ, of the parameter space Θ ⊆ R p . On the other hand, the case where θ T ∈ ∂Θ := Θ -Θ is a boundary value of Θ, has not been studied in a systematical way until present. Moreover, it turns out that, for the above-mentioned estimators, the asymptotic normality of √ n θ -θ T , may fail to hold for θ T ∈ ∂Θ; indeed, under some regularity conditions, when θ is univariate, we can prove that the limit law is the distribution of Z1 (Z≥0) where Z is a centred normal variable, and that the limit law of the generalized pseudo-likelihood ratio statistic is a mixture of chi-square laws with one degree of freedom and Dirac measure at zero; see [START_REF] Bouzebda | A test of independence in some copula models[END_REF]. Furthermore, when the parameter is multivariate, the derivation of the limit distributions under the null hypothesis of independence, becomes much more complex; see [START_REF] Self | Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions[END_REF]. Also, the limit distributions are not standard which yields formidable numerical difficulties to calculate the critical value of the test. We cite below some examples of parametric copulas, for which marginal independence is verified for some specific values of the parameter θ, on the boundary ∂Θ of the admissible parameter set Θ. We start with examples for which θ varies within subsets of R. Such is the case for the extreme value copulas, namely

C A (u 1 , u 2 ) := exp log u 1 u 2 A log u 1 log u 1 u 2 , (1.1)
where

A(•) is a convex function on [0, 1], satisfying -A : [0, 1] → [1/2, 1] such that max(t, 1 -t) ≤ A(t) ≤ 1 for all 0 ≤ t ≤ 1. For A(t) := A θ (t) = (t θ + (1 -t) θ ) 1/θ ; θ ∈ [1, ∞[ (1.2)
we have [START_REF] Gumbel | Bivariate exponential distributions[END_REF] family of copulas, which is one of the most popular model used to model bivariate extreme values. For

A θ (t) = 1 -(t -θ + (1 -t) -θ ) -1/θ ; θ ∈ [0, ∞[ (1.3)
we obtain [START_REF] Galambos | Order statistics of samples from multivariate distributions[END_REF] family of copulas. Finally for

A θ (t) = tΦ θ -1 + 1 2 θ log t 1 -t + (1 -t)Φ θ -1 - 1 2 θ log t 1 -t , (1.4)
where θ ∈ [0, ∞[ and Φ(•) denoting the standard normal N(0, 1) distribution function, we obtain the [START_REF] Hüsler | Maxima of normal random vectors: between independence and complete dependence[END_REF] family of copulas. A useful family of copulas, due to [START_REF] Joe | Parametric families of multivariate distributions with given margins[END_REF], is given, for 0 < u 1 , u 2 < 1, by

C θ (u 1 , u 2 ) := 1 -(1 -u 1 ) θ + (1 -u 2 ) θ -(1 -u 1 ) θ (1 -u 2 ) θ 1/θ ; θ ∈ [1, ∞[. (1.5)
The Gumbel-Barnett copulas are given, for 0 < u 1 , u 2 < 1, by

C θ (u 1 , u 2 ) := u 1 u 2 exp {-(1 -θ)(log u 1 )(log u 2 )} ; θ ∈ [0, 1]. (1.6)
The Clayton copulas of positive dependence are such that, for 0 < u 1 , u 2 < 1,

C θ (u 1 , u 2 ) = u -θ 1 + u -θ 2 -1 -1/θ ; θ ∈]0, ∞[. (1.7)
Parametric families of copulas with parameter θ varying in R p , for some p ≥ 2, include the following classical examples. Below, we set

θ = θ 1 , θ 2 ⊤ ∈ R 2 . C θ (u 1 , u 2 ) := 1 + (u -θ 1 1 -1) θ 2 + (u -θ 1 2 -1) θ 2 1/θ 2 -1/θ 1 , θ ∈]0, ∞[×[1, ∞[; (1.8) C θ (u 1 , u 2 ) := exp -θ 2 -1 log exp -θ 2 (log u 1 ) θ 1 (1.9) + exp -θ 2 (log u 2 ) θ 1 -1 1/θ 1 , θ ∈ [1, ∞[×]0, ∞[.
For other examples of the kind, we refer to [START_REF] Joe | Multivariate models and dependence concepts[END_REF].

For each of the above examples, the independence case

C θ T (u 1 , u 2 ) = u 1 u 2 (or A(t) = 1)
occurs at the boundary of the parameter space Θ, i.e., when θ T = 1 for the models (1.2), (1.5) and (1.6), θ T = 0 for the models (1.3), (1.4) and (1.7), θ T = (0, 1) ⊤ for the bivariate parameter model (1.8), and θ T = (1, 0) ⊤ for the bivariate parameter model (1.9). In the sequel, we will denote by θ 0 the value of the parameter (when it exists), corresponding to the independence of the marginals, i.e., the value of the parameter for which we have

C θ 0 (u 1 , u 2 ) := u 1 u 2 , for all (u 1 , u 2 ) ∈ (0, 1) 2 .
Hence, θ 0 = 1 for the models (1.2), (1.5) and (1.6), θ 0 = 0 for the models (1.3), (1.4) and (1.7), θ 0 = (0, 1) ⊤ for the model (1.8), and θ 0 = (1, 0) ⊤ for the model (1.9). Note that for the models (1.3), (1.4), (1.7), (1.8) and (1.9), C θ 0 (u 1 , u 2 ) = u 1 u 2 is naturally defined to be the limit of C θ (•, •) when θ tends to θ 0 with values in Θ. We denote c θ (•,

•) := ∂ 2 ∂u 1 ∂u 2 C θ (•, •) the density of C θ (•,
•) and we define c θ 0 (•, •) to be the limit of c θ (•, •) when θ tends to θ 0 with values in Θ. Hence, we can show that for all the above models c θ 0 (u 1 , u 2 ) = 1 for all 0 < u 1 , u 2 < 1.

In contrast with the preceding examples, where θ 0 ∈ ∂Θ is a boundary value of Θ, the case where θ 0 is an interior point of Θ may, at times, occur, but is more seldom. An example where θ 0 ∈ Θ is given by the Farlie-Gumbel-Morgenstern (FGM) copula, defined by

C θ (u 1 , u 2 ) := u 1 u 2 + θu 1 u 2 (1 -u 1 )(1 -u 2 ), θ ∈ Θ := [-1, 1],
(1.10) and for which

θ 0 = 0 ∈ Θ =] -1, 1[.
In the present article, we will treat parametric estimation of θ T , and tests of the independence assumption θ T = θ 0 . We consider both the case where θ 0 ∈ Θ is an interior point of Θ, and the case where θ 0 ∈ ∂Θ is a boundary value of Θ. To treat this case, we propose a new inference procedure, based on an estimation of χ 2 -divergence by duality technique. This method may be applied independently of the dimension of the parameter space. Also the limit law of the estimate of the parameter is normal and the limiting distribution of the proposed test statistic is χ 2 under independence, either when θ 0 is an interior point, or when θ 0 is a boundary point of Θ. The idea is to include the parameter domain Θ into an enlarged space, say Θ e , in order to render θ 0 an interior point of the new parameter space, Θ e . The conclusion is then obtained through an application of χ 2 -divergence and duality technique.

Our methods rely on the fact that, under appropriate assumptions, the definition of the density c θ (•,

•) := ∂ 2 ∂u 1 ∂u 2 C θ (•, •) of C θ (•,
•), pertaining to the models we consider, may be extended beyond the standard domain of variation Θ of θ. On the other hand, the definition of c θ (•, •) which corresponds to these extensions, is then, in general, no longer a density, and may, at times, become negative. For example, such is the case for the parametric models (1.2), (1.5), (1.3) and (1.4), for which c θ (•, •) is meaningful for some θ ∈ Θ, but then, becomes negative over some non-negligible (with respect to Lebesgue's measure) subsets of (0, 1) 2 . This implies that the log-likelihood of the data is not properly defined on the whole space Θ e . For this reason, we will use the χ 2 -divergence between signed finite measures. We will discuss this problem in more details, below, in section 2.

