Xavier Milhaud

Denys Pommeret

Yahia Salhi

Pierre Vandekerkhove

Shape constraint free two-sample contamination model testing

Keywords: AMS 2000 subject classifications. Primary 62G05, 62G20; secondary 62E10 finite mixture model, semiparametric estimation, Cramér-von Mises, mortality

come

Introduction

Let us consider the semiparametric two-component mixture model with cumulative distribution function (cdf)

L(x) = (1 -p)G(x) + pF (x), x ∈ R, (1)
where G is a known cdf and where the unknown parameters are the mixture proportion p ∈]0, 1[ and the cdf F which is not supposed to belong to any parametric family. This model, sometimes so-called contamination or admixture model, has been widely investigated in the last decades, see for instance Bordes and Vandekherkove [START_REF] Bordes | Semiparametric two-component mixture model when a component is known: an asymptotically normal estimator[END_REF], Matias and Nguyen [START_REF] Nguyen | On efficient estimators of the proportion of true null hypotheses in a multiple testing setup[END_REF], Cai and Jin [START_REF] Cai | Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing[END_REF] or Celisse and Robin [START_REF] Robin | A cross-validation based estimation of the proportion of true null hypotheses[END_REF] among others. Numerous applications of model ( 1) can be found in topics such as: i) genetics regarding the analysis of gene expressions from microarray experiments as done in Broët et al. [START_REF] Broët | A mixture model-based strategy for selecting sets of genes in multiclass response microarray experiments[END_REF]; ii) the false discovery rate problem (used to assess and control multiple error rates such as in Efron and Tibshirani [START_REF] Efron | Empirical Bayes methods and false discovery rates[END_REF]), see McLachlan et al. [START_REF] Mclachlan | A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays[END_REF]; iii) astronomy, in which this model arises when observing variables such as metallicity and radial velocity of stars as in Walker et al. [START_REF] Walker | Clean kinematic samples in dwarf spheroidals: An algorithm for evaluating membership and estimating distribution parameters when contamination is present[END_REF]; iv) biology to model trees diameters, see Podlaski and Roesch [START_REF] Podlaski | Modelling diameter distributions of two-cohort forest stands with various proportions of dominant species: A two-component mixture model approach[END_REF]; v) kinetics to model plasma data, see Klingenberg et al. [START_REF] Klingenberg | A consistent kinetic model for a twocomponent mixture with an application to plasma[END_REF], vi) genomics to represent populations formed by admixture between ancestral founding populations as in Chakraborty and Weiss [START_REF] Chakraborty | Admixture as a tool for finding linked genes and detecting that difference from allelic association between loci[END_REF], among many other fields of applications. We recommend also the excellent survey on semiparametric mixture models by Xiang et al [START_REF] Xiang | An overview of Semiparametric Extensions of finite Mixture Models[END_REF] to have a panoramic view on this last generation of mixture models.

In this paper, the data of interest is made of two i.i.d. samples X 1 = (X 1,1 , . . . , X 1,n 1 ) and X 2 = (X 2,1 , . . . , X 2,n 2 ) with respective cdfs:

L 1 (x) = (1 -p 1 )G 1 (x) + p 1 F 1 (x), x ∈ R L 2 (x) = (1 -p 2 )G 2 (x) + p 2 F 2 (x), x ∈ R, (2)
where p 1 , p 2 are the unknown mixture proportions and F 1 , F 2 are unknown cdfs component we propose to name nodular distributions. For simplicity matters, we denote n = n 1 and consider that n 2 = κn where κ ≥ 1. In this work, similarly to Patra and Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF], we will consider situations where the G i 's and F i 's distributions are: i) absolutely continuous with respect to the Lebesgue measure, supported over R, R + or intervals of R; ii) finite discrete or N-discrete distributions such as Binomial or Poisson; iii) a mixture of a discrete and an absolutely continuous distribution. All our results will be still valid in such frameworks. Given the above model, our goal is now to answer the following statistical problem:

H 0 : F 1 is equal to F 2 against H 1 : F 1 is different from F 2 (3)
without assigning any specific parametric family to the F i 's.

Such a problem arises in various applications. For instance, studying the genetic make up of populations has been the subject of much attention. The admixture behaviour is of paramount importance in comparing the populations with known ancestors and exhibits linkage relationships between them, see Loh et al. [START_REF] Loh | Inferring admixture histories of human populations using linkage disequilibrium[END_REF]. In a more general sense, such a testing problem arises when a known random phenomenon is contaminated by an unknown "nodular" random effect, which may correspond to non-observed heterogeneity. This can be also the case, for example, during crisis where some populations are clearly impacted when others are not concerned yet. The objective of the test is then to compare two independent populations that are contaminated. The case of the coronavirus disease's example is appealing in the sense that the excess of mortality is clearly affecting populations over the world in a different manner. Its nodular impact, among others, can be treated under the statistical problem exposed in [START_REF] Bordes | Semiparametric estimation of a two components mixture model[END_REF], where the known component refers the baseline mortality observed during the recent years.

The testing strategy we propose here is very different from the one proposed in Milhaud et al. [START_REF] Milhaud | Semiparametric twosample mixture components comparison test[END_REF] where a semiparametric penalized χ 2 -type test is used. The latter test is based on a √ n-consistent estimation p n = ( p 1,n , p 2,n ) of p = (p 1 , p 2 ) along with a pairwise comparison of the F i 's p n -plugged in orthogonal polynomial basis expansion coefficients estimation. The √ nconsistency of p n is satisfied under both H 0 or H 1 when the zero-symmetry of the f i 's, pdf of the F i 's is assumed, according to Bordes and Vandekerkhove [START_REF] Bordes | Semiparametric two-component mixture model when a component is known: an asymptotically normal estimator[END_REF]. However when considering the general Patra and Sen setup (see [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF], Theorem 3), the √ n-consistency is not theoretically achieved.

Despite that, Milhaud et al. [START_REF] Milhaud | Semiparametric twosample mixture components comparison test[END_REF] proposed interestingly to plug-in the p n estimate of Patra and

Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF] in their testing approach to study its numerical performance in practice. To overcome the lack of √ n-consistency under H 0 or H 1 in the Patra and Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF] setup, and after all get a complete valid asymptotic theory, we decided to rethink from scratch the two-sample testing problem [START_REF] Bordes | Semiparametric estimation of a two components mixture model[END_REF].

Our idea relies basically on the definition of two parametric function families:

F i = F i (x, L i , p i ) := L i (x) -(1 -p i )G i (x) p i , p i ∈ Θ i , x ∈ R , i = 1, 2, (4)
where Θ i is a compact set of R + \ {0}. For simplicity matters and without loss of generality we will consider Θ 1 = Θ 2 = [δ 1 , δ 2 ], where 0 < δ 1 < 1 < δ 2 < +∞. It is important to notice here that the F i 's are not constrained to contain exclusively cumulative distribution functions and that the parametric space Θ i associated to the p i 's is not necessarily a [δ, 1 -δ]-type subset, 0 < δ < 1, of the natural ]0, 1[ mixture proportion support anymore. On the other hand, the key point is that the true F i 's belong respectively to these classes by picking the true value of the parameters p * i ∈ Θ i , when δ ≥ δ 1 > 0 are taken small enough, i.e.

F i (x) = F i (x, L i , p * i ), x ∈ R. (5)
For convenience we introduce F the set of all the probability cumulative distribution functions.

Consider now the following discrepancy measure

d(θ) = R (F 1 (x, L 1 , p 1 ) -F 2 (x, L 2 , p 2 )) 2 dU (x), (6)
with θ = (p 1 , p 2 ) ∈ Θ = [δ 1 , δ 2 ] 2 (because these quantities are not specifically viewed as mixture proportions anymore), measuring possible departures between the functions F 1 (•, L 1 , p 1 ) and F 2 (•, L 2 , p 2 ). We denote θ = (p 1 , p 2 ) contrarily to p previously to stress out the fact that the fitting in (p 1 , p 2 ) is done jointly now when it was done F i -wisely, i = 1, 2, in Patra and Sen setup [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF] or Bordes and Vandekerkhove [START_REF] Bordes | Semiparametric two-component mixture model when a component is known: an asymptotically normal estimator[END_REF]. The integrating cdf U should obviously be ideally chosen in order to focus (highly weight) on domains where the F i (•, L i , p i )'s cleary depart from each other to help on the final test decision. Nevertheless since the structure of the F i 's is constantly changing as the p i 's vary in the parametric space, we propose in practice to consider for U rather flat distributions encompassing the support of the observations.

It is worth to notice that under H 0 , there exists

p 1 = p * 1 and p 2 = p * 2 such that d(θ * ) = d(p * 1 , p * 2 ) = 0.
Suppose now that under some regularity and identifiability-type conditions we could prove that:

arg min θ∈Θ d(θ) = θ * d(θ * ) = 0, under H 0 , and arg min θ∈Θ d(θ) = θ c d(θ c ) > 0, under H 1 , (7)
we would directly have under

H 1 : inf θ∈[δ,1-δ] 2 : F i (•,L i ,p i )∈F ,i=1,2 d(θ) ≥ inf θ∈Θ d(θ) = d(θ c ) > 0. ( 8
)
Note that the search of the infimum in the left hand side of ( 8) matches what we would normally expect in a classical semiparametric estimation problem (mixing proportions in ]δ, 1 -δ[, for δ > 0 small enough, and F i 's in the cdfs range), when the second infimum have much more relaxed constraints (mixing proportions in a compact set Θ of (R + ) 2 embedding ]δ, 1 -δ[ 2 and no specific constraints on the F i 's) that we claim to be sufficient to solve our two sample testing problem.

This relaxation in the optimisation problem, see [START_REF] Robin | A cross-validation based estimation of the proportion of true null hypotheses[END_REF][START_REF] Chakraborty | Admixture as a tool for finding linked genes and detecting that difference from allelic association between loci[END_REF], that was a blocking point to achieve the √ n-consistency of the estimator p n in Patra and Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF] -method returning a parameter p ∈]0, 1[ and a true isotonic regression-based cdf estimate for F , is the first key idea of our paper. It is important to notice at this stage that if we let the parameters p i , i = 1, 2, go together to infinity in the parametric expressions (4), the functions F i (•, L i , p i ) mecanically flatten to 0 which makes the discrepancy measure d(θ) → 0, see expression [START_REF] Cai | Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing[END_REF]. As illustrated in Appendix B, we then have to face two types of situations: i) there exists a local minima of d(θ), θ * under H 0 or θ c under H 1 , in the interior of the parametric space, and then the testing problem is non-trivial and should be addressed, or ii) the optimization of d(θ) shows that we bump into the boundaries of the parametric space, i.e. one of the component of θ c is equal to δ 2 because the main way to reduce the contrast d(θ) is to make θ large, and then the testing problem is not even worth to be adressed because there is no "reasonable" θ c = (p c 1 , p c 2 ) close to the probability weights domain [0, 1] 2 that make

F 1 (x, L 1 , p c 1 ) close to F 2 (x, L 2 , p c 2 ).
Now the empirical estimate d n (•) of d(•) obtained by replacing the L i 's by the accessible empirical cdfs L i in [START_REF] Cai | Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing[END_REF], would naturally lead us to find, respectively under H 0 or H 1 , the true value of the parameter θ * , respectively the (F 1 , F 2 )-models distance minimizer θ c , by looking at:

θ n = arg min θ∈Θ d n (θ). ( 9
)
We propose to call IBM-method the semiparametric estimation strategy based on the "Inversion" step (4) and the "Best Matching" step (9) between the F 1 and F 2 families to look at the closest they can possibly be. At this stage of the reasoning it still looks hard to figure out how the finding of an "only" H 0 -consistent estimation method could solve our two-sample testing problem (3). The next part of the intuition consists in looking at the asymptotic behavior of the stochastic process

(a n d n ( θ n )) n≥1
, for a well chosen increasing sequence of real numbers (a n ) n≥1 such that a n → +∞ as n → +∞. This way we could ideally expect a clear hypothesis separation coming from:

a n d n ( θ n ) = a n [d n ( θ n ) -d(θ * )] L → Z 0 , under H 0 a n d n ( θ n ) = a n [d n ( θ n ) -d(θ c )] + a n d(θ c ) a.s. → +∞ under H 1 ,
with Z 0 an identified limiting random variable (which distribution could be at least tabulated).