The remainder of the present paper is organized as follows. In section 2, we present our semiparametric inference procedure, based upon optimization of the χ 2 -divergence between the model (C θ , θ ∈ Θ e ) and the empirical copula associated to the data, and by using the dual representation of χ 2 -divergence. We then derive the asymptotic limiting distribution of the proposed estimator. It will become clear later on from our results, that the asymptotic normality of the estimate holds, even under the independence assumption, when, either, θ 0 is an interior, or a boundary point of Θ. The proposed test statistic of independence is also studied, under the null hypothesis H 0 of independence, as well as under the alternative hypothesis. The limiting asymptotic distribution of the test statistic under the alternative hypothesis is used to derive an approximation to the power function. An application of the forthcoming results will allow us to evaluate the sample size necessary to guarantee a pre-assigned power level, with respect to a specified alternative. Finally, section 4 reports a short simulation results, to illustrate the performance of the proposed test statistic. The proofs of these results will be postponed to the appendix.

2 A semiparametric estimation procedure through χ 2divergence As mentioned earlier, the problem of estimating θ, when θ ∈ ∂Θ, has not been systematically considered in the scientific literature; and the classical asymptotic normality property of the estimators is no longer satisfied. To overcome this difficulty, in what follows, we enlarge the parameter space Θ into a wider space Θ e ⊃ Θ. This is tailored to let θ 0 become an interior point of Θ e . Naturally, we assume that the definition of the function c θ (•, •) may be extended to Θ e . The difficulty associated with this construction is that, subject to a proper definition, the densities c θ (•, •) of C θ (•, •) with respect to the Lebesgue's measure, may become negative on some non negligible subsets of I := (0, 1) 2 (in this case c θ (•, •) becomes the density of a signed measure, see remark 3.2). Note that just as Deheuvels's empirical copula is not a copula, c θ (•, •) for θ ∈ Θ e is not necessarily a copula density and fail to integrate to 1. When such is the case, a semiparametric estimation of θ T via log-likelihood cannot be used. To circumvent this difficulty, we introduce a new inference procedure, based on χ 2 -divergence method, and duality technique. Recall that the χ 2 -divergence between a bounded signed measure Q, and a probability P on D, when Q is absolutely continuous with respect to P, is defined by

χ 2 (Q, P ) := D ϕ dQ dP dP, where ϕ : x ∈ R → ϕ(x) := 1 2 (x -1) 2 . (2.1)
In the sequel, we denote by χ 2 (θ 0 , θ T ) the χ 2 -divergence between C θ 0 (•, •) and C θ T (•, •). Applying the dual representation of φ-divergence obtained by [START_REF] Broniatowski | Minimization of φ-divergences on sets of signed measures[END_REF] Theorem 4.4, we readily obtain that χ 2 (θ 0 , θ T ) can be rewritten as

χ 2 (θ 0 , θ T ) := sup f ∈F I f dC θ 0 - I ϕ * (f ) dC θ T , (2.2)
where ϕ * (•) is used to denote the convex conjugate of ϕ(•), namely, the function defined by

ϕ * : t ∈ R → ϕ * (t) := sup x∈R {tx -ϕ(x)} = t 2 2 + t,
and F is an arbitrary class of measurable functions, fulfilling the following conditions

∀f ∈ F ; |f | dC θ 0 is finite and ϕ ′ (dC θ 0 /dC θ T ) = ϕ ′ (1/c θ T ) ∈ F . Furthermore, the sup in (2.2) is unique and achieved at f = ϕ ′ (1/c θ T ).
Define the new parameter space Θ e of θ as follows. Set

Θ e := θ ∈ R d such that |ϕ ′ (1/c θ (u 1 , u 2 ))| du 1 du 2 < ∞ . (2.3)
By choosing the class of functions F , via

F := {(u 1 , u 2 ) ∈ I → ϕ ′ (1/c θ (u 1 , u 2 )) -1 ; θ ∈ Θ e } ,
we infer from (2.2) the relation

χ 2 (θ 0 , θ T ) := sup θ∈Θe I 1 c θ (u 1 , u 2 ) -1 du 1 du 2 - I 1 2 1 c θ (u 1 , u 2 ) 2 - 1 2 dC θ T (u 1 , u 2 ) .(2.4)
It turns out that the supremum in (2.4) is reached iff θ = θ T . Moreover, in general, θ T is an interior point of the new parameter space Θ e , especially under the null hypothesis of independence, namely, when

θ T = θ 0 . Set m(θ, u 1 , u 2 ) := I 1 c θ (u 1 , u 2 ) -1 du 1 du 2 - 1 2 1 c θ (u 1 , u 2 ) 2 - 1 2 . (2.5) Consider a random sample {(X 1k , X 2k ); k = 1, . . . , n} from the distribution of (X 1 , X 2 ) de- noted by F θ T (x 1 , x 2 ) := C θ T (F 1 (x 1 ), F 2 (x 2 )) = P(X 1 ≤ x 1 , X 2 ≤ x 2 ). In what follows, we propose to estimate the χ 2 -divergence χ 2 (θ 0 , θ T ) between C θ 0 (•, •) and C θ T (•, •), by χ 2 (θ 0 , θ T ) := sup θ∈Θe I m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ), (2.6)
and to estimate the parameter θ T by

θ n := arg sup θ∈Θe I m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ) , (2.7)
where C n (•, •) is the modified empirical copula, defined by

C n (u 1 , u 2 ) := 1 n n k=1 1 {F 1n (X 1k )≤u 1 } 1 {F 2n (X 2k )≤u 2 } , (u 1 , u 2 ) ∈ I, (2.8)
where

F jn (t) := 1 n n k=1 1 ]-∞,t] (X jk ), j = 1, 2,
and 1 A stands for the indicator function of the event A.

Remark 2.1 The choice of the χ 2 -divergence among various divergences is motivated by the following statements:

-Recall that the φ-divergences between a bounded signed measure Q, and a probability P on D, when Q is absolutely continuous with respect to P, is defined by

D φ (Q, P) := D φ dQ dP dP, where φ is a proper closed convex function from ] -∞, ∞[ to [0, ∞[ with φ(1) = 0
and such that the domain domφ := {x ∈ R : φ(x) < ∞} is an interval with end points a φ < 1 < b φ .The Kullback-Leibler, modified Kullback-Leibler, χ 2 , modified χ 2 , and Hellinger divergences are examples of φ-divergences; they are obtained respectively for φ

(x) = x log x -x + 1, φ(x) = -log x + x -1, φ(x) = 1 2 (x -1) 2 , φ(x) = 1 2 (x-1) 2
x , and φ(x) = 2( √ x -1) 2 . We observe that Kullback-Leibler, modified Kullback-Leibler, modified χ 2 , and Hellinger divergences are infinite when dQ/dP takes negative values on non negligible (with respect to P) subset of D, since the corresponding φ(•) is infinite on (-∞, 0). This problem does not hold in the case of χ 2 -divergence, indeed the corresponding φ(•) is finite on R.

-We give an example of copulas for which the likelihood-based procedure fails. We consider the Gumbel copulas C θ (•, •) given in (1.2), it's corresponding density copula is defined by

c θ (u 1 , u 2 ) := C θ (u 1 , u 2 )(u 1 u 2 ) -1 ( u 1 u 2 ) (θ-1) ( u 1 θ + u 2 θ ) (2-1/θ) ( u 1 θ + u 2 θ ) (1/θ) + θ -1 , (2.9)
where x = -log x. We can show that c θ (•, •) may takes negative values for some θ ∈ Θ e .

In fact c 0.7 (u 1 , u 2 ) is negative for (u 1 , u 2 ) ∈ [0.9, 1] 2 , hence the likelihood function is not well defined.

Remark 2.2 The set Θ e defined in (2.3) is generally with non empty interior Θe . In particular, we may check that θ 0 (the value corresponding to independence) belongs to Θe , since the integral in (2.3) is finite; it is equal to zero when θ = θ 0 , for any copula density c θ (•, •). For example, in the case of FGM copulas (1.10), it is easy to show that Θ e = R. However, it is hard to determine the whole set Θ e for some copulas, but in order to test the independence, we need only to prove the existence of a neighborhood N(θ 0 ) of θ 0 for which the integral in (2.3) is finite since we calculate the estimate θ n in (2.7) by Newton-Raphson algorithm using θ 0 as initial point. The explicit calculation of the integral in (2.3) may be complicated for some copulas, in such cases we use the Monte Carlo method to compute this integral.