Unfortunately since under H 1 we could not probably get access to the limiting distribution of a Z 0reference distribution to build-up our test we could try to figure out a way to exhibit a distribution under H 1 that could be in the regime of a Z 0 distribution (to decide which magnitude of deviation from 0 could be considered as excessive or not in a test perspective). This is the second key idea of our paper. By taking a n = n and by making a closer analysis of nd n ( θ n ) under H 1 we observe the following asymptotic behavior (see also the end of Appendix A):

nd n ( θ n ) = U 0 n L → Z(θ * , L 1 , L 2 ), under H 0 nd n ( θ n ) = U 1 n + V 1 n , with U 1 n L → Z(θ c , L 1 , L 2 ) and V 1 n a.s. → +∞, under H 1 ,
where the random variables Z(θ * , L Finally, by considering an empirical sample-based (1 -α)-quantile of the stochastic integral

Z( θ n , L 1 , L 2 )
, denoted q 1-α , we decide to consider the following H 0 -rejection rule:

nd n ( θ n ) ≥ q 1-α ⇒ H 0 is rejected. (10)
The above decision rule expresses the following principle: if the test statistic nd n ( θ n ) is too far from the inner convergence regime we could legitimately suspect a difference between F 1 and F 2 , as illustrated in the right side of Fig. under θ we obtain for all x ∈ R:

F 1 (x, L 1 , p 1 ) = L 1 (x) -(1 -p 1 )G 1 (x) p 1 and F 2 (x, L 2 , p 2 ) = L 2 (x) -(1 -p 2 )G 2 (x) p 2 . ( 11
)
Let us investigate now the situations where possibly F 1 (x, p 1 ) = F 2 (x, p 2 ) :

F 1 (x, L 1 , p 1 ) = F 2 (x, L 2 , p 2 ) ⇔ L 1 (x) -(1 -p 1 )G 1 (x) p 1 = L 2 (x) -(1 -p 2 )G 2 (x) p 2 ⇔ (p 1 -p * 1 )G 1 (x) + p * 1 F 1 (x) p 1 = (p 2 -p * 2 )G 2 (x) + p * 2 F 2 (x) p 2 ⇔ p 1 -p * 1 p 1 G 1 (x) = p 2 -p * 2 p 2 G 2 (x) + p * 2 p 2 F 2 (x) - p * 1 p 1 F 1 (x). ( 12
)
Under H 0 , F 1 = F 2 = F , we simply obtain

p 1 -p * 1 p 1 G 1 (x) = p 2 -p * 2 p 2 G 2 (x) + p * 2 p 2 - p * 1 p 1 F (x). Hence, if G 1 / ∈ span(G 2 , F ),
θ c = arg min θ∈Θ d(θ), where d(θ) = R D 2 (x, L 1 , L 2 , θ)dU (x), (13)
with

D(x, L 1 , L 2 , θ) = F 1 (x, L 1 , p 1 ) -F 2 (x, L 2 , p 2 ), and
F i (x, L i , p i ) = L i (x) -(1 -p i )G i (x) p i , i = 1, 2,( 14
)
where U is a continuous distribution which support encompasses the support of the L i 's. For simplicity, we will denote hereafter D(x, θ) = D(x, L 1 , L 2 , θ) and

F i (x, p i ) = F i (x, L i , p i ), i = 1, 2,
except when the role of the L i 's is central in our study.

Asymptotic results

To look at the proofs related to our theoretical results, the reader is referred to Appendix A. We introduce here two assumptions connected to the identifiability and definite positiveness of the d-Hessian matrix.

(I) The cdfs family

E 1 = {G 1 , F 1 , G 2 , F 2 } is linearly independent.
(II) The family of functions

E 2 = {G 1 G 2 , F 1 F 2 , G 2 i , F 2 i , G i F i ; i = 1, 2} is linearly independent.
We consider in the above assumptions that, by convention, under

H 0 (F 1 = F 2 = F ) the cdfs family E 1 reduces to {G 1 , F, G 2 } and E 2 reduces to {G 1 G 2 , F 2 , G 2 i , G i F ; i = 1, 2}.
We assume now the following technical assumption: In order to consistently estimate θ * under H 0 and make sure that under H 1 (F 1 = F 2 ) the functions F 1 (•, p 1 ) and F 2 (•, p 2 ) will also differ from each other for any value of θ ∈ Θ, we consider the following estimator:

(A) θ * under H 0 (θ c = θ * ), or θ c under H 1 (θ c = θ * ),
θ n = arg min θ∈Θ d n (θ), (15)
where

d n (θ) = R D 2 (x, L 1 , L 2 , θ)dU (x), with L i (x) = 1 n i n i k=1 I X i,k ≤x i = 1, 2.

Theoretical results

In the sequel, we denote by ˙ (ϑ) and ¨ (ϑ) the gradient vector and hessian matrix of any real function (when it makes sense) with respect to argument ϑ ∈ R 2 . The notation A T refers to the transpose matrix of matrix A.

Lemma 1. (i) The mapping θ → d(θ) is C 2 over Θ both under H 0 or H 1 .
(ii) Assume that conditions (I) and (A) hold. If U is strictly increasing on an interval I U that encompasses the support of the L i 's and G i 's, i = 1, 2, then under H 0 , d is a contrast function, (iv) We have under H 0 or H 1 that:

sup θ∈Θ |d n (θ) -d(θ)| = o a.s. (n -1/2+α ), for all α > 0. ( 16
)
(v) Assume that conditions (I), (II), and (A) hold. Then under H 0 we have:

d(θ * ) = 2 R Ḋ(x, θ * ) ḊT (x, θ * )dU (x) > 0. ( 17
)
(vi) Assume that conditions (I), (II), and (A) hold. Then under H 1 we have:

d(θ c ) = 2 R D(x, θ c )D(x, θ c ) + Ḋ(x, θ c ) Ḋ(x, θ c ) T dU (x) > 0.
Let us denote • 2 the Euclidean distance in R 2 , and θ c = θ * if the assumption H 0 is specified.

Theorem 1. If conditions (I), (II) and (A) hold, we have under

H 0 or H 1 that θ n -θ c 2 = o a.s. (n -1/4+α
) for all α > 0.

Theorem 2. i) If conditions (I), (II) and (A) hold, we have under H 0 or H 1 :

√ n   p 1 -p c 1 p 2 -p c 2 D n (•) -D(•)   L → W (θ c , •), as n → +∞, ( 18
)
where

D n (•) = D(•, L 1 , L 2 , θ n ), and W (θ c , •) = (W 1 (θ c ), W 2 (θ c ), W 3 (θ c , •)) T is a centered 3-dimensional
Gaussian process with covariance matrix

Σ W = M (θ c , •)Σ L (•, •)M (θ c , •) T where M (θ c , •) is defined in (38) and Σ L (•, •) in (41).
ii) If conditions (I), (II) and (A) hold, we have respectively as n → +∞:

nd n ( θ n ) = U 0 n L → Z(θ * ) = R (W 3 (θ * , x)) 2 dU (x), under H 0 , (19)
nd n ( θ n ) = U 1 n + V 1 n , with U 1 n L → Z(θ c ) = R (W 3 (θ c , x)) 2 dU (x) and V 1 n = O a.s. (n), under H 1 . (20)
Let us remind that the above stochastic integrals distribution can be simulated by standard Monte Carlo methods, see for instance [START_REF] Higham | An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations[END_REF], and thus fully tabulated. As detailed in the Introduction the use of the above theorem in our testing perspective consists in rejecting H 0 if the statistic

nd n ( θ n ) exceeds q 1-α , where q 1-α is the approximated (1 -α)-quantile of the limiting random variable R (W 3 (θ c , x)) 2 dU (x) (given that by convention θ c = θ * under H 0 ).
We propose, in order to prevent from obvious not competing situations, to check if a θ c (1 -α)-domain of confidence, denoted I 1-α (θ c ), intersects somehow the ]0, 1[ 2 proportions domain. Let us consider

I 1-α (θ c ) = I 1-α/2 (p c 1 ) × I 1-α/2 (p c 2 ), where I 1-α/2 (p c i ) = I - i,1-α/2 , I + i,1-α/2 ,( 21
)
with

I - i,1-α/2 = p i - Σ W ( θn) [i, i] √ n φ 1 - α 4 and I + i,1-α/2 = p i + Σ W ( θn) [i, i] √ n φ 1 - α 4 ,
where φ(•) denotes the quantile function of the N (0, 1) distribution, and Σ[i, i] the i-th diagonal term of matrix Σ. In the sequel we denote either A the complimentary of a generic probability event A and I the complimentary of a generic domain

I of R d , d ≤ 2.
Noticing that according to Therorem 2 we have

P (I 1-α/2 (p c i )) ≈ 1 -α/2 as n → +∞, i = 1, 2, we can write P (θ c ∈ I 1-α (θ c )) = P (θ c ∈ I 1-α (θ c )) ≤ P (p c 1 ∈ I 1-α/2 (p c 1 )) + P (p c 2 ∈ I 1-α/2 (p c 2 )) = P (p c 1 ∈ I 1-α/2 (p c 1 )) + P (p c 2 ∈ I 1-α/2 (p c 2 )) ≈ α,
which leads to

P (θ c ∈ I 1-α (θ c )) ≥ 1 -α approximately as n → +∞. ( 22
)
Finally a simple "green light" criterion to proceed to the test could be the checking of the condition:

I - i,1-α/2 < 1, i = 1, 2 (green light testing criterion). ( 23
)
Remark 3. Although the same underlying idea can be used, the case where G 1 = G 2 is slightly different and requires a "picking trick" as the parameters are no longer identifiable even under H 0 (see Appendix E for further details).

Convergence Monte Carlo assessment

Introduce the random vector (P

1 , P 2 , D z ) T = √ n ( p 1 -p c 1 , p 2 -p c 2 , D n (z) -D(z)) T at any point z ∈ support(X 1 , X 2 ).
Recall that z represents one location point of the empirical process trajectory.

Theorem 2 states that this vector is asymptotically Gaussian, and that its first two components are consistent towards θ c (both under H 0 or H 1 ). Our goal is to check this by comparing numerical approximations of our theoretical expressions to Monte Carlo experiments. To this aim, we consider K = 200 simulations of two samples X 1 and X 2 with cdfs given by (2), both following twocomponent mixtures of Gaussian distributions. More precisely, the k-th simulation provides X k 1 and X k 2 (k = 1, ..., K), where X k 1 and X k 2 are respectively drawn from mixtures with parameters

n 1 = n 2 = 5,000, p * 1 = 0.4, p * 2 = 0.6, F 1 = F 2 are N (1, 1) cdfs, when G 1 , G 2 are respectively N (2, 0.7) and N (3, 1.
2) cdfs. Note that we are here under the null, but remember that such comparisons were also made on very different setups involving H 1 -type frameworks, with n 1 = n 2 and distributions supported over R + , N or bounded intervals of R.