Statistics of the form Ψ

n := I ψ(u 1 , u 2 ) dC n (u 1 , u 2 ),
belong to the general class of multivariate rank statistics. Their asymptotic properties have been investigated at length by a number of authors, among whom we may cite [START_REF] Ruymgaart | Asymptotic normality of nonparametric tests for independence[END_REF], [START_REF] Ruymgaart | Asymptotic normality of nonparametric tests for independence[END_REF] and [START_REF] Rüschendorf | Asymptotic distributions of multivariate rank order statistics[END_REF]. In particular, the previous authors have provided regularity conditions, imposed on ψ(•, •), which imply the asymptotic normality of Ψ n . The corresponding arguments have been modified by [START_REF] Genest | A semiparametric estimation procedure of dependence parameters in multivariate families of distributions[END_REF], [START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF] and [START_REF] Fermanian | Weak convergence of empirical copula processes[END_REF], as to establish almost sure convergence of the estimators they consider. In the same spirit, the limiting behavior, as n tends to the infinity, of our estimator and test statistic T n , will make an instrumental use of the general theory of multivariate rank statistics. Using some similar arguments as in [START_REF] Qin | Empirical likelihood and general estimating equations[END_REF], the existence and consistency of our estimator and test statistic will be established through an application of the law of the iterated logarithm for empirical copula processes, in combination with general arguments from multivariate rank statistics theory. In the sequel, without loss of generality, we will limit ourselves to the case where the parameter is univariate. The extensions of our result in a multivariate framework may be achieved, at the price of additional technicalities, and under similar assumptions. We will make use of the following definitions.

Definition 2.1 (i) Let Q be the set of continuous functions q on [0, 1] which are positive on (0, 1), symmetric about 1/2, increasing on [0, 1/2] and satisfy

1 0 {q(t)} -2 dt < ∞.
(ii) A function r : (0, 1) -→ (0, ∞) is called u-shaped if it is symmetric about 1/2 and increasing on (0, 1/2].

(iii) For 0 < β < 1 and u-shaped function r, we define

r β (t) := r(βt) if 0 < t ≤ 1/2; r{1 -β(1 -t)} if 1/2 < t ≤ 1.
If for β > 0 in a neighborhood of 0, there exists a constant M β , such that r β ≤ M β r on (0, 1), then r is called a reproducing u-shaped function. We denote by R the set of reproducing u-shaped functions.

Typical examples of elements in Q and R are given by

q(t) = [t(1 -t)] ζ , 0 < ζ < 1/2, r(t) = ̺ [t(1 -t)] -ς , ς ≥ 0, ̺ ≥ 0.
We will describe the asymptotic properties of the proposed estimate θ n under the following conditions.

(C.1) There exist functions r 1,k , r 2,k ∈ R such that

∂ k ∂θ k m(θ T , u 1 , u 2 ) ≤ r 1,k (u 1 )r 2,k (u 2 ), for k = 1, 2, and ∂ 3 ∂θ 3 m(θ, u 1 , u 2 ) ≤ r 1,3 (u 1 )r 2,3 (u 2 ) on a neighborhood N(θ T ) of θ T , where I {r 1,k (u 1 )r 2,k (u 2 )} 2 dC θ T (u 1 , u 2 ) < ∞ for k = 1, 2, 3; (C.2) The function (u 1 , u 2 ) ∈ I → ∂ ∂θ m(θ T , u 1 , u 2
) is of bounded variation on I;

(C.3) For each θ, the function ∂ ∂θ m(θ, u 1 , u 2 ) : I -→ R is continuously differentiable, and there exist functions

r i ∈ R, r i ∈ R and q i ∈ Q for i = 1, 2, such that ∂ 2 ∂θ∂u i m(θ, u 1 , u 2 ) ≤ r i (u i )r j (u j ), i, j = 1, 2 and i = j,
and

I {q i (u i ) r i (u i )r j (u j )} dC θ T (u 1 , u 2 ) < ∞, i, j = 1, 2 and i = j. Set Ξ := -E ∂ 2 ∂θ 2 m (θ T , F 1 (X 1k ), F 2 (X 2k )) , (2.10) and Σ 2 := Var ∂ ∂θ m (θ T , F 1 (X 1 ), F 2 (X 2 )) + W 1 (θ T , X 1 ) + W 2 (θ T , X 2 ) , (2.11)
where

W i (θ T , X i ) := I 1 {F i (X i )≤u i } ∂ 2 ∂θ∂u i m (θ T , u 1 , u 2 ) c θ T (u 1 , u 2 ) du 1 du 2 , i = 1, 2.
We can see that Ξ and W i (X i ) can be defined, respectively, by

Ξ = E ∂ ∂θ m (θ T , F 1 (X 1k ), F 2 (X 2k )) 2 and, for i = 1, 2, W i (θ T , X i ) = - I 1 {F i (X i )≤u i } ∂ ∂θ m (θ T , u 1 , u 2 ) ∂ ∂u i m (θ T , u 1 , u 2 ) c θ T (u 1 , u 2 ) du 1 du 2 .
The following theorem describes the asymptotic behavior of the estimate θ n given in (2.7).

From now on, d -→ denotes the convergence in distribution.

Theorem 2.1 Assume that the conditions C.1-C.3 hold.

(a) Let B(θ T , n -1/3 ) := θ ∈ Θ e , |θ -θ T | ≤ n -1/3 . Then, as n → ∞, with probability one, the function θ → m (θ, u 1 , u 2 ) dC n (u 1 , u 2 ) reaches its maximum value at some point θ n in the interior of the interval B(θ T , n -1/3 ). As a consequence, the estimate is consistent almost surely and satisfies

I ∂ ∂θ m θ n , u 1 , u 2 dC n (u 1 , u 2 ) = 0. (b) As n → ∞, √ n( θ n -θ T ) d -→ N(0, Σ 2 /Ξ 2 ).
The proof of Theorem 2.1 is postponed to section 6.

3 A test based on "χ 2 -divergence"

One of the motivations of the present work is to build a statistical test of independence, based on χ 2 -divergence. In the framework of the parametric copula model, the null hypothesis, namely, the independence case C θ (u 1 , u 2 ) = u 1 u 2 corresponds to the condition that H 0 : θ T = θ 0 . We consider the alternative composite hypothesis H 1 : θ T = θ 0 . The corresponding generalized pseudo-likelihood ratio statistic is then given by

S n (θ 0 , θ) = 2 log sup θ∈Θ n k=1 c θ ( F 1n (X 1k ), F 2n (X 2k )) n k=1 c θ 0 ( F 1n (X 1k ), F 2n (X 2k ))
, where, for j = 1, 2, F jn stands for n/(n+1) times the marginal empirical distribution function of the j-th variable X j . The rescaling by the factor n/(n + 1), avoids difficulties arising from potential unboundedness of log c θ (u 1 , u 2 ), when either u 1 or u 2 tends to one. Since, θ 0 is a boundary value of the parameter space Θ, we can see, that the convergence in distribution of S n to a χ 2 random variable is not likely to hold. In order to bring a solution to this problem, we introduce the statistic

T n (θ 0 , θ n ) := 2n sup θ∈Θe I m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ). (3.1)
Below, we will show that, under the null hypothesis H 0 , the just-given statistic T n (θ 0 , θ n ) converges in distribution to a χ 2 random variable. This property allows us to build a test of H 0 against H 1 , asymptotically of level α. The limit law of T n (θ 0 , θ n ) will also be given under the alternative hypothesis H 1 . The following additional conditions will be needed for the statement of our results.

(C.4) We have lim

θ→θ 0 ∂ ∂u i m(θ, u 1 , u 2 ) = 0,
and there exist M 1 > 0 and δ 1 > 0 such that, for all θ in some neighborhood of θ 0 , one has, for i = 1, 2,

∂ ∂θ m(θ, u 1 , u 2 ) ∂ ∂u i m(θ, u 1 , u 2 )c θ (u 1 , u 2 ) < M 1 r(u i ) -1.5+δ 1 r(u 3-i ) 0.5+δ 1 ,
where r(u) := u(1 -u) for u ∈ (0, 1);

(C.5) There exist a neighborhood N(θ T ) of θ T , and functions r i ∈ R such that for all θ ∈ N(θ T ), we have

∂ ∂θ m(θ, u 1 , u 2 ) ≤ r 1 (u 1 )r 2 (u 2 ) with I {r 1 (u 1 )r 2 (u 2 )} 2 dC θ T (u 1 , u 2 ) < ∞; (C.6) There exist functions r i , r i ∈ R, q i ∈ Q, i = 1, 2 such that m(θ T , u 1 , u 2 ) ≤ r 1 (u 1 )r 2 (u 2 ), ∂ ∂u i m(θ, u 1 , u 2 ) ≤ r i (u i )r j (u j ), i, j = 1, 2 and i = j, with I {r 1 (u 1 )r 2 (u 2 )} 2 dC θ T (u 1 , u 2 ) < ∞,
and

I {q i (u i ) r i (u i )r j (u j )} dC θ T (u 1 , u 2 ) < ∞, i, j = 1, 2 and i = j.
Theorem 3.1 Assume that conditions C.1-C.4 hold. Then, under the null hypothesis H 0 , the statistic T n converges in distribution to a χ 2 1 random variable (with 1 degree of freedom).

The proof of Theorem 3.1 is postponed until section 6.

Theorem 3.2 Assume that conditions C. 1-C.3 and C.5-C.6 hold. Then, under the alternative hypothesis H 1 , we have

√ n T n 2n -χ 2 (θ 0 , θ T )
converges to a centered normal random variable with variance

σ 2 χ 2 := Var [m (θ T , F 1 (X 1k ), F 2 (X 2k )) + Y 1 (θ T , X 1 ) + Y 2 (θ T , X 2 )] , (3.2) where Y i (θ T , X i ) := I 1 {F i (X i )≤u i } ∂ ∂u i m (θ T , u 1 , u 2 ) c θ T (u 1 , u 2 ) du 1 du 2 .
The proof of Theorem 3.2 is postponed until section 6.

Remark 3.1 An application of Theorem 3.1, leads to reject the null hypothesis H 0 : θ T = θ 0 , whenever the value of the statistic T n exceeds q 1-α , namely, the (1 -α)-quantile of the χ 2 1 law. The test corresponding to this rejection rule is then, asymptotically of level α, when n → ∞. Accordingly, the critical region is given by

CR := {T n > q 1-α } . (3.3)
The fact that this test is consistent follows from Theorem 3.2. Further, this theorem can be used to give an approximation to the power function θ T → β(θ T ) := P θ T {CR} in a similar way to [START_REF] Keziou | Test of homogeneity in semiparametric two-sample density ratio models[END_REF] and [START_REF] Keziou | On empirical likelihood for semiparametric twosample density ratio models[END_REF]. We so obtain that

β(θ T ) ≈ 1 -Φ √ n σ χ 2 q 1-α 2n -χ 2 (θ 0 , θ T ) , (3.4)
where Φ denotes, as usual, the distribution function of N(0, 1) standard normal random variable. A useful consequence of (3.4) is the possibility of computing an approximate value of the sample size ensuring a specified power β(θ T ), with respect to some pre-assigned alternative θ T = θ 0 . Let n 0 be the positive root of the equation

β = 1 -Φ √ n σ χ 2 q 1-α 2n -χ 2 (θ 0 , θ T ) , (3.5)
which can be rewritten as

n 0 = (a + b) -a(a + 2b) 2χ 2 (θ 0 , θ T ) , (3.6)
where

a := σ 2 χ 2 (Φ -1 (1 -β))
2 and b := q 1-α χ 2 (θ 0 , θ T ). The sought-after approximate value of the sample size is then given by

n * := ⌊n 0 ⌋ + 1,
where ⌊u⌋ denote the integer part of u.

Remark 3.2 If the parameter θ does not belong to Θ, i.e., θ ∈ Θ e -Θ, the densities c θ (•, •), with respect to the Lebesgue's measure, associated to C θ (•, •), may become negative on some non negligible subsets of I. So, in this case

C θ (u 1 , u 2 ) := (0,u 1 )×(0,u 2 ) c θ (u 1 , u 2 ) dλ(u 1 , u 2 ); ∀u ∈ I, (3.7)
may be a signed measure (and not a copula) for some θ ∈ Θ e -Θ. Hence, the formula (2.7) may lead to a value of the estimator θ n belonging to Θ e -Θ, and the corresponding C θn is not necessarily a copula. So, for point estimation, the pseudo-maximum likelihood estimator (restricted to vary in the admissible domain) should be used instead of (2.7). The latter estimator may not be meaningful, and is likely to have a larger mean square error. The main advantage of our formula (2.6), where the supremum is taken on the extended space Θ e , instead of the admissible domain Θ, is that it permits easily to build a test of independence even when the dimension of the parameter space Θ is larger than one. We can show that the proposed test statistic, based on the formula (2.6), has a χ 2 (p) limit law with p = dim(Θ) degrees of freedom.

Remark 3.3 The asymptotic variances (2.11) and (3.2) may be consistently estimated respectively by the sample variances of

∂ ∂θ m θ n , F 1n (X 1,k ), F 2n (X 2,k ) + W 1 ( θ n , X 1,k ) + W 2 ( θ n , X 2,k ), k = 1, . . . , n, (3.8)
m θ n , F 1n (X 1,k ), F 2n (X 2,k ) + Y 1 ( θ n , X 1,k ) + Y 2 ( θ n , X 2,k ), k = 1, . . . , n, (3.9)
as was done in [START_REF] Genest | A semiparametric estimation procedure of dependence parameters in multivariate families of distributions[END_REF]. Similarly, the parameter Ξ in (2.10) may be consistently estimated by the sample mean of

∂ ∂θ m θ n , F 1n (X 1,k ), F 2n (X 2,k ) 2 , k = 1, . . . , n. (3.10)

Simulation results

In this section, we present some simulation results aiming to illustrate the theoretical results of Theorem 3.1 and Theorem 3.2.

The chi square approximation

We illustrate the accuracy of the approximation of the statistic T n by its limit law χ 2 1 under the null hypothesis of independence of marginals; see Theorem 3.1. We consider the Clayton, FGM and Gumbel copulas given in (1.7), (1.10) and (1.2) respectively. We use the Q-Q plot of the empirical quantiles of the proposed statistic T n versus the quantiles of the χ 2 1 law. In Figure 1, the Q-Q plots are obtained from 1000 independent runs of samples with sizes n = 100 and n = 500 for Clayton and FGM copulas, and from 500 independent runs of samples with sizes n = 100 and n = 500 for Gumbel copula. We observe that the approximation is good even for moderate sample sizes. The integral in the expression of m(θ, •, •) is calculated by the Monte Carlo method, and the supremum in (3.1) is considered on the extended space Θ e ; it has been computed on a neighborhood of θ 0 := 0 by the Newton-Raphson algorithm taking θ 0 := 0 as an initial point.

τ n := 2 n 2 -n 1≤i<j≤n sign(R i -R j )sign(S i -S j ).
Here, (R 1 , S 1 ), . . . , (R n , S n ) are the paired rank statistics pertaining to the sample (X 1,1 , X 2,1 ), . . . , (X 1,n , X 2,n ). ρ n is the Spearman ρ statistic, defined by

ρ n := -3 n + 1 n -1 + 12 n 3 -n n i=1 R i S i .
We consider the FGM copula and the marginals are exponential with parameter one. The power functions of the three statistics are plotted as a function of θ ∈ [0, 0.5] for sample sizes n = 100 and n = 500. Each power entry was obtained from 1000 independent runs. Looking at Figure 2, we can see that the test based on T n is superior to τ n and ρ n . On the right panel of Figure 2, for large samples (n=500), we see that the power of the tests based on Kendall's tau and T n are almost equal.

Conclusion

We have proposed a new test procedure based on χ 2 -divergence and duality technique in the framework of parametric copulas. It seems that the procedure introduced here is particularly well adapted to the boundary problem. For point estimation, the estimator based on χ 2 -divergence when we extend the parameter space, may not have a meaningful interpretation and most probably has a larger mean square error. However, Theorem 2.1 may be useful to construct the confidence region for the parameter of interest in connection with the intersection method (see [START_REF] Feng | Statistical inference using maximum likelihood estimation and the generalized likelihood ratio when the true parameter is on the boundary of the parameter space[END_REF]) which is easy to implement, while the asymptotic distribution of the maximum likelihood estimator is difficult to use, since the limiting distributions are complex when incorporating boundary constraints.

For clarity, our arguments are presented in the bivariate context. Note, however, that a d-variate generalization is possible with obvious changes of notation and modification assumptions. We mention that the popular multivariate Archimedean copulas

C θ (u 1 , . . . , u d ) = φ -1 {φ(u 1 ) + • • • + φ(u d )}
can only admit positive dependence, so that as long as the independence copula belongs to a given Archimedean family, θ 0 is on the boundary. Note, also that the proposed test of independence is not omnibus but depends on the hypothesis that the dependence structure belong to a certain family.

Appendix

First we give a technical Lemma which we will use to prove our results. Lemma 6.1 Let F(•, •) be continuous and C(•, •) have continuous partial derivatives. Assume that  is a continuous function, with bounded variation on I.