Estimating θ = (p 1 , p 2 ) by ( 15) from each of the K simulated couples (X 1 , X 2 ), we obtain (0.402, 0.601) as empirical mean of θ n = ( p 1 , p 2 ), illustrating the asymptotic consistency of our estimators. Kolmogorov-Smirnov tests on the components of the vector (P 1 , P 2 , D z ) T validate that the three estimators are asymptotically Gaussian, with p-values always greater than 0.7. To validate the explicit covariance structure between the estimators, it is necessary to fix z and to compare the empirical covariances (computed from the Monte Carlo simulations) to the theoretical ones. Appendix C shows the obtained results in the aforementioned parametric setup for z = 2.

U 1 n + V 1 n (dotted).
Clearly, all the tests made through these comparisons for different values of z show the validity of formulas ( 38)-( 41), which confirms the theoretical consistency given in part i) of Theorem 2.

It now remains to have a closer look at the behaviour of the statistic nd n ( θ n ), see formulas [START_REF] Mclachlan | A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays[END_REF] and [START_REF] Milhaud | Semiparametric twosample mixture components comparison test[END_REF]. The theorem states that the empirical distribution of U 0 n under H 0 (obtained through the Monte Carlo procedure providing as many realizations of nd n ( θ n ) as the K experiments) should converge to some explicit random variable Z(θ * ). Also, the same kind of regime for U 1 n should be observed under H 1 . However, in the latter case, the discrepancy measure dramatically increases due to the term V 1 n , exhibiting the departure from the null hypothesis. This phenomenon is well illustrated by Figure 1, where one can see that the empirical distributions suit the expected behaviours provided by the random variables Z(θ * ) and Z(θ c ). Indeed, under H 1 , the empirical distribution of nd n ( θ n ) is far from Z(θ c ), showing the impact of the drift V 1 n . This way, the tabulated distributions Z(θ * ) and Z(θ c ) and their appropriate (1 -α)-quantile can be used to fruitfully answer our testing problem.

Test performance

In this section, we study the empirical levels and powers of the test in various situations. To this aim, we generate X 1 and X 2 from (2) on various supports, and the behaviour of the test is investigated in more or less challenging setups. Depending on the case under study, mixture components can be easily detected or not, either because of the importance of the mixture weight p i , i = 1, 2, or due to the specified mixture components features. The idea is to get some insights about the strengths and weaknesses of our testing procedure. Each time we evaluate the empirical level (respectively power) of the test, the 95-th percentile of the test was previously assessed using 150 trajectories in the computation of the stochastic integral appearing on the right-hand side of ( 19) and [START_REF] Milhaud | Semiparametric twosample mixture components comparison test[END_REF]. Then the testing procedure [START_REF] Genest | Testing for independence in arbitray distribution[END_REF] is performed K times to get the result, through the K simulations of the k-th samples X k 1 and X k 2 and the associated test statistic nd n ( θ k n ) (k = 1, ..., K). Here, we take K = 100, fix similar sample sizes n 1 = n 2 = n for conciseness, and make n vary.

Empirical levels (F

1 = F 2 = F )
In terms of distributions depending on the support, we consider Gaussian-Gaussian mixtures on R, Gamma-Exponential ones on R + , Negative-Binomial-Poisson on N, and Logit-Uniform on [0, 1].

To check the significance of the test in real-life situations, we have chosen to make the component weights (p i ) i=1,2 vary from 10% to 60%. The asymptotic properties of the test can be checked by considering different values for the number n of observations (ranging from 500 to 10,000).

However, our experiments show that the number of observations does not have a big impact on the level of the test, provided that there are at least around 300 observations for the mixture component to test. That is why we have chosen to present here only the results corresponding to n = 2, 000 observations, which lightens our results presentation.

For each support (R, R + , N, [0, 1]), four very different frameworks are studied (see Figure 9 in Appendix F.1, with corresponding mixture parameters stored in Table 2). We will denote from setups in all (4 supports, 4 cases, and 9 combinations for p 1 and p 2 ). We remind that for each of these 144 possibilities, the testing procedure ( 10) is performed 100 times, which leads to give an approximation of the empirical level of the test in all of the aforementioned situations.

The overall results are summarized thanks to the heatmap in Figure 2, with dark zones indi-

Empirical powers

In the same spirit, one can analyse the heatmap that illustrates the empirical power of the test in here that the most difficult case to detect is the latter one. Here, the sample size has a major impact on the results, which explains why the heatmap is provided for results corresponding to a sensitively higher sample size n = 3, 000. To understand how crucial the number of observations is, Figure 4 depicts the connection between the empirical power of the test and n. In fact we can observe very heterogeneous behaviours depending on the support and component weights, especially in the case where alternatives are very difficult to distinguish, that is, in the case where 3 of Appendix F.2, case (e), see also Fig. 11).

F

Application to COVID-19 excess mortality

There is an abundant literature investigating the impact of the 2019 coronavirus disease (COVID-19) on the mortality across countries, see for instance Beaney et al. [START_REF] Beaney | Excess mortality: the gold standard in measuring the impact of COVID-19 worldwide?[END_REF]. We generally witness a wide variation in mortality across countries, leading to questioning the extent to which one can proceed to pairwise comparative studies. In our application, we will be looking at the nodular impact of the COVID-19 and compare the latter across a panel of European countries. Formally, we investigate the age distribution of deaths (the distribution of the proportion of deaths per age group among all deaths) and we study the changes between 2019 and 2020 for France, Belgium, Germany, Italy, Netherlands and Spain from the Short-Term Mortality Fluctuations (STMF) data series compiled by the Human Mortality Database (HMD). The datasets contain death records aggregated over age groups: 0-14, 15-64, 65-74, 75-85 and 85+. We restrain our study to the first 25 weeks (and ages over 15 years-old) of each considered year as shown in Figure 5. Figure 6 shows the distribution of the proportion of deaths per age class for years 2019 and 2020 (total of proportions equals to 1), indicating the empirical probability for a death to be in each age class.

It is assumed that the differences in the observed mortality between 2019 and 2020 is imputed can be regarded as a measure that encompasses all causes of death and provides a metric of the overall mortality impact in 2020.

Week

Death Records 800 100012001400160018002000

W 1 W 4 W 7 W 1 0 W 1 3 W 1 6 W 1 9 W 2 2 W 2 5
BEL 9000 11000 13000 DEUTNP 4000 6000 8000 10000

W 1 W 4 W 7 W 1 0 W 1 3 W 1 6 W 1 9 W 2 2 W 2 5
ESP 5000 6000 7000 8000 9000 In Table 1 we report the outputs of the testing procedure developed in this paper for the aforementioned countries. The known component G i is described as the multinomial distribution computed in 2019 for each country. We shall stress out that in this application we are clearly in presence of two distinct known cdfs, i.e. G 1 = G 2 , which is our basic assumption to implement our procedure. Also, in this case, we choose the discrete uniform distribution for the integrating cdf U , see [START_REF] Higham | An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations[END_REF]. In order to avoid any departure of the estimated proportion p i to the infinity and thus make the discrepancy measure go to 0, we bounded the parametric space over which we optimize. Here, we set the upper bound equal to δ 2 = 3, which seems to be large enough given the Netherlands and Italy (56%), on the other hand, shares the same excess mortality profile. However, the equality between Spain and Netherlands is rejected. This result can be interpreted as follows:

FRATNP W 1 W 4 W 7 W 1 0 W 1 3 W 1 6 W 1 9 W 2
the estimation procedure tends to find a common cdf, that is given in Figure 7 where it can be seen a slight difference between the two last classes which are less numerous and therefore more sensible with a larger variability. We also have to take into account that the Netherlands second component estimation is based on only 7.22% of the observations.

F 1 = F 2 = F ,
In conclusion, the excess mortality Italian profile seems to be very similar to the Spain one with a very large p-value. A part of this excess mortality seems to be similar to the Netherlands one, with a lower p-value. But our test procedure do not retain the equality between Netherlands and Spain excess mortalities.

Eventually, the proportion for Spain and Italy approximate respectively 30.58% and 14.04%, which is consistent with the reported statistics [START_REF] Beaney | Excess mortality: the gold standard in measuring the impact of COVID-19 worldwide?[END_REF][START_REF] Mannucci | Factors associated with increased all-cause mortality during the COVID-19 pandemic in Italy[END_REF]. The discussion of such a behaviour is, however, beyond the scope of this paper. Instead, we can refer to the various discussions in the literature that intended to understand the differential impact of the COVID-19 crisis over countries looking at the socio-economic and demographic variables. In our case, we depict the mortality during this first wave of the pandemic for countries validating the null hypothesis H 0 (Spain, Italy and Netherlands), see the left panel of Figure 7. We see that these countries exhibit a comparable behavior but which cannot be visually validated.

Death Distribution

IBM-method and further models

In the next two sections we propose to highlight on the range of our method, to describe challenging situations (involving dependencies) in which our semiparametric IBM-method could provide interesting results. These are ongoing works which are beyond the scope of the current paper.

Independence, concordance and discordance

Let us consider for simplicity a bivariate contamination model (extension to the d-variate setup, d ≥ 3, being straightforward):

L(x 1 , x 2 ) = pG(x 1 , x 2 ) + (1 -p)F (x 1 , x 2 ), (x 1 , x 2 ) ∈ R 2 , ( 24
)
where L is the common cdf of an i.i.d. sample (X 1 , . . . , X n ), G is a known cdf when the mixture proportion p and the cdf F are both unknown. By splitting the observation vector X into 2 components X = (X 1 , X 2 ) T , we have respective marginal cdfs

L i (x) = pG i (x) + (1 -p)F i (x), x ∈ R, i = 1, 2. ( 25
)
An interesting problem is then to test the mutual independence of the nodular components X 1 and X 2 , i.e.

H 0 : F = F 1 ⊗ F 2 against H 0 : F = F 1 ⊗ F 2 , (26)
where G = G 1 ⊗ G 2 on a µ-non null set to avoid trivial testing situations (otherwise independence on the L-components would then reflect the independence on the F -components). Given the above remarks we can define two parametric families (Inversion step):

F 1 = F (u 1 , u 2 ; p) = L(u 1 , u 2 ) -pG(u 1 , u 2 ) 1 -p , p ∈]0, 1[ , and
F 2 = F 1×2 (u 1 , u 2 ; p) = F 1 (u 1 ; p)F 2 (u 2 ; p), F i (•; p) = L i (•) -pG i (•) 1 -p , i = 1, 2, p ∈]0, 1[ ,
and build a contrast function (Best Matching step) in the spirit of (13-14)

d(p) = R×R (F (u 1 , u 2 ; p) -F 1×2 (u 1 , u 2 ; p)) 2 dU (u 1 , u 2 ).
Using copula techniques to handle global and marginal empirical processes, as it is classically done in the "direct" (not mixture component testing) Cramér-von Mises independence testing literature, see Genest et al. [START_REF] Genest | Testing for independence in arbitray distribution[END_REF] for recent results and bibliography, we reasonably think that asymptotic decision results similar to ii) in Theorem 2 could be established on the test statistic nd n ( p n ) where d n is the empirical version of d and p n is the minimum argument of d n over ]0, 1[. Note that such accomplishment would also help in answering/testing the complete concordance/discordance problem arising in z-score analysis, see Lai et al. [START_REF] Lai | A mixture model approach to the tests of concordance and discordance between two large scale experiments with two-sample groups[END_REF] and Lai et al. [START_REF] Lai | An efficient concordant integrative analysis of multiple large-scale two-sample expression data sets[END_REF], where basically model [START_REF] Pommeret | Semiparametric density testing in the contamination model[END_REF] can take (among others) two basic forms:

L = pG 1 (x 1 ) ⊗ G 2 (x 2 ) + (1 -p)F 1 ⊗ F 2 , (complete concordance), L = (p 1 G 1 + (1 -p 1 )F 1 ) ⊗ (p 2 G 2 + (1 -p 2 )F 2 ), (complete discordance).
In fact in the above models, slightly more complex contrast functions d, based on F -inversions and comparison inspired from the previous independence testing strategy, can also be proposed and proved to provide a fully tractable Cramér-von Mises test in the spirit of Theorem 2.