Then I (u 1 , u 2 ) d (C n (u 1 , u 2 ) -C(u 1 , u 2 )) = O n -1/2 (log log n) 1/2 (a.s.). (6.1)
Proof of Lemma 6.1. Recall that the modified empirical copula C n (•, •), is slightly different from the empirical copula C n (•, •), introduced by Deheuvels (1979a), and defined by

C n (u 1 , u 2 ) = F n F -1 1n (u 1 ), F -1 2n (u 2 ) for (u 1 , u 2 ) ∈ (0, 1) 2 , (6.2)
where F -1 1n (•) and F -1 2n (•) denote the empirical quantile functions, associated with F 1n (x 1 ) = F n (x 1 , ∞) and F 2n (x 2 ) = F n (∞, x 2 ), respectively, and defined by

F -1 jn (t) := inf{x ∈ R | F jn (x) ≥ t}, j = 1, 2. (6.3)
Here, F n (•, •) denotes the joint empirical distribution function, associated with the sample {(X 1k , X 2k ); k = 1, . . . , n}, defined by

F n (x 1 , x 2 ) = 1 n n k=1 1 {X 1k ≤x 1 } 1 {X 2k ≤x 2 } , -∞ < x 1 , x 2 < ∞. (6.4) We know that C n (•, •) and C n (•, •) coincide on the grid {(i/n, j/n) , 1 ≤ i ≤ j ≤ n} .
|C n (u) -C n (u)| ≤ max 1≤i,j≤n C n i n , j n -C n i -1 n , j -1 n ≤ 2 n . (6.5)
Using integration by parts, as in Fermanian et al. (2004a), we see that

√ n I (u 1 , u 2 ) d(C n -C)(u 1 , u 2 ) = I √ n(C n -C)(u 1 , u 2 ) d(u 1 , u 2 ) - I √ n(C n -C)(u 1 , 1) d(u 1 , u 2 ) - I √ n(C n -C)(1, u 2 ) d(u 1 , u 2 ) - [0,1] √ n(C n (u 1 , 1) -u 1 ) d(u 1 , 1) - [0,1] √ n(C n (1, u 2 ) -u 2 ) d(1, u 2 ).
Hence,

√ n I (u 1 , u 2 ) d(C n -C)(u 1 , u 2 ) ≤ 5 √ n sup u∈I |(C n -C)(u)| I d |(u)| .
From this and (6.5), applying Theorem 3.1 in Deheuvels (1979a), we obtain the following result

I (u 1 , u 2 ) d(C n -C)(u 1 , u 2 ) = O n -1/2 (log log n) 1/2 (a.s.).
Proof of Theorem 2.1. (a) Under the assumption (C.1), a straightforward calculus yields

I ∂ ∂θ m(θ T , u 1 , u 2 ) dC θ T (u 1 , u 2 ) = 0. (6.6)
Under the assumptions (C.1), and by applying Proposition 2.2 in [START_REF] Genest | A semiparametric estimation procedure of dependence parameters in multivariate families of distributions[END_REF], we can see that, as n → ∞,

I ∂ ∂θ m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) -→ I ∂ ∂θ m(θ T , u 1 , u 2 ) dC θ T (u 1 , u 2 ) = 0, (6.7) and I ∂ 2 ∂θ 2 m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) -→ I ∂ 2 ∂θ 2 m(θ T , u 1 , u 2 ) dC θ T (u 1 , u 2 ) := -Ξ < 0, (6.8) almost surely. Now, for any θ = θ T + vn -1/3 , with |v| ≤ 1, consider a Taylor expansion of m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ) in θ in a neighborhood of θ T , using (C.1) part 2, one finds n I m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ) -n I m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) (6.9) = n 2/3 v I ∂ ∂θ m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) + n 1/3 v 2 2 I ∂ 2
∂θ 2 m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) + O(1) (a.s.), uniformly in v with |v| ≤ 1. On the other hand, under condition (C.2), by Lemma 6.1, we have I ∂ ∂θ m(θ T , u 1 , u 2 )dC n (u 1 , u 2 ) = O(n -1/2 (log log n) 1/2 ) (a.s.).

Therefore, using (6.8) and (6.9), we obtain for any θ = θ T + vn -1/3 with |v| = 1, n I m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ) -n I m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 )

= O(n 1/6 (log log n) 1/2 ) -2 -1 Ξn 1/3 + O(1) (a.s.). (6.10)

Observe that the right-hand side vanishes when θ = θ T , and that the left-hand side, by (6.8), becomes negative for all n sufficiently large. Thus, by the continuity of θ → m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ), it holds that as n → ∞, with probability one, θ → m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ) reaches its maximum value at some point θ n in the interior of the interval B(θ T , n -1/3 ). Therefore, the estimate θ n satisfies (6.14) This completes the proof.

Proof of Theorem 3.1. Assume that θ T = θ 0 . From (6.12), using (6.8) and ( 6 and making use of a Taylor expansion of Tn(θ 0 , θn)

2n

, with respect to θ n , in a neighborhood of θ T , under (C.5) to obtain Finally, under condition (C.6), application once more of Proposition 3 page 362 in [START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF], concludes the proof.

Figure 1 :

 1 Figure 1: The Q-Q Plots of the statistic T n versus its limit law χ 2 1

Figure 2 :

 2 Figure 2: Power comparison

  θ n , u 1 , u 2 ) dC n (u 1 , u 2 ) = 0 and | θ n -θ T |= O(n -1/3 ). (6.11) (b) Making use of the first part of Theorem 2.1, and once more, by a Taylor expansion ofI ∂ ∂θ m( θ n , u 1 , u 2 ) dC n (u 1 , u 2 ),with respect to θ n , in the neighborhood of θ T , we obtain that0 = I ∂ ∂θ m( θ n , u 1 , u 2 ) dC n (u 1 , u 2 ) = I ∂ ∂θ m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) + ( θ n -θ T ) I ∂ 2 ∂θ 2 m(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) + o(n -1/2 ). (θ T , u 1 , u 2 ) dC n (u 1 , u 2 ) -1 √ nW n (θ T ) + o(θ T , u 1 , u 2 ) dC n (u 1 , u 2 ).Applying Proposition 3 page 362 in[START_REF] Tsukahara | Semiparametric estimation in copula models[END_REF], under assumptions (C.1) part 1 and (C.3), as n → ∞, we have√ nW n (θ T ) d -→ N 0, Σ 2 . (6.13)Finally, by combining (6.13) and (6.8) in connection with Slutsky's Theorem, as n → ∞,we conclude that √ n( θ n -θ T ) d -→ N(0, Σ 2 /Ξ 2 ).

  T , u 1 , u 2 ) dC n (u 1 , u 2 ) + o P (n -1T , u 1 , u 2 ) dC n -I m(θ T , u 1 , u 2 ) dC θ T +o P (1).

  The subtle difference lies in the fact that C n (•, •) is left-continuous with right-hand limits, whereas C n (•, •) on the other hand is right continuous with left-hand limits. The difference between C n (•, •) and C n (•, •), however, is small sup u∈I

  Expanding in Taylor series T n (θ 0 , θ n ) in θ n around θ T , we getT n (θ 0 , θ n ) = 2nW n (θ T )( θ n -θ T ) -Σn( θ n -θ T ) 2 + o P (1). When θ T = θ 0 , under Assumption (C.4), we can see that Σ 2 in (6.18) is equal to 1/Ξ; see Proposition 2.2 in[START_REF] Genest | A semiparametric estimation procedure of dependence parameters in multivariate families of distributions[END_REF]. Combining this with (6.17) to conclude that

								.13), we
	obtain	√	n θ n -θ T = -	1 Σ	√	nW n (θ T ) + o P (1).	(6.15)
								(6.16)
	Now, use (6.15) combined with (6.16) to obtain
			T T n	d -→ χ 2 1 ,
	under the null hypothesis H 0 .
	Proof of Theorem 3.2. Rewriting Tn 2n as
		T n (θ 0 , θ n ) 2n	θ∈Θe I = sup	m(θ, u 1 , u 2 ) dC n (u 1 , u 2 ),	(6.19)

n (θ 0 , θ n ) = 1 Σ nW n (θ T ) 2 + o P (1).

(6.17) By (6.13), as n → ∞, we have

√ nW n (θ T ) d -→ N 0, Σ 2 (6.18)

in distribution.


MD François Angoulvant
email: francois.angoulvant@rdb.aphp.fr

MD Florentia Kaguelidou

MD Stephane Dauger

MD Corinne Alberti

Fewer infants than older patients in paediatric randomised controlled trials

Keywords: Age groups, Randomised Control Trial, Infant, Newborn, Research Design

published or not. The documents may come

INTRODUCTION

Clinical trials have led to major changes in clinical practice. In paediatrics, a wellknown example is childhood acute lymphoblastic leukaemia, in which 5-year survival has improved from 25% to more than 80% as a result of management changes tested in multicenter trials. [START_REF] O'leary | Progress in childhood cancer: 50 years of research collaboration, a report from the Children's Oncology Group[END_REF] Unfortunately, the number of paediatric trials remains small. [START_REF] Caldwell | Clinical trials in children[END_REF] At least 50% [START_REF] Conroy | Survey of unlicensed and off label drug use in paediatric wards in European countries. European Network for Drug Investigation in Children[END_REF][START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF][START_REF] Shah | Off-label drug use in hospitalized children[END_REF] of drugs prescribed in children are used off-label or unlicensed, and many data on drug pharmacokinetics and toxicity are extrapolated from adult studies to paediatric populations. [START_REF] Conroy | Survey of unlicensed and off label drug use in paediatric wards in European countries. European Network for Drug Investigation in Children[END_REF][START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF][START_REF] Shah | Off-label drug use in hospitalized children[END_REF] Awareness of this gap in paediatric research [START_REF] Caldwell | Clinical trials in children[END_REF][START_REF] Wilson | An update on the therapeutic orphan[END_REF][START_REF] Salazar | Pediatric clinical trial experience: government, child, parent and physician's perspective[END_REF] led to legislative and regulatory changes in the USA, [START_REF]Best Pharmaceuticals for Children Act[END_REF] Europe, [START_REF]Better Medicines for Children -proposed regulatory actions in paediatric medicinal products[END_REF] and Australia, [START_REF] Aamc | Principles for Protecting Integrity in the Conduct and Reporting of Clinical Trials[END_REF] with the goal of encouraging paediatric research. Nevertheless, many studies indicate that the number of high-quality randomised controlled trials (RCTs) remains lower in paediatric patients than in adults. [START_REF] Martinez-Castaldi | Child versus adult research: the gap in high-quality study design[END_REF][START_REF] Klassen | Children Are Not Just Small Adults: The Urgent Need for High-Quality Trial Evidence in Children[END_REF][START_REF] Cohen | Child vs adult randomized controlled trials in specialist journals: a citation analysis of trends, 1985-2005[END_REF] For example, among the 32 RCTs investigating antiepileptic drugs for the add-on treatment of drug-resistant partial epilepsy, only 5 involved evaluations in both children and adults. [START_REF] Rheims | Greater response to placebo in children than in adults: a systematic review and meta-analysis in drug-resistant partial epilepsy[END_REF] Studies that focus specifically on the paediatric population are crucial. Paediatric patients differ from adults in many ways, including body composition, body-surface-area to weight ratio, and maturation of organ systems and enzymatic pathways. Furthermore, considerable differences exist across the paediatric population; adolescents (12-18 years), [START_REF] Emea | Note for Guidance on Clinical investigation of medicinal products in the Paediatric Population (E11)[END_REF][START_REF] Department | Guidance for Industry E11 Clinical investigation of medicinal products in the Pediatric Population[END_REF] for instance, are very different from neonates (0-27 days) or infants (28 days-23 months). [START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF] A case in point is the expression of drug-metabolising enzymes such as UDP glucuronosyltransferase, which changes with age. [START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF] The greatest differences in drug pharmacokinetics and pharmacodynamics compared to adults occur for neonates and infants. [START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF] Thus, there is an urgent need to improve the number and quality of studies conducted in paediatric populations, [START_REF] Rivara | Randomized controlled trials and pediatric research[END_REF] particularly in neonates and infants. [START_REF] Klassen | Children Are Not Just Small Adults: The Urgent Need for High-Quality Trial Evidence in Children[END_REF][START_REF]The International Forum of Standards for Research with Children[END_REF] We hypothesised that neonates and infants were less likely to be included in paediatric RCTs (PRCTs) than older paediatric patients. To test this hypothesis, we reviewed a subset of PRCTs published in 34 leading journals in 2006. We evaluated study design, objective, sample size, and geographic location. To assess the impact of recent regulations designed to encourage paediatric drug trials, we retrieved the PRCTs published in the same journals during the last 4 months of 2009.

METHODS

Literature Search

An electronic search of MEDLINE was conducted using PubMed to identify all RCTs published in 2006 in 34 high-impact-factor journals. We chose the general medical journals (GMJs), paediatric medical journals (PMJs), and specialist medical journals (SMJs) that had the highest impact factors in their category, as a high impact factor is a good predictor of impact in the field of paediatrics and of high study quality. [START_REF] Lee | Association of journal quality indicators with methodological quality of clinical research articles[END_REF] Information on the journals, including impact factors, was obtained from the Institute for Scientific Information's Journal Citation Reports® for 2005 (http://www.isiwebofknowledge.com/). We required an impact factor of 7.0 or higher for GMJs and SMJs. Since PMJs had lower impact factors, we arbitrarily chose a lower cut-off of 3.0 corresponding to the 90 th percentile of PMJ impact factors listed in the 2005 Journal Citation Reports ® . Furthermore, we chose journals that published at least ten RCTs in patients belonging to any age group, in 2006. We conducted our search with the keyword "randomised controlled trial". Limits, in addition to 2006 for year of publication, were "Humans" and "All child: 0-18 years". Letters, comments, and editorials were excluded. We identified 417 articles labelled as PRCTs including 90 (22%) in GMJs, 176 (42%) in PMJs, and 151 (36%) in SMJs. We arbitrarily chose to select a random sample of 35% of the PRCTs published in 2006, in keeping with an earlier study, [START_REF] Gow | A review of five cardiology journals found that observer variability of measured variables was infrequently reported[END_REF] instead of a 3-month sample. [START_REF] Martinez-Castaldi | Child versus adult research: the gap in high-quality study design[END_REF] We used a computer-based random selection strategy stratified on journal category. This strategy selected 31 articles in GMJs, 60 in PMJs, and 53 in SMJs (144 articles in all). The numbers of articles with the journal categories, names, and impact factors are shown in Table 1. One of us (FA) read the titles and abstracts of the 144 articles. When the information in the title and abstract suggested that the study was a PRCT, the full-length article was read. Articles that included at least one patient aged 0 to 18 years and that constituted first trial reports were selected. We excluded studies that included no paediatric patients (even when the endpoint was assessed in a child, i.e., in newborns of pregnant women included in the study) and articles that supplied additional information on a previously reported trial. For each excluded article, we recorded the reason for exclusion. Figure 1 is the flow chart of the articles. We developed a standardized data collection form based on a review of the relevant literature then calibrated it based on 10 articles. Data were then extracted from the selected articles to the form by one of us (FA), who is a certified paediatrician. Among the data extracted from each article were the characteristics of the population, with special attention to age, for which we recorded the central tendency (mean or median), range, and dispersion (standard deviation or interquartile range). We also recorded the number of patients, male-to-female ratio, topic of the study (determined using a list published by the 10 th International Statistical Classification of Diseases and Related Health Problems, [START_REF] Who | International Statistical Classification of Diseases and Related Health Problems 10th Revision Version for[END_REF] continent where the study was done, type of intervention (treatment, diagnosis, or prevention), and whether age-related subgroup analyses were performed. We classified the studies based on the age groups included, using the following definitions: neonates, 0-27 days; infants, 28 days-23 months; children, 2-11 years; and adolescents, 12-18 years [START_REF] Emea | Note for Guidance on Clinical investigation of medicinal products in the Paediatric Population (E11)[END_REF][START_REF] Department | Guidance for Industry E11 Clinical investigation of medicinal products in the Pediatric Population[END_REF]. Preterm babies were defined as babies born before 37 weeks of gestation [START_REF] Emea | Note for Guidance on Clinical investigation of medicinal products in the Paediatric Population (E11)[END_REF][START_REF] Department | Guidance for Industry E11 Clinical investigation of medicinal products in the Pediatric Population[END_REF]. If the mean or median age, or at least part of the age range, was contained within the interval defining an age group, the study was classified as including at least that age group. Incomplete age data were handled as follows. When only one end of the age range was known, we used this value and the interval separating it from the mean to identify the age group. When some of the included patients were younger than 18 years, but the minimum age was missing, and when the maximum and mean ages were either missing or greater than 18 years, the study was classified as a PRCT in 6/25 an unspecified age group. We did not evaluate whether the treatments studied in the older age groups were relevant to the younger age groups. We tested interobserver variability of our data extraction method by having one of us (FK), who was blinded to the results of the initial data extraction, independently extract data from a random sample of 35% of the 144 articles.