Blending process

As mentioned earlier, the testing methodology we introduce in this paper can be extended to temporal contamination models we propose to name blending process. This type of model are especially interesting to analyze situations in which a phenomenon has been observed with a good stability for a long period of time but turns out to be contaminated by a new trend which importance becomes more and more prominent. This type of model would be especially relevant to analyze temporal mortality datasets during the COVID-19 crisis as described in Section 5 (collections of mortality datasets over time would be required instead of one single sample collected during a given period of time). By denoting G the cdf of the well known phenomenon and by p t , respectively F t , the proportion, resp. the cdf, of the new trend at time t, the distribution of a generic i.i.d sample X t = (X t 1 , . . . , X t nt ) at time t ∈ N could be expressed as follows:

L t (x) = p(t)G(x) + (1 -p(t))F t (x), x ∈ R. ( 27
)
In that setup it could be interesting, following the identifiability and parameter picking strategy presented in Section E when G 1 = G 2 , to test the consistency in time of the trend distribution,

i.e.

H 0 (t i , t j ) :

F t i = F t j against H 1 (t i , t j ) : F t i = F t j , i = j ∈ {1, . . . , T } . ( 28
)
Note that if the testing problem ( 28) is very mostly answered positively we could possibly assess that F is independent from t and then estimate nonparametrically the mixing proportion function p(t) based on the condition F t = F , t ∈ N and Remark 5 of Section E. The main technical difficulty here is to handle correctly the possible dependencies between samples X t i and X t j , for i = j, especially when t i and t j are close. Note that the analog of ( 59) and (66) need to established with a limiting bivariate Gaussian process B with no longer independent coordinates since the source samples X t i and X t j are dependent. The paper by Gribovka and Lopez [START_REF] Gribovka | Non-parametric copula estimation under bivariate censoring[END_REF] on non-parametric copula estimation under bivariate censoring loooks to provide interesting ideas to solve this problem.

Conclusion

In this work, we address the comparison testing of the unknown components of a two sample contamination model. We introduce the so-called IBM (Inversion-Best Matching) approach that results into a relaxed semi-parametric Cramér-von Mises type two-sample test with very minimal assumptions about the unknown components. Indeed, we do not require any shape constraints on the unknown distributions, such as symmetry, tail conditions etc. which are very often key technical identifiability conditions arising in univariate semiparametric mixture models. We establish in particular a functional joint central limit theorem on the proportion parameters (with consistency under H 0 ) along with the best fitted differences between the unknown cdfs, which is unachievable in the basic univariate case as shown by Patra and Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF]. An intensive numerical study has been carried out from a large range of simulation setups to illustrate the asymptotic properties of our test. This includes examples using Gaussian distributions but also more challenging distributions supported on R + , N or [0, 1] which are considered as very non-standard in the mixture models literature. Finally, our testing procedure is applied to a real-life case attempting to fill the gap in understanding the disparities of the excess of mortality during the COVID-19 crisis, which allows to test pairwise the excess of mortality across a panel of European countries.

This work could be extended in many interesting ways, among which the case of paired samples which would particularly be interesting for time-varying models consideration (as in [START_REF] Ghattas | Data driven smooth test for paired populations[END_REF]). We should, indeed, consider realizations of such samples while tackling the problem of the underlying dependence structure. Also, coming back to the COVID-19 case, it is fortunate to develop a more adapted scheme to pairwise testing for the underlying populations. In fact, a clustering procedure would be beneficial along with a K-sample testing procedure based on the result of this paper. This could bring in a new challenging problem. Finally, given the ability of the test to accommodate very different frameworks, an upcoming R package implementing a wide variety of two-sample testing methods for contaminated models will be very soon available to researchers as well as practitioners.

A Proofs

Proof of Lemma 1. Note that, since p i ∈ Θ i = [δ 1 , δ 2 ], 0 < δ 1 < 1 < δ 2 < +∞, we have: (ii) Under H 0 , if θ = θ * then we have d(θ) = 0. To prove the reciprocal let us remark that d(θ) = 0 implies that F 1 (•, θ) = F 2 (•, θ) µ-almost everywhere (µ-a.e.) because U is strictly increasing over I U that includes the support of the L i 's and G i 's. Now according to (I) and ( 12) we necessarily have:

|F i (x, p i )| ≤ 1 + δ δ 1 , (x, p i ) ∈ R × Θ i , i = 1, 2 where δ = max(|1 -δ 1 |, |1 -δ 2 |). ( 29
     (p 1 -p 1 * )/p 1 = 0 (p 2 -p * 2 )/p 2 = 0 p * 2 /p 2 -p * 1 /p 1 = 0, ⇔ p 1 = p * 1 and p 2 = p * 2 ,
which concludes the proof.

(iii) Under H 1 (F 1 = F 2 ), if we assume that F 1 (•, p 1 ) = F 2 (•, p 2 ) then we similarly obtain:

         (p 1 -p * 1 )/p 1 = 0 (p 2 -p * 2 )/p 2 = 0 p * 2 /p 2 = 0 p * 1 /p 1 = 0,
which is impossible since the p i 's and p * i 's are supposed to be strictly greater than 0. Hence, for any θ ∈ Θ, we have d(θ) > 0 which implies the wanted result. iv) Let consider θ and θ two distinct points in Θ. We have according to (29):

|d(θ) -d(θ )| ≤ R |D(x, θ) + D(x, θ )| × |D(x, θ) -D(x, θ )| dU (x) ≤ 4(1 + δ) δ 1 R |D(x, θ) -D(x, θ )| dU (x).
Since for i = 1, 2 we have

|F i (x, p i ) -F i (x, p i )| = |L i -G i | p i -p i p i p i ≤ 2 δ 2 1 |p i -p i |, it comes |D(x, θ) -D(x, θ )| ≤ 2 i=1 |F i (x, p i ) -F i (x, p i )| ≤ 2 δ 2 1 2 i=1 |p i -p i | ≤ 4 δ 2 1 θ -θ 2 ,
which leads to

|d(θ) -d(θ )| ≤ 16(1 + δ) δ 3 1 θ -θ 2 ,
and proves the Lipschitz property for

θ → d(θ) over Θ. Let us denote D(x, θ) = D(x, L 1 , L 2 , θ)
where D(•) is defined in [START_REF] Klingenberg | A consistent kinetic model for a twocomponent mixture with an application to plasma[END_REF], and notice that

|d n (θ) -d(θ)| ≤ R D 2 (x, θ) -D 2 (x, θ) dU (x).
For all x ∈ R, we have:

D 2 (x, θ) -D 2 (x, θ) ≤ D(x, θ) + D(x, θ) × D(x, θ) -D(x, θ) ≤ 2(1 + δ) δ 1 D(x, θ) -D(x, θ) ≤ 2(1 + δ) δ 1 2 i=1 L i (x, θ) -L i (x, θ) ,
which leads to

|d n (θ) -d(θ)| ≤ 2(1 + δ) δ 1 2 i=1 L i (•, θ) -L i (•, θ) ∞ . Noticing that, for i = 1, 2, L i (•, θ) -L i (•, θ) ∞ = O a.s.
( n i -1 log log(n i )) (see Shorack and Wellner [START_REF] Shorack | Empirical Processes with Applications to Statistics[END_REF], p. 766), we obtain the wanted result.

v) For any θ ∈ Θ, we have

d(θ) = 2 R D(x, θ)D(x, θ) + Ḋ(x, θ) Ḋ(x, θ) T dU (x) = 2 R M (x, θ)dU (x),
where, as detailed in Appendix D, M (x, θ) is a 2 × 2 symmetric real-valued matrix for all x fixed in R. As a consequence, we have for all x ∈ R and any vector v ∈ R 2 :

v T M (x, θ)v ≥ 0 ⇒ v T d(θ)v = 2 R v T M (x, θ)vdU (x) ≥ 0, but also v T d(θ)v = 0 ⇒ M 1,1 (x, θ)v 2 1 + 2M 1,2 (x, θ)v 2 v 1 + M 2,2 (x, θ)v 2 2 = 0 µ -a.e.
Now according to assumption (II) and decomposition (48), the above nullity condition implies in particular that the coefficients associated to Proof of Theorem 1 By Lemma 1 v) or vi) there exists γ > 0 such that for all v ∈ R 2 ,

F 2 i 's are null, i.e. a 3 (p i , v i )(p * i ) 2 = 3v 2 i (p * i ) 2 /p 4 i = 0, i = 1, 2, which implies v 1 = v 2 = 0
v T d(θ c )v > γ||v|| 2 2
. By a two order Taylor expansion of d at point θ c ∈ o Θ we can find η > 0 such that for all v satisfying v < η and θ c + v ∈

• Θ , we have

d(θ c + v) ≥ γ 4 v 2 2 . (30)
Let us consider now B(θ c , η n ) the Euclidean ball centered at point θ c with radius η n > 0. Following the proof in Bordes et al. [START_REF] Bordes | Semiparametric estimation of a two components mixture model[END_REF] we show the following events inclusion:

lim sup n θ n / ∈ B(θ c , η n ) ⊆ lim sup n inf θ∈Θ\B(θ c ,ηn) d(θ) < ξ n ∪ lim sup n ξ n ≤ 2 sup θ∈Θ |d n (θ) -d(θ)| ,
for any arbitrary sequence ξ n . Choosing now ξ n = n -1/2+α and η n = n -1/4+β/2 , with 0 < α < β taken arbitrarily small, it follows from (30) and the uniform almost sure rate of d n given in Lemma 1 (iv), that

P lim sup n inf θ∈Θ\B(θ c ,ηn) d(θ) < ξ n = 0,
and

P lim sup n ξ n ≤ 2 sup θ∈Θ |d n (θ) -d(θ)| = 0.
In conclusion, θ n converges almost surely towards θ c at rate n -1/4+α , α > 0 chosen arbitrarily small. Proof of Theorem 2. i) By a Taylor expansion of ḋn about θ c ∈ o Θ we have:

dn ( θ n ) √ n( θ n -θ c ) = - √ n ḋn (θ c ), (31)
where θ n lies in the line segment with extremities θ n and θ c . Now writing that ḋ(θ) = ( ḋ1 (θ), ḋ2 (θ)) T ,

ḋ1 (θ) = 2E U (2 -p 1 )G 1 L 1 p 3 1 - (1 -p 1 )G 2 1 p 3 1 - L 2 1 p 3 1 - G 1 L 2 p 2 1 p 2 + (1 -p 2 )G 1 G 2 p 2 1 p 2 + L 1 L 2 p 2 1 p 2 - (1 -p 2 )L 1 G 2 p 2 1 p 2 ḋ2 (θ) = 2E U (2 -p 2 )G 2 L 2 p 3 2 - (1 -p 2 )G 2 2 p 3 2 - L 2 2 p 3 2 - G 2 L 1 p 2 2 p 1 + (1 -p 1 )G 2 G 1 p 2 2 p 1 + L 2 L 1 p 2 2 p 1 - (1 -p 1 )L 2 G 1 p 2 2 p 1 , we look at ḋ1,n (θ) -ḋ1 (θ) = 2 2 -p 1 p 3 1 T 1,1 - 1 -p 1 p 3 1 T 1,2 - 1 p 3 1 T 1,3 - 1 p 2 1 p 2 T 1,4 + 1 -p 2 p 2 1 p 2 T 1,5 + 1 p 2 1 p 2 T 1,6 - 1 -p 2 p 2 1 p 2 T 1,7 ḋ2,n (θ) -ḋ2 (θ) = 2 2 -p 2 p 3 2 T 2,1 - 1 -p 2 p 3 2 T 2,2 - 1 p 3 2 T 2,3 - 1 p 2 2 p 1 T 2,4 + 1 -p 1 p 2 2 p 1 T 2,5 + 1 p 2 2 p 1 T 2,6 - 1 -p 1 p 2 2 p 1 T 2,7 ,
where