We also performed a MEDLINE search using the same strategy to identify PRCTs published in the same 34 journals between 1 st September 2009 and 31 st December 2009. We chose a 4month period to obtain about one-third of PRCTs published in 2009, as we included 35% of PRCTs published in during 2006 in the same journals. To maximize the likelihood of detecting an impact of the recent regulations, we chose the latest possible time period, that is, the last 4 months of 2009. We identified 66 PRCTs, for which we recorded the age groups and topics.

Statistical Analysis

Qualitative variables were described as numbers (percentages) and quantitative variables as medians (range and interquartile range). Our assessment of interobserver variability showed perfect agreement for the identification of PRCTs (Kappa =1) and substantial agreement for the main extracted items (Kappa =0.70; 95% confidence interval [95%CI], 0.62-0.79). We used Fisher's test to compare age-group distributions in PRCTs published in 2006 and 2009.

RESULTS

Selected Articles

Of the 144 articles, 82 were found to be PRCTs. The 62 other articles were done in adults or were not first reports of RCTs (Figure 1); therefore, they were excluded from the analysis.

Study characteristics

Reported age characteristics in the 82 articles are shown in Table 2 Most of the 12 trials on rare diseases included both paediatric patients and adults. Of the 66 therapeutic trials, 53 evaluated drugs, most of which had already been used in paediatric patients or adults. Among the remaining trials, 14 assessed preventive or educational interventions and 2 assessed diagnostic tools. Of the 82 trials, 41 (50%) were done in North Australasia, and 4 (5%) in Latin America. The total exceeds 82 because seven studies involved at least two regions. In four articles, the geographic region where the study was done was not clearly described. For 58 of the 66 PRCTs published in the last four months of 2009, inclusion started before 1 st January 2007 suggesting little or no impact of regulations enacted in 2007.

DISCUSSION

PRCTs published in 34 leading journals in 2006 and during the last four months of 2009 were less likely to include neonates and infants than children or adolescents. A review of articles published in 2005 in six leading general and specialist journals showed that studies performed in adults were significantly more likely than paediatric studies to be RCTs, systematic reviews, or studies of therapeutic interventions. [START_REF] Martinez-Castaldi | Child versus adult research: the gap in high-quality study design[END_REF] Compared to studies in adults, studies in paediatric patients are fewer, less well designed, and less well reported. [START_REF] Klassen | Children Are Not Just Small Adults: The Urgent Need for High-Quality Trial Evidence in Children[END_REF] In addition to this previously described paucity of high-quality paediatric studies, we found that few paediatric studies focused on the youngest patients. Drug pharmacokinetics and pharmacodynamics differ significantly between adults and paediatric patients and across paediatric age groups. [START_REF] Rakhmanina | Pharmacological research in pediatrics: From neonates to adolescents[END_REF][START_REF] Johnson | The development of drug metabolising enzymes and their influence on the susceptibility to adverse drug reactions in children[END_REF] Drugs that are effective and safe in adults or older paediatric patients may be neither in neonates and infants. For instance, the administration in the 1950s of chloramphenicol to neonates in a total daily dosage based on experience with adults caused the potentially fatal side effect known as grey baby syndrome related to liver enzyme immaturity. [START_REF] Weiss | Chloramphenicol in the newborn infant. A physiologic explanation of its toxicity when given in excessive doses[END_REF] However, many drugs are not tested in paediatric patients, who must therefore be given drugs off-label. In a multicenter study of 344,094 inpatients in tertiary paediatric hospitals in the United States, at least one drug was used off-label in 78.7% of the overall study population. [START_REF] Shah | Off-label drug use in hospitalized children[END_REF] The younger the child, the greater the difference with adults.

9/25

Both the nature and the dosage of drugs must be adapted to the specific characteristics of each paediatric age group. About 20% of drug prescriptions written for paediatric patients are written for infants, [START_REF] Jong | Unlicensed and off-label prescription of respiratory drugs to children[END_REF] and the use of unlicensed drugs is highest among children younger than 1 year. [START_REF] Schirm | Risk factors for unlicensed and off-label drug use in children outside the hospital[END_REF] Among drugs used in neonates, up to 90% may be given off-label. [START_REF] Barr | Unlicensed and off-label medication use in a neonatal intensive care unit: a prospective study[END_REF] The prevalences of the various health conditions in each age group were estimated based on several indicators. In a study of annual outpatient visit rates between 1993-1995 in the USA, rates were highest among patients younger than 1 year of age (800 visits per 100 children versus 200 visits per 100 children aged 5 to 14 years). [START_REF] Freid | Ambulatory health care visits by children: principal diagnosis and place of visit[END_REF] The main diagnoses in this study were middle ear infections (especially in patients younger than 1 year), injury (especially in patients older than 5 years), asthma (with similar rates in infants, preschoolers, and school-age children), and attention deficit disorders. [START_REF] Freid | Ambulatory health care visits by children: principal diagnosis and place of visit[END_REF] In 2006 in the USA, 155 000 children were admitted for asthma and contributed 5.6% of all admitted children; there were also 593 000 emergency room visits of children for asthma, contributing 2.3% of all paediatric emergency room visits. [START_REF] Akinbami | Status of childhood asthma in the United States, 1980-2007[END_REF] The numbers of admissions and visits were highest among the youngest patients and decreased with advancing age, although the prevalence of asthma was higher in school-age children and adolescents. [START_REF] Akinbami | Status of childhood asthma in the United States, 1980-2007[END_REF] A hospital discharge survey performed in the USA in 2005 showed that 45% of neonates had at least one illness or risk-related diagnosis. [START_REF] Defrances | National Hospital Discharge Survey: 2005 annual summary with detailed diagnosis and procedure data[END_REF] Common diagnoses were perinatal jaundice (20%), respiratory conditions (11%), disorders related to prematurity (8%), and congenital anomalies (8%). [START_REF] Defrances | National Hospital Discharge Survey: 2005 annual summary with detailed diagnosis and procedure data[END_REF]. The prevalence of chronic conditions was studied over 6 years in the USA in children aged 2 years at inclusion. Prevalences were 2% to 3.6% for asthma, 11.9% to 13.3% for obesity, 1% to 4.7% for mental and behavioural disorders, and 3.9% to 5.7% for other physical conditions. [START_REF] Van Cleave | Dynamics of obesity and chronic health conditions among children and youth[END_REF] Fewer data are available on the worldwide scale. A study of major causes of death in children younger than 5 years [START_REF] Who | Global health risks: mortality and burden of disease attributable to selected major risks[END_REF] showed that diarrhoea and pneumonia each contributed 17% of deaths, followed by other infections (12%), severe neonatal infections (11%), prematurity (11%), birth asphyxia and trauma (8%), malaria (7%), measles (4%), injuries (4%), and nutritional deficiencies.

10/25

Thus, chronic health conditions predominate in older children, whereas acute and severe illnesses are more common in the youngest patients. Therefore, although neonates and infants contribute a smaller proportion of the paediatric population than do older age groups, their specific characteristics require a large amount of research attention. Clearly, there is an urgent need for high-quality studies in neonates and infants. Nevertheless, despite a few initiatives to promote research in neonates, [START_REF] Anand | Summary proceedings from the neonatal pain-control group[END_REF] most paediatric studies are done in children and adolescents, as shown by a study of 253 trials conducted to obtain paediatric exclusivity extensions. [START_REF] Benjamin | Peerreviewed publication of clinical trials completed for pediatric exclusivity[END_REF] Drugs studied in these trials are mainly used in older children and adolescents for conditions such as hypertension, systemic autoimmune disorders, and gastrointestinal disorders. [START_REF] Benjamin | Peerreviewed publication of clinical trials completed for pediatric exclusivity[END_REF] We identified important deficiencies in the reporting of age characteristics of populations included in PRCTs. In 17% of the PRCTs published in 2006, at least one major age characteristic was missing. Furthermore, 40% trials included more than one paediatric age group. In this situation, detailed information on the number of participants in each age group is crucial. Inadequate information on age characteristics limits the external validity and therefore the usefulness of PRCTs. [START_REF] Klassen | Children Are Not Just Small Adults: The Urgent Need for High-Quality Trial Evidence in Children[END_REF] Finally, of the 13 trials that included both adults and paediatric patients, 12 did not specify the number of each. Only 6 PRCTs involved age-related subgroup analyses. In many PRCTs, sample sizes were too small for valid subgroup analyses.