T 1,1 (G 1 , L 1 ) = E U G 1 L 1 -L 1 T 1,2 (G 1 ) = E U G 2 1 -G 2 1 = 0 T 1,3 (L 1 ) = E U L 2 1 -L 2 1 = E U ( L 1 -L 1 )( L 1 + L 1 ) = E U ( L 1 -L 1 )(2L 1 + o a.s. ( 1)
)

T 1,4 (G 1 , L 2 ) = E U (G 1 ( L 2 -L 2 )) T 1,5 (G 1 , G 2 ) = E U (G 1 G 2 ) -E U (G 1 G 2 ) = 0 T 1,6 (L 1 , L 2 ) = E U L 1 L 2 -L 1 L 2 = E U L 1 ( L 2 -L 2 ) + L 2 ( L 1 -L 1 ) = E U (L 1 + o a.s. (1)( L 2 -L 2 ) + E U L 2 ( L 1 -L 1 ) T 1,7 (G 2 , L 1 ) = E U G 2 ( L 1 -L 1 ) = T 1,4 (G 2 , L 1 ) T 2,1 (G 2 , L 2 ) = E U G 2 ( L 2 -L 2 ) = T 1,1 (G 2 , L 2 ) T 2,2 (G 2 ) = E U G 2 2 -E U G 2 2 = 0 T 2,3 (L 2 ) = E U ( L 2 -L 2 )( L 2 + L 2 ) = E U ( L 2 -L 2 )(2L 2 + o a.s. (1)) = T 1,3 (L 2 ) T 2,4 (L 1 , G 2 ) = E U G 2 ( L 1 -L 1 ) = T 1,4 (L 1 , G 2 ) T 2,5 (G 1 , G 2 ) = E U (G 2 G 1 ) -E U (G 2 G 1 ) = 0 T 2,6 (L 1 , L 2 ) = E U L 2 L 1 -L 2 L 1 = T 1,6 (L 1 , L 2 ) T 2,7 (G 1 , L 2 ) = E U G 1 ( L 2 -L 2 ) .
For a generic cdf Y and a generic N -sample based empirical process

V = √ N ( V -V ), define ϕ(Y, V) = R Y (x)V(x)dU (x). Introducing S = (G 1 , G 2 , L 1 , L 2 ), let us consider Ψ 1,1 (S, θ) = 2 2 -p 1 p 3 1 G 1 - 2 p 3 1 L 1 + 1 p 2 1 p 2 L 2 - 1 -p 2 p 2 1 p 2 G 2 Ψ 1,2 (S, θ) = 2 1 p 2 1 p 2 (L 1 -G 1 ) Ψ 2,1 (S, θ) = 2 2 -p 2 p 3 2 G 2 - 2 p 3 2 L 2 + 1 p 2 2 p 1 L 1 - 1 -p 1 p 2 2 p 1 G 1 Ψ 2,2 (S, θ) = 2 1 p 2 2 p 1 (L 2 -G 2 ) ,
and

Φ 1,1 (L 1 , θ) = ϕ(Ψ 1,1 (S, θ), L 1 ), Φ 1,2 (L 2 , θ) = ϕ(Ψ 1,2 (S, θ), L 2 ), Φ 2,1 (L 2 , θ) = ϕ(Ψ 2,1 (S, θ), L 2 ), Φ 2,2 (L 1 , θ) = ϕ(Ψ 2,2 (S, θ), L 1 ).
Note that the first and fourth, respectively the second and third, expression depends only on the randomness of L 1 , resp. L 2 . We resume the above remarks into the following basic expression:

√ n( ḋn (θ) -ḋ(θ)) = Φ(L 1 , L 2 , θ) + o a.s (1), (32)
where, according to

√ n = √ κn/ √ κ = ζ √ n 2 with ζ = 1/ √ κ: Φ(L 1 , L 2 ) = Φ 1,1 (L 1 , θ) + ζΦ 1,2 (L 2 , θ) ζΦ 2,1 (L 2 , θ) + Φ 2,2 (L 1 , θ). (33)
Since the empirical processes L 1 and L 2 are independent, by the Donsker Theorem [27, The-

orem 19.3, p. 266], the vector [L 1 , L 2 ] converges in distribution in the space D[-∞, ∞] to a bi-dimensional zero-mean Gaussian process B, i.e. L 1 L 2 B = B 1 B 2 , (34)
where B is a bi-dimensional gaussian process with diagonal correlation matrix ρ = diag(ρ 1 , ρ 2 ),

where

ρ 1 (x, y) = L 1 (x ∧ y)(1 -L 1 (x ∨ y)) and ρ 2 (x, y) = L 2 (x ∧ y)(1 -L 2 (x ∨ y)).
Moreover,

√ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ c )] = √ n[F 1 (x, L 1 , p 1 ) -F 1 (x, L 1 , p c 1 )] - √ n[(F 2 (x, L 2 , p 2 ) -F 2 (x, L 2 , p c 2 ))]. (35)
Let decompose closely, for i = 1, 2, the terms

F i (x, L i , p i ) -F i (x, L i , p c i ): √ n[F i (•, L i , p i ) -F i (•, L i , p c i )] = √ n L i p i - L i p c i - 1 -p i p i - 1 -p c i p c i G i = √ n L i -L i p i + √ n p c i -p i p c i p i (L i -G i ) = ζ i 1 p c i L i - L i -G i (p c i ) 2 √ n[ p i -p c i ] + o P (1), (36)
where by convention ζ 1 = 1 and

ζ 2 = ζ = 1 √ κ .
It is also easy to prove that dn ( θ n ) a.s.

→ d(θ c ) > 0, as n → +∞. Indeed, let us now consider the decompositions (45), ( 46) and (47) in Appendix D, we have

[ dn ( θ n ) -d(θ c )] i,j ≤ R M i,j (x, L 1 , L 2 , G 1 , G 2 , θ n ) -M i,j (x, L 1 , L 2 , G 1 , G 2 , θ) dU (x) ≤ R M i,j (x, L 1 , L 2 , G 1 , G 2 , θ n ) -M i,j (x, L 1 , L 2 , G 1 , G 2 , θ n ) dU (x) + R M i,j (x, L 1 , L 2 , G 1 , G 2 , θ n ) -M i,j (x, L 1 , L 2 , G 1 , G 2 , θ) dU (x) ≤ C P( θ n ) 2 i=1 L i -L i ∞ + |P( θ n ) -P(θ c )| , (37)
where

P(θ) = 4 k=0 p -k 1 p -4+k
2 is a R 2 → R continuous mapping. Now by using on (37) the Glivenko-Cantelli theorem and the almost sure convergence of θ n towards θ c , we obtain the wanted result.

In order to synthetically summarize results (31), ( 32), ( 33) and (35-36) for the Central Limit Theorem relative to our quantities of interest, we define the following matrix-type relation:

√ n   p 1 -p c 1 p 2 -p c 2 D n (•) -D(•)   = M (θ c , •)         Φ 1,1 (L 1 , θ c ) Φ 2,2 (L 1 , θ c ) L 1 Φ 2,1 (L 2 , θ c ) Φ 1,2 (L 2 , θ c ) L 2         + o a.s. (1), with M (θ c , •) = L(•, θ c ) J -1 (θ c ) C (38)
where

C =     -1 0 0 0 -ζ 0 0 -1 0 -ζ 0 0 0 0 1 0 0 0 0 0 0 0 0 1     , J(θ) = d(θ) 0 2×2 0 2×2 Id 2×2 , L(•, θ) =    1 0 0 0 0 1 0 0 -L 1 (•)-G 1 (•) p 2 1 L 2 (•)-G 2 (•) p 2 2 1 p 1 -ζ p 2    .( 39
)
We finally have

        Φ 1,1 (L 1 , θ c ) Φ 2,2 (L 1 , θ c ) L 1 Φ 2,1 (L 2 , θ c ) Φ 1,2 (L 2 , θ c ) L 2         L → Z =         Φ 1,1 (B 1 , θ c ) Φ 2,2 (B 1 , θ c ) B 1 Φ 2,1 (B 2 , θ c ) Φ 1,2 (B 2 , θ c ) B 2         , ( 40
)
where Z is Gaussian random vector of R 6 with covariance matrix Σ L = E(ZZ T ). Since B 1 and B 2 are two independent (limiting) Gaussian processes, we have:

Σ L (x, y) = Σ 1 (x, y) 0 3×3 0 3×3 Σ 2 (x, y) . ( 41
) Because E R f 1 (x)B(x)dU (x) × R f 2 (y)B(y)dU (y) = E R 2 f 1 (x)f 2 (y)B(x)B(y)dU (x)dU (y) = R 2 f 1 (x)f 2 (y)E(B(x)B(y))dU (x)dU (y) = R 2 f 1 (x)f 2 (y)cov(B(x), B(y))dU (x)dU (y) = R 2 f 1 (x)f 2 (y)ρ(x, y)dU (x)dU (y),
it comes that Σ 1 (x, y) = (σ 1 (i, j; x, y)) 1≤i,j≤3 where and Σ 2 = (σ 2 (i, j; x, y)) 1≤i,j≤3 where

σ 1 (1, 1; x, y) = σ 1 (1, 1) = R 2 Ψ 1,1 (S, θ, u)Ψ 1,1 (S, θ, v)ρ 1 (u, v)dU (u)dU (v) σ 1 (1, 2; x, y) = σ 1 (1, 2) = σ 1 (2, 1) = R 2 Ψ 1,1 (S, θ, u)Ψ 2,2 (S, θ, v)ρ 1 (u, v)dU (u)dU (v) σ 1 (1, 3; x, y) = σ 1 (1, 3; y) = R Ψ 1,1 (S, θ, u)ρ 1 (u, y)dU (u) σ 1 (2, 2; x, y) = σ 1 (2, 2) = R 2 Ψ 2,2 (S, θ, u)Ψ 2,2 (S, θ, v)ρ 1 (u, v)dU (u)dU (v)
σ 2 (1, 1; x, y) = σ 2 (1, 1) = R 2 Ψ 2,1 (S, θ, u)Ψ 2,1 (S, θ, v)ρ 2 (u, v)dU (u)dU (v) σ 2 (1, 2; x, y) = σ 2 (1, 2) = σ 2 (2, 1) = R 2 Ψ 2,1 (S, θ, u)Ψ 1,2 (S, θ, v)ρ 2 (u, v)dU (u)dU (v) σ 2 (1, 3; x, y) = σ 2 (1, 3; y) = R Ψ 2,1 (S, θ, u)ρ 2 (u, y)dU (u) σ 2 (2, 2; x, y) = σ 2 (2, 2) = R 2 Ψ 1,2 (S, θ, u)Ψ 1,2 (S, θ, v)ρ 2 (u, v)dU (u)dU (v) σ 2 (2, 3; x, y) = σ 2 (2, 3; y) = R Ψ 1,2 (S, θ, u)ρ 2 (u, y)dU (u) σ 2 (3, 3; x, y) = ρ 2 (x, y) σ 2 (3, 1; x, y) = σ 2 (3, 1; x) = σ 2 (1, 3; x) σ 2 (3, 2; x, y) = σ 2 (3, 2; x) = σ 2 (2, 3; x).
Note that all the above matrices can be estimated consistently, see Appendix D.2.

ii) Let us now decompose nd n ( θ n ) :

nd n ( θ n ) = R n(D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ c ) + D(x, L 1 , L 2 , θ c )) 2 dU (x) = R ( √ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ c )]) 2 dU (x) +2 √ n R √ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ c )]D(x, L 1 , L 2 , θ c )dU (x) +n R D 2 (x, L 1 , L 2 , θ c )dU (x). ( 42
)
Note that under H 0 , θ c = θ * , and we simply have:

nd n ( θ n ) = R ( √ nD(x, L 1 , L 2 , θ n )) 2 dU (x) = R ( √ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ * )] 2 dU (x) = U 0 n , (43)
since D(•, L 1 , L 2 , θ * ) = 0 almost everywhere. Next, notice that it is easy to show that the map U n (D) is Hadamard differentiable from the domain of càd-làg functions of bounded variation into R [27, Lemma 20.10]. This combined with the result for weak convergence of the empirical process D, in the third row of [START_REF] Mannucci | Factors associated with increased all-cause mortality during the COVID-19 pandemic in Italy[END_REF], yields the desired result.

Under H 1 , we have R D 2 (x, L 1 , L 2 , θ c )dU (x) > 0, which leads to

nd n ( θ n ) = R ( √ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , θ c )]) 2 dU (x) +n R D 2 (x, L 1 , L 2 , θ c )dU (x) + o P (n) = U 1 n + V 1 n . (44)
Given the asymptotic convergence analysis under both H 0 or H 1 , the random variable within brackets involved in ( 42) and (44) can be analyzed closely. Being able to tabulate its asymptotic distribution whatever H 0 or H 1 is true, we will be able to accept H 0 if the test statistic nd n ( θ n ) is lower than the (1 -α)-quantile of that asymptotic distribution, otherwise reject H 0 .

B Optimization of the discrepancy measure

As discussed in Section 1, we can face two types of situations when looking for the solution of our optimization problem (15) : i) there exists a local minima of d(θ), θ * under H 0 or θ c under H 1 , in the interior of the parametric space, and then the testing problem is non-trivial and should be addressed, or ii) the optimization of d(θ) shows that we bump into the boundaries of the parametric space, i.e. one of the component of θ c is equal to the upper-bound of the parametric space because the main way to reduce the contrast d(θ) is to make θ large. In the latter case, the testing problem is not even worth addressing (we clearly are under H 1 ). This phenomenon is illustrated on Figure 8, where the two first graphs show the situation i) and the last one depicts the case ii). It is clear that a unique global minimizer belonging to the natural parametric space ]0, 1[ exists under H 0 .

On the contrary, under H 1 , local minima out of this natural parametric space can exist and lead to select a minimizer tending to infinity.

C Validation of the estimators variance-covariance

D Further details for computations D.1 Derivatives

Let us compute the gradient vector Ḋ(θ) and the Hessian matrix D(θ) for any θ ∈ Θ,

Ḋ(θ) = G 1 (x) -L 1 (x) p 2 1 , - G 2 (x) -L 2 (x) p 2 2 T D(θ) = diag -2 G 1 (x) -L 1 (x) p 3 1 , 2 G 2 (x) -L 2 (x) p 3 2 .
Dropping for simplicity the dependence on (x, θ) it comes now that

Ḋ(x, θ) Ḋ(x, θ) T = (G 1 (x)-L 1 (x)) 2 p 4 1 -(G 1 (x)-L 1 (x))(G 2 (x)-L 2 (x)) p 2 1 p 2 2 -(G 1 (x)-L 1 (x))(G 2 (x)-L 2 (x)) p 2 1 p 2 2 (G 2 (x)-L 2 (x)) 2 p 4 2 , and
D(θ)D(θ) = diag -2 G 1 (x) -L 1 (x) p 3 1 , 2 G 2 (x) -L 2 (x) p 3 2 × L 1 (x) -(1 -p 1 )G 1 (x) p 1 - L 2 (x) -(1 -p 2 )G 2 (x) p 2 .
Let us compute now the terms of the matrix M (x, θ) = Ḋ(x, θ) Ḋ(x, θ) T + D(θ)D(θ). Dropping for simplicity the dependence on (x, θ) It comes now that

M 1,1 = G 2 1 -2G 1 L 1 + L 2 1 p 4 1 - 2 p 4 1 G 1 L 1 + 2(1 -p 1 ) p 4 1 G 2 1 + 2 p 3 1 p 2 G 1 L 2 - 2(1 -p 2 ) p 3 1 p 2 G 1 G 2 + 2 p 4 1 L 2 1 - 2(1 -p 1 ) p 4 1 L 1 G 1 - 2 p 3 1 p 2 L 1 L 2 + 2(1 -p 2 ) p 3 1 p 2 L 1 G 2 = -6 + 2p 1 p 4 1 G 1 L 1 + 3 -2p 1 p 4 1 G 2 1 + 2 p 3 1 p 2 G 1 L 2 - 2(1 -p 2 ) p 3 1 p 2 G 1 G 2 + 3 p 4 1 L 2 1 - 2 p 3 1 p 2 L 1 L 2 + 2(1 -p 2 ) p 3 1 p 2 L 1 G 2 , (45)

D.2 Matrices consistent estimation

The notations used here are taken from the notations introduced throughout the paper. All the matrices involved in our theoretical results can be estimated consistently as follows.

J 1,1 (θ c ) = M 1,1 ( S, θ n ), J 1,2 (θ c ) = M 1,2 ( S, θ n ) = J 2,1 (θ c ), J 2,2 (θ c ) = M 2,2 ( S, θ n ) L 3,1 (θ c ) = I 1 ( θ), L 3,2 (θ c ) = -I 2 ( θ), L 3,3 (θ c ) = 1 p 1 , L 3,4 (θ c ) = I 1 ( θ), L 3,5 (θ c ) = -I 2 ( θ), L 3,6 (θ c ) = 1 p 2 ,
and

σ 1 (1, 1) = R 2 Ψ 1,1 (S, θ, x)Ψ 1,1 (S, θ, y) ρ 1 (x, y)dU (x)dU (y) σ 1 (1, 2) = σ 1 (2, 1) = R 2 Ψ 1,1 (S, θ, x)Ψ 2,2 (S, θ, y) ρ 1 (x, y)dU (x)dU (y) σ 1 (1, 3) = σ 1 (3, 1) = R 2 Ψ 1,1 (S, θ, x)D(S, θ, y) ρ 1 (x, y)dU (x)dU (y) σ 1 (2, 2) = R 2 Ψ 2,2 (S, θ, x)Ψ 2,2 (S, θ, y) ρ 1 (x, y)dU (x)dU (y) σ 1 (2, 3) = σ 1 (3, 2) = R 2 Ψ 2,2 (S, θ, x)D(S, θ, y) ρ 1 (x, y)dU (x)dU (y) σ 1 (3, 3) = R 2 D(S, θ, x)D(S, θ, y) ρ 1 (x, y)dU (x)dU (y),
and Σ 2 = (σ 2 (i, j)) 1≤i,j≤3 where

σ 2 (1, 1) = R 2 Ψ 2,1 (S, θ, x)Ψ 2,1 (S, θ, y) ρ 2 (x, y)dU (x)dU (y) σ 2 (1, 2) = σ 1 (2, 1) = R 2 Ψ 2,1 (S, θ, x)Ψ 1,2 (S, θ, y) ρ 2 (x, y)dU (x)dU (y) σ 2 (1, 3) = σ 1 (3, 1) = R 2 Ψ 2,1 (S, θ, x)D(S, θ, y) ρ 2 (x, y)dU (x)dU (y) σ 2 (2, 2) = R 2 Ψ 1,2 (S, θ, x)Ψ 1,2 (S, θ, y) ρ 2 (x, y)dU (x)dU (y) σ 2 (2, 3) = σ 1 (3, 2) = R 2 Ψ 1,2 (S, θ, x)D(S, θ, y) ρ 2 (x, y)dU (x)dU (y) σ 2 (3, 3) = R 2 D(S, θ, x)D(S, θ, y) ρ 2 (x, y)dU (x)dU (y), where ρ 1 (x, y) = L 1 (x ∧ y)(1 -L 1 (x ∨ y) and ρ 2 (x, y) = L 2 (x ∧ y)(1 -L 2 (x ∨ y).
The above quantities are consistent estimates of their closed form counterparts given in the proof of Theorem 4. The proof of consistency for each of these terms is lengthy but straightforward. They basically use the dominated convergence theorem noticing that for all (i, j) ∈ {1, 2} 2 and (x, y) fixed in R 2 , the terms Ψ i,j (S, θ, x), D(S, θ, y), are bounded and almost surely consistent estimates Ψ i,j (S, θ c , x), D(S, θ c , x) according to Theorem 1, and ρ i (x, y) is an almost surely uniformly consistent estimate of ρ i (x, y) due to the Glivenko-Cantelli theorem.

Similarly to Section 2, we propose the following natural estimator of λ c in (51), i.e.:

λ n = arg min λ∈[δ,A] d n (λ), where d n (λ) = R D 2 (x, L 1 , L 2 , λ)dU (x). (52)
According to the above identifiability discussion, Lemma 1 and Theorem 1 in Section 3 are still valid and we can establish the counterpart of Theorem 2. To answer the testing problem in the case G 1 = G 2 , we would thus use the following result.

(I') The cdfs family

E 1 = {G, F 1 , F 2 } is linearly independent. (II') The family of functions E 2 = {G 2 , F 1 F 2 , F 2 i , GF i ; i = 1, 2} is linearly independent. ( A
nd n ( λ n ) = U 1 n + V 1 n , with U 1 n L → R (W 2 (λ c , x)) 2 dU (x) and V 1 n = O P (n), under H 1 . ( 55
)
Remark 5. It is important to notice that if we have at our disposal a consistent estimate of the proportion p 1 , let say by using p 1 the √ n-consistent estimator of Bordes et al. [START_REF] Bordes | Semiparametric two-component mixture model when a component is known: an asymptotically normal estimator[END_REF] in the symmetric case or the p 1 estimator of Patra and Sen [START_REF] Patra | Estimation of a Two-component Mixture Model with Applications to Multiple Testing[END_REF] in the general case, we can use it to calibrate a consistent value for p1 and finally obtain a consistent couple (p 1 , λ n ) → (p * 1 , p * 2 ), with according rates of convergence under H 0 .

Proof. i) By a Taylor expansion of ḋn around λ c we have

dn ( λ n ) √ n( λ n -λ c ) = - √ n ḋn (λ c ), ( 56
)
where λ n lies in the line segment with extremities λ n and λ c . In particular, we have

ḋ(λ) = 2E U (2 -λ)GL 2 λ 3 - (1 -λ)G 2 λ 3 - L 2 2 λ 3 - GL 1 λ 2 p1 + (1 -p1 )G 2 λ 2 p1 + L 2 L 1 λ 2 p1 - (1 -p1 )L 2 G λ 2 p1
, and ḋn (λ) -ḋ(λ) = 2 2 -λ λ 3 T 1 -

1 -λ λ 3 T 2 - 1 λ 3 T 3 - 1 λ 2 p1 T 4 + 1 -p1 λ 2 p1 T 5 + 1 λ 2 p1 T 6 - 1 -p1 λ 2 p1 T 7 .
where

T 1 (G, L 2 ) = E U G( L 2 -L 2 ) T 2 (G) = E U G 2 -E U G 2 = 0 T 3 (L 2 ) = E U ( L 2 -L 2 )( L 2 + L 2 ) = E U ( L 2 -L 2 )(2L 2 + o a.s. ( 1
)
)

T 4 (L 1 ) = E U G( L 1 -L 1 ) T 5 (G) = E U G 2 -E U G 2 = 0 T 6 (L 1 , L 2 ) = E U L 2 L 1 -L 2 L 1 T 7 (G, L 2 ) = E U G( L 2 -L 2 ) .
We can also denote by S = (G, L 1 , L 2 ) and consider Note that the first, respectively the second, above expression depends only on L 2 , resp. L 1 .

Ψ 1 (S, λ) = 2 2 -λ λ 3 G - 2 λ 3 L 2 + 1 λ 2 p1 L 1 - 1 -p1 λ 2 p1 G , Ψ 2 (S, λ) = 2 1 λ 2 p1 (L 2 -G) ,
√ n( ḋn (λ) -ḋ(λ)) = Φ(L 1 , L 2 , λ) + o a.s (1),

where according to

√ n = √ κn/ √ κ = ζ √ n 2 with ζ = 1/ √ κ: Φ(L 2 , L 1 , λ) = Φ 2 (L 1 , λ) + ζΦ 1 (L 2 , λ). ( 58
)
Since the empirical processes L 1 and L 2 are independent, by the Donsker theorem:

L 1 L 2 B = B 1 B 2 , ( 59
)
where B is a bi-dimensional gaussian process with diagonal correlation matrix ρ = diag(ρ 1 , ρ 2 ) with ρ 1 (x, y) = L 1 (x ∧ y)(1 -L 1 (x ∨ y) and ρ 2 (x, y) = L 2 (x ∧ y)(1 -L 2 (x ∨ y). Moreover,

√ n[D n (x, L 1 , L 2 , λ n ) -D(x, L 1 , L 2 , λ c )] = √ n[F 1 (x, L 1 , p1 ) -F 1 (x, L 1 , p1 )] - √ n[(F 2 (x, L 2 , λ) -F 2 (x, L 2 , λ))]. (60)
Let analyze separately the two terms in the right hand side of the above equality.

√ n[F 1 (•, L 1 , p1 ) -F 1 (•, L 1 , p1 )] = 1 p1 L 1 , (61)
and

√ n[F 2 (•, L 2 , λ) -F 2 (•, L 2 , λ c )] = √ n L 2 λ - L 2 λ c - 1 -λ λ - 1 -λ c λ c G = √ n L 2 -L 2 λ + √ n λ c -λ λ c λ (L 2 -G) = ζ λ c L 2 - L 2 -G (λ c ) 2 √ n[ λ -λ c ] + o P (1). ( 62
)
It is also easy to prove, similarly to the proof of Therorem 2, that dn ( λ n ) a.s.

→ d(λ c ) > 0, as n → +∞.

In order to synthetically summarize results (56), ( 57), ( 58) and (60-62) and build up the Central Limit Theorem relative to our quantities of interest, we define the following matrix-type relation:

√ n λ -λ c D n (•) -D(•) = M (λ c , •)     Φ 2 (L 1 , λ c ) L 1 Φ 1 (L 2 , λ c ) L 2     + o a.s. ( 1
Finally, we have

    Φ 2 (L 1 , λ c ) L 1 Φ 1 (L 2 , λ c ) L 2     L → Z =     Φ 2 (B 1 , λ c ) B 1 Φ 1 (B 2 , λ c ) B 2     , (65)
where Z is Gaussian random vector of R 4 with covariance matrix Σ L = E(ZZ T ).

Since B 1 and B 2 are two independent (limit) Gaussian processes we have

Σ L (x, y) = Σ 1 (x, y) 0 2×2 0 2×2 Σ 2 (x, y) , (66)
where Σ 1 (x, y) = (σ 1 (i, j; x, y)) 1≤i,j≤3 with Under H 1 , we have R D 2 (x, L 1 , L 2 , λ c )dU (x) > 0, which leads to

nd n ( λ n ) = R ( √ n[D(x, L 1 , L 2 , λ n ) -D(x, L 1 , L 2 , λ c )]) 2 dU (x) +n R D 2 (x, L 1 , L 2 , λ c )dU (x) + o P (n) = U 1 n + V 1 n . (69)
Given the asymptotic convergence analysis under both H 0 or H 1 , the random variable within brackets involved commonly in (67) and (69) can once again be analyzed closely, leading to answer our statistical test.

  i.e. for all θ ∈ Θ, d(θ) ≥ 0 and d(θ) = 0 if and only if θ = θ * ∈ o Θ . (iii) Assume that conditions (I) and (A) hold. If U is strictly increasing on an interval I U that encompasses the support of the L i 's and G i 's, i = 1, 2, and if for any given case under H 1 there exists one single point θ c ∈ o Θ such that θ c = arg min θ∈Θ d(θ), then d(θ c ) > 0.

Figure 1 :

 1 Figure 1: On the left panel, theoretical distribution Z(θ * ) (solid) and empirical version U 0 n (dotted). On the right, distribution Z(θ c ) (solid) and empirical contrast distribution U 1 n + V 1 n (dotted).

  (a) to (d) these four different cases, corresponding to: (a) G 1 not so different from G 2 , and G 1 and G 2 close to F ; (b) G 1 very different from G 2 , with both distributions close to F ; (c) G 1 not so different from G 2 , with both distributions far from F ; (d) G 1 very different from G 2 , with G 1 close to F and G 2 far from F . The global simulation scheme thus encompasses 144 different

Figure 3 ,

 3 Figure 3, still considering the same mixture distributions as previously (Gaussian-Gaussian on R, Gamma-Exponential on R + , Negative-Binomial-Poisson on N, and Logit-Uniform on [0, 1]). However, the difference lies in the different frameworks studied, illustrated by Figure 10 in Appendix F.2. Hereafter, we denote them as follows: case (a) F 1 and F 2 have the same distribution, with very different means; case (b) F 1 and F 2 have the same distribution, with close means; case (c) F 1 and F 2 have the same distribution, with same means but very different variances; case (d) F 1 and F 2 have the same distribution, with same means and close variances. We obviously expect

Figure 4 :

 4 Figure4: Empirical power depending on the sample size (n = 300; 3,000; 10,000; 25,000 in logarithmic scale), on various supports (R, R + , N, [0, 1]), when F 1 and F 2 have same mean and variance (other parameters are listed in Table3of Appendix F.2, case (e), see also Fig.11).

(

  directly or indirectly) to the COVID-19. The 2020 population is then a two-component mixture composed by the previous 2019 population plus a latent population subject to the impact of the COVID-19 crisis. In other words, model (1) has an appealing application to capture the excess of mortality due the COVID-19. It is then legitimate to assume a second unknown nodular component driving the mortality due to the COVID-19 during the considered period. More precisely, we will assume that the known cdf is the one observed over 2019, i.e. the multinomial distribution G, and look to compare the distribution F of the excess mortality across countries. This excess mortality

Figure 5 :Figure 6 :

 56 Figure 5: Total death records of individuals aged over 15 years-old across years 2017, 2018, 2019 (gray curves) and 2020 (black curve).

Figure 7

 7 Figure 7 where we can see the very close patterns of these two multinomial distributions. In the Netherlands-Italy comparison case, the common component is slightly different. It represents only 15.23% of the Italian deaths population and 7.22% of the Netherlands one. Their estimations are

Figure 7 :

 7 Figure 7: The nodular distribution F i due to the COVID-19 crisis for the countries with similar impacts on the mortality : Netherlands-Italy (left) and Spain-Italy (right)

  ) (i) Noticing that according to (29) the mapping θ → |D(x, θ)| is bounded over Θ by 2(1 + δ)/δ 1 and continuous at any point θ ∈ Θ for all fixed x ∈ R, the wanted result is a direct consequence of the Lebesgue dominated convergence Theorem. The same type of proof, straightforward but painfull, hold on the gradient and the Hessian of D(x, θ), see Section D for close form expressions.

  and concludes the proof of the definite-positiveness of d(θ) for any θ ∈ Θ.

σ 1 ( 2 , 3 ;

 123 x, y) = σ 1 (2, 3; y) = R Ψ 2,2 (S, θ, u)ρ 1 (u, y)dU (u) σ 1 (3, 3; x, y) = ρ 1 (x, y) σ 1 (3, 1; x, y) = σ 1 (3, 1; x) = σ 1 (1, 3; x) σ 1(3, 2; x, y) = σ 1 (3, 2; x) = σ 1 (2, 3; x),

Figure 8 :

 8 Figure 8: Example of the surface of the discrepancy d(θ) where p i (i = 1, 2) belongs to Θ i . From left to right: under H 0 , under H 1 with a solution belonging to the natural parametric space ]0, 1[, and under H 1 where p 2 (minimizer of d(θ)) is very likely to be very far from the interval ]0, 1[.

R(W 2

 2 ') λ * under H 0 (λ c = θ * ), or λ c under H 1 (λ c = λ * ), belong to ]δ, A[ the interior of the compact parametric space Θ .Theorem 4. i) If conditions (I'), (II') and (A') hold, we have under both H 0 or H 1 :√ n λ -λ c D n (•) -D(•) L → W (λ c , •), as n → +∞,(53)whereW (λ c , •) = (W 1 (λ c ), W 2 (λ c )) T is a 2-dimensional centered Gaussian process with covariance matrix Σ W = M (λ c , •)Σ L (•, •)M (λ c , •) T where M (λ c , •) is defined in (63) and Σ L (•, •) in (66).ii) Under assumptions (I) and (II) we have respectively as n → +∞:nd n ( λ n ) = U 0 n L → (λ * , x)) 2 dU (x), under H 0 (54)

along with Φ 1 (

 1 L 2 , λ) = ϕ(Ψ 1 (S, λ), L 2 ), Φ 2 (L 1 , λ) = ϕ(Ψ 2 (S, λ), L 1 ).

  ), with M (λ c , •) = L(•, λ c ) J -1 (λ c ) C,

σ 1 ( 1 , 1 ; 2 Ψ 2 ( 2 Ψ 1 (D 2

 11122212 x, y) = σ 1 (1, 1) = R S, θ, u)Ψ 2 (S, θ, v)ρ 1 (u, v)dU (u)dU (v) σ 1 (1, 2; x, y) = σ 1 (1, 2; y) = R Ψ 2 (S, θ, u)ρ 1 (u, y)dU (u) σ 1 (2, 2; x, y) = ρ 1 (x, y) σ 1 (2, 1; x, y) = σ 1 (2, 1; x) = σ 1 (1, 2; x) and Σ 2 = (σ 2 (i, j; x, y)) 1≤i,j≤3 with σ 2 (1, 1; x, y) = σ 2 (1, 1) = R S, θ, u)Ψ 1 (S, θ, v)ρ 2 (u, v)dU (u)dU (v) σ 2 (1, 2; x, y) = σ 1 (1, 3; y) = R Ψ 1 (S, θ, u)ρ 2 (u, y)dU (u) σ 2 (2, 2; x, y) = ρ 2 (x, y) σ 2 (2, 1; x, y) = σ 2 (2, 1; x) = σ 2 (1, 2; x). ii) Let us now decompose nd n ( λ n ). nd n ( λ n ) = R n(D(x, L 1 , L 2 , λ n ) -D(x, L 1 , L 2 , λ c ) + D(x, L 1 , L 2 , λ c )) 2 dU (x) = R ( √ n[D(x, L 1 , L 2 , θ n ) -D(x, L 1 , L 2 , λ c )]) 2 dU (x) +2 √ n R √ n[D(x, L 1 , L 2 , λ n ) -D(x, L 1 , L 2 , λ c )]D(x, L 1 , L 2 , λ c )dU (x) +n R (x, L 1 , L 2 , λ c )dU (x). (67)Note that under H 0 , λ c = λ * , and we simply get:nd n ( λ n ) = R ( √ nD(x, L 1 , L 2 , λ n )) 2 dU (x) = R ( √ n[D(x, L 1 , L 2 , λ n ) -D(x, L 1 , L 2 , λ * ]) 2 dU (x) = U 0 n .(68)since D(•, L 1 , L 2 , λ * ) = 0 almost everywhere. Next, notice that it is easy to show that the map U n (D) is Hadamard differentiable from the domain of càd-làg functions of bounded variation into R [27, Lemma 20.10]. This combined with the result for weak convergence of the empirical process D, in the second row of (53), yields the desired result.

  [ 2 the true proportions parameter value. By isolating the expressions of F 1 and F 2

	2 Identifiability under G 1 = G 2
	Consider models (2) with generic proportions parameter θ = (p 1 , p 2 ) ∈ Θ and denote by θ * =
	(p * 1 , p * 2 ) ∈ ]0, 1
	to COVID-19
	across a panel of European countries. Finally, Section 6 contains a discussion where we present two
	further leads of research based on dependent two-sample models: i) we introduce the contaminant
	distribution independence component testing along with the complete concordance/discordance
	testing problem arising in z-scores modeling, ii) we introduce the homogeneity testing problem
	in the so-called blending process (temporal contamination model). All the proofs and technical
	material are relegated in Appendix sections A-E. Note that Appendix E is devoted to the non
	identifiable situation where G 1 = G 2 , in which the testing problem (3) can still surprisingly be
	addressed by using a parametrization trick.

1, and then reject H 0 .

Our paper is organized as follows: In Section 2 we analyze the model (2) identifiability, and suggest an IBM-parameter picking principle relevant under both H 0 and H 1 . Section 3 is dedicated to assumptions and asymptotic results showing the theoretical validity of our testing procedure under the condition G 1 = G 2 . In Section 3.2 we numerically validate the finite sample size properties of the central limit theorem stated in Theorem 2. In section 4 we investigate the empirical levels and powers of our test through Monte Carlo simulations. In Section 5 we present an original application in which we compare pairwise the excess of mortality due

  which at least requires G 1 = G 2 and frames our present study, we

	necessarily have p 1 = p * 1 and p 2 = p * 2 . On the other hand, under H 1 , F 1 = F 2 , if the cdfs family
	{G 1 , G 2 , F 1 , F 2 } is linearly independent, equation (12) is impossible since it would imply p * 1 = 0
	and p * 2 = 0 which is in contradiction with θ * ∈]0, 1[ 2 , and therefore F 1 (x, p 1 ) = F 2 (x, p 2 ) for all
	θ ∈ Θ. That being said, in order to consistently pick the right θ * under H 0 and select under H 1 a
	θ such that F 1 (x, p 1 ) = F 2 (x, p 2 ) (the property being actually true for all θ ∈ Θ), we propose to
	investigate the location of the minimum contrast parameter θ c :

  unsatisfactory results. For one given support, the four panels from top left to bottom right correspond to cases (a) to (d). For instance, case (c) with mixtures of Gaussian distributions is the bottom left 3x3 square of the heatmap. One can see that most of the setups under study lead to satisfactory empirical levels of the test, close to the theoretical 5%. Indeed, since each simulation enables to compare the empirical test statistic to the 95th percentile of the calibrated distribution U 0 , it is expected that the level of the test fluctuates around 5%. In practice only 12 over 144 approximations of the level exceed 10%, which means that less than 9% of the setups under study provide mixed conclusions. Looking more carefully at the results, the problematical situations mostly arise when at least one of the proportions p i equals 10%. It is very likely that the main reason explaining this drop of efficiency is the lack of observations to perform the test about the unknown components. The low component weight affected to the unknown part of the distribution leads to underrepresent the observations useful for the test to be efficient. In some very rare frameworks, although p 1 and p 2 equal at least 30%, the empirical level remains "high"

		0								0.05							0.1							0.15
													Empirical level
	0.09 0.07 0.03 0.05 0.06 0.07 0.05 0.08 0.03 0.03 0.02 0.03 0.05 0.05 0.08 0.06 0.06 0.07 0.02	0.1	0.03 0.07 0.03 0.01	p1=0.1
	0.04 0.06 0.04 0.02 0.03 0.04	0.1	0.07 0.07 0.02 0.05 0.03 0.02 0.03 0.06 0.08 0.08	0	0.06 0.03 0.04 0.07 0.03 0.07	p1=0.3
	0.05 0.05 0.03 0.01 0.05 0.07 0.07 0.06 0.04 0.09 0.06 0.06 0.02 0.05 0.07 0.25 0.06 0.08	0.1	0.02 0.05 0.11 0.08 0.08	p1=0.6
	0.05 0.06 0.06 0.08 0.05 0.06 0.13 0.05 0.06 0.03 0.05 0.08 0.03 0.05 0.04 0.04 0.12 0.04 0.05 0.02 0.13 0.04 0.08 0.14	p1=0.1
	0.04 0.04 0.05 0.03 0.16 0.05 0.04 0.04 0.02 0.04 0.07 0.05 0.05 0.04 0.09 0.05 0.03 0.12 0.06 0.05 0.05 0.03 0.06 0.04	p1=0.3
	0.03 0.05 0.01 0.04 0.05 0.07 0.12 0.14 0.02 0.04 0.06 0.04 0.06 0.11 0.07 0.03 0.03 0.11 0.09 0.05 0.02	0.1	0.02 0.05	p1=0.6
	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6
		Real						Positive				Integer					Bounded
	Figure 2: Heatmap of empirical level (under H

0 ) for different supports, different components weights, and different parameters for component distributions. For each support, cases (a) to (d) are given from top left to bottom right. cating (e.g. the case of mixing Negative Binomial and Poisson distributions, case (d), with p 1 = 0.3 and p 2 = 0.6

, where the empirical level equals 12%). In such cases, the choice of the mixture components parameters (Table

2

in Appendix F.1) has a crucial impact and can affect the estimation of the component weights, which spreads to the overall quality of the test.

  [START_REF] Beaney | Excess mortality: the gold standard in measuring the impact of COVID-19 worldwide?[END_REF] and F 2 have the same two first order moments (see case (e) in Table3of Appendix F.2 for further details about mixture distributions and parameters). Not surprisingly, departures from the null hypothesis can be detected provided that the number of observations is large enough, otherwise the power of the test remains low (especially when F 1 and F 2 are very similar, see cases (b) and (d)). Indeed, low proportions p i (i = 1, 2) leads to deteriorate the accuracy of the estimates p i , which favour situations where θ n can be "far" from θ c (minimization of the contrast is solved

		0					0.2				0.4					0.6				0.8				1
												Empirical power
																									p1=0.1
																									p1=0.3
	1	1	1	0.63	1	1	1	1	1	0.92	1	1	1	1	1	0.72 0.84 0.92	1	1	1	0.76	1	1	p1=0.6
	0.45	1	1	0.28 0.42 0.77 0.09 0.47 0.61 0.43 0.64 0.73 0.38 0.44	1	0.04 0.06 0.08	1	1	1	0.37	1	1	p1=0.1
	1	1	1	0.45 0.99	1	0.39	1	1	0.55 0.82 0.99 0.39	1	1	0.12 0.93	1	1	1	1	0.96	1	1	p1=0.3
	1	1	1	0.73	1	1	0.66	1	1	0.71 0.89 0.97 0.94	1	1	0.1	1	1	1	1	1	1	1	1	p1=0.6
	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6	p2=0.1	p2=0.3	p2=0.6
		Real						Positive				Integer					Bounded

by escaping from ]0, 1[ 2 ). The natural consequence of this phenomenon is that extreme quantiles (e.g. 95th percentile) of the tabulated random variable U 1 tend to be larger, which mechanically lowers the power of the test. Figure 3: Heatmap of empirical power (under H 1 ) for different supports, different components weights, and different parameters for component distributions. For each support, cases (a) to (d) are given from top left to bottom right.

Table 1 :

 1 Pairwise testing of the excess of mortality behavior across some European countries. simulation results. First, we can see from Table1that some estimated proportions in the pairwise analysis bump into this boundary, which clearly indicates to reject the null hypothesis H 0 . Among these countries, we see that only Spain and Italy (with p-value equal to 76%), on one hand, and

	Population	p 1	p 2	Green light Test statistic 95% quantile p-value Decision
	1	2
	Spain	Italy	0.1404	0.3058	1.3471	14.6930	0.75	H 0
	Spain	France	3	0.7388	-	-	-	H 1
	Spain	Germany	3	0.0997	-	-	-	H 1
	Spain	Netherlands 0.6701	3	-	-	-	H 1
	Spain	Belgium	3	3	-	-	-	H 1
	Netherlands	Italy	0.0722	0.1523	20.3887	90.3917	0.65	H 0
	Netherlands	France	3	0.3031	-	-	-	H 1
	Netherlands	Germany	0.94	0.1638	14.9862	2.8878	-	H 1
	Netherlands	Belgium	3	0.6364	-	-	-	H 1
	Italy	France	0.32	3	-	-	-	H 1
	Italy	Germany	3	0.1710	-	-	-	H 1
	Italy	Belgium	3	3	-	-	-	H 1
	Belgium	France	3	3	-	-	-	H 1
	Belgium	Germany	0.5742 0.1908354	1.7297	1.344	-	H 1

  To validate the explicit covariance structure between the estimators, it is necessary to fix z and to compare the empirical covariances obtained by Monte Carlo simulations to the theoretical ones. Here, we illustrate the case of two-component Gaussian mixtures with parameters given in We can see that the approximations of both quantities are very close, which ensures the validity of i) in Theorem 2 in this case. Of course, we checked that for many different values of z, as well as many different frameworks in terms of mixture distributions.

	Section 3.2. Moreover, we take z = 2 in formulas (38)-(41). Hereafter, we provide the numerical
	approximations for the theoretical version of the covariance structure:
		Var(P 1 )	Cov(P 1 , P 2 ) Cov(P 1 , D z )				2.0086503	1.1209473 -0.1861036 
		Cov(P 2 , P 1 )	Var(P 2 )	Cov(P 2 , D z )	 =		1.1209473	1.6979287 -0.2714744  ,
	Cov(D z , P 1 ) Cov(D z , P 2 )		Var(D z )			-0.1861036 -0.2714744 0.5082710
	to be compared to its Monte Carlo version:
			Var(P 1 )	Cov(P 1 , P 2 ) Cov(P 1 , D z )			2.031565 1.091873 -0.170923 
		  Cov(D z , P 1 ) Cov(D z , P 2 ) Cov(P 2 , P 1 ) Var(P 2 )	Cov(P 2 , D z ) Var(D z )	  =	-0.170923 -0.29527 0.5158633  1.091873 1.639897 -0.29527	 .

1 0.49 0.98 1 1 1 1 0.8 1 1 1 1 1 0.49 0.62 0.79 1 1 1 0.38 1 1 1 1 1 0.3 0.71 1 1 1 1 0.42 0.97 0.94 1 1 1 0.3 0.43 0.68 1 1 1 0.14 0.5 0.9

Funding

This work was conducted within the Research Chair DIALog under the aegis of the Risk Foundation, a joint initiative by CNP Assurances and ISFA, Université Claude Bernard Lyon 1 (UCBL).

Y. Salhi was supported by the BNP Paribas Cardif Chair "New Insurees, Next Actuaries" as well as the CY Initiative of Excellence (grant "Investissements d'Avenir" ANR-16-IDEX-0008), Project "EcoDep" PSI-AAP2020-0000000013.

Since now the term

we obtain the following developped expression:

This decomposition relates to condition (II) insuring the positive definiteness of d under both H 0 or H 1 .

E Special case:

In this section we propose to investigate how the case G 1 = G 2 , previously analyzed in Milhaud et al. [START_REF] Milhaud | Semiparametric twosample mixture components comparison test[END_REF] (under the F i 's symmetry assumption), can be handled simply using the IBM-method described in Sections 1 and 2. First, the basic identifiability conditions in this particular setup are slightly simplified but also weakened. In fact, rewriting conditions [START_REF] Ghattas | Data driven smooth test for paired populations[END_REF] and [START_REF] Gribovka | Non-parametric copula estimation under bivariate censoring