The three health conditions most often studied in PRCTs were respiratory diseases, infectious diseases, and mental and behavioural disorders, in keeping with earlier data. [START_REF] Benjamin | Peerreviewed publication of clinical trials completed for pediatric exclusivity[END_REF] Asthma and vaccines contributed 29% of the trials in our study. Studies of asthma were done chiefly in children and studies of vaccines in infants. Although surgery is a high-cost intervention, only 2 trials investigated surgical treatments. Similarly, there were few studies of diagnostic procedures. We excluded 62 of the 144 initially selected articles. Many of the excluded articles reported studies in adults, although they were associated with paediatric keywords (e.g., "adolescent"). Differences in age-group definitions further complicate the identification of PRCTs on Medline. For example, the age range for adolescence is 12 to 18 years according 11/25 to the ICH definition used by the US Department of Health [START_REF] Department | Guidance for Industry E11 Clinical investigation of medicinal products in the Pediatric Population[END_REF] and European Medicines Agency [START_REF] Emea | Note for Guidance on Clinical investigation of medicinal products in the Paediatric Population (E11)[END_REF] and 13 to 18 years according to the Medical Subject Headings (MeSH, United States National Library of Medicine (http://www.nlm.nih.gov/mesh/meshhome.html) used for PubMed search limits. There is clearly a need for standardizing age-group definitions. [START_REF] Klassen | Children Are Not Just Small Adults: The Urgent Need for High-Quality Trial Evidence in Children[END_REF] Reporting of age characteristics in PRCTs also needs to be improved, to ensure easy identification of paediatric age groups included in trials. Nevertheless, we were able to determine the age groups in most of the trials selected for our study. We excluded RCTs in pregnant women. Although RCTs performed during pregnancy are numerous and often relevant to paediatrics, for instance in the field of HIV infection, they are not influenced by new legislation about paediatric research. Low publication rates of PRCTs done to obtain exclusivity extensions have been found. [START_REF] Benjamin | Peerreviewed publication of clinical trials completed for pediatric exclusivity[END_REF] PRCTs of interventions whose use is well established in adults may be unlikely to be accepted for publication by peer-reviewed journals. Publication bias may occur when the studies are conducted in narrow fields or with small effect sizes or when they fail to show efficacy of the study intervention. [START_REF] Ioannidis | Why most published research findings are false[END_REF] These sources of publication bias are often present in paediatric trials, especially in the youngest age-groups. Thus, our focus on published RCTs is a limitation of our study.

Because we focused on paediatric journals having high impact factors, most subspecialty paediatric journals were excluded. PRCTs may have been published in these excluded journals. However, only data published in peer-reviewed journals are likely to have a major impact on clinical practice. [START_REF] Martinez-Castaldi | Child versus adult research: the gap in high-quality study design[END_REF][START_REF] Benjamin | Peerreviewed publication of clinical trials completed for pediatric exclusivity[END_REF] Labelling issues constitute only the most visible part of the far-reaching deficiencies that characterize paediatric clinical research. Because the rules applying to drug use are more restrictive than in other fields (e.g., there is no equivalent of labelling in surgery), the imbalance in labelling between adults and children and across paediatric age groups is more easily recognizable, compared to other problems affecting paediatric research. Labelling requires published studies, and labelling inadequacies constitute a marker for the more extensive problems that affect paediatric clinical research.

12/25

Only 13 of the 148 trials identified in our study were done in preterm babies. Survival is increasing among extremely immature newborns, in whom drug handling and toxicity differ markedly from those in older patients. High-quality studies in immature babies are needed to determine the risk/benefit profile of drugs and the mechanisms of drug toxicity. [START_REF] Ward | Basic and translational research in neonatal pharmacology[END_REF] Limitations on blood sampling are among the specific challenges raised by pharmacodynamic, pharmacokinetic, and pharmacogenetic studies in preterm babies. [START_REF] De Wildt | Challenges for drug studies in children: CYP3A phenotyping as example[END_REF] However, methods have been devised to circumvent this problem. [START_REF] Thingvoll | Observational trial of a 48-hour gentamicin dosing regimen derived from Monte Carlo simulations in infants born at less than 28 weeks' gestation[END_REF] The need for special attention to paediatric drugs was recognized in the 1990s. The

  Food and Drug Administration Modernization Act of 1997 (FDAMA) required the FDA to specify which drugs must carry paediatric labelling. The Best Pharmaceuticals for Children Act (BPCA) passed in 2002 encourages more studies in children and promotes the development of treatments for children; it offers extended exclusivity to pharmaceutical companies that perform studies in children. The Pediatric Research Equity Act (PREA) of 2003 allows the FDA to require paediatric drug trials sponsored by pharmaceutical companies when such studies are not performed despite incentives and publicly funded mechanisms.[38] The Food and Drug Administration Amendments Act of 2007 (FDAAA) extends applicability of the BPCA and PREA authority to 2012. The paediatric exclusivity program in the USA provided paediatric data for 59 drugs between 2002 and 2004, and the cost of the PRCTs was offset by the net returns from the additional 6 months of exclusive marketing rights.[39] This program seems effective for high-cost diseases that also exist in adults, such as asthma,[40, 41] but may be less useful for conditions not seen in adults. Other types of program are needed to encourage research in neonates and infants. Since 2006 (the year chosen for our first study), new laws and regulations (FDAAA, EMEA paediatric initiative, and FP-7) have been enacted to promote clinical research in paediatrics. However, our evaluation of PRCTs published in late 2009 found no changes in age-group distribution compared to 2006. However, only 12% 13/25 of the 2009 PRCTs had inclusion periods that started in or after 2007. Thus, the time interval may have been too short to detect an effect of the new regulations. CONCLUSION Paediatric health issues vary across age group: infectious diseases and the need for immunizations are crucially important in the youngest children, whereas mental and behavioural disorders chiefly affect older children. Newborns and infants were less often included in RCTs published in 2006 and in the 4 last months of 2009 than older paediatricpatients, although their specific characteristics entail a need for specific studies. Additional efforts are needed to promote research in neonates and infants. Further studies designed to evaluate the impact of recent regulations may provide insight into the best means of encouraging PRCTs conducted in the youngest age groups and published in high impact factor journals. Furthermore, age group definitions must be standardized and improvements made in the reporting of age characteristics in PRCTs.[START_REF] Begg | Improving the quality of reporting of randomized controlled trials. The CONSORT statement[END_REF]

Figure 1 .Figure 1 GMJs

 11 Figure 1. Flow of articles in the literature search

Table 5 .

 5 . A single paediatric age group was included in 46 studies, two paediatric age groups in 29 studies, and three paediatric age groups in 4 studies. Of the 13 studies that included both adults and paediatric patients, three did not provide sufficient data to determine the age group of the paediatric participants. Among the 44 PRCTs published in 2006 that included either patients from at least two paediatric age-groups or both adults and children, 6 reported age-related subgroup analyses. The distribution of age groups is shown in Table 3 for the 2006 sample of PRCTs and in Table 4 for the 2009 sample. Far fewer studies were done in neonates and infants than in older paediatric patients. There was no statistically significant difference in age-group distribution between 2006 and 2009 (p=0.77). Sample size ranged from 22 to 69,274, with a median of 196. The male-to-female ratio was available for 67 studies and ranged from 0.1 to 0.91 with a median of 0.53. Topics of the PRCTs published in 2006 and 2009 are listed in The most widely studied topics were infectious diseases (15 on vaccines), respiratory diseases (including 19 on asthma), and mental and behavioural disorders. Only 2 studies dealt with surgical conditions. In 2006, trials on asthma and vaccines contributed 29% of all trials.

=> 144 full-length articles assessed for eligibility

		417 articles
		published in 2006
	identified as PRCTs in MEDLINE
		using PubMed
	GMJs: n=90
	PMJs: n=176
	SMJs: n=151
	Random sample of 35% in
		each journal category
	13 Exclusions	14 Exclusions	35 Exclusions
	13 adult RCTs	3 adult RCTs	20 adult RCTs
		5 previously	10 previously
		reported RCTs	reported RCTs
		5 not RCTs	5 not RCTs
		1 other
	GMJs 18 PRCTs included	PMJs 46 PRCTs included	SMJs 18 PRCTs included
		82 articles studied

ACKNOWLEDGMENT:We thank A Wolfe MD for helping to prepare the manuscript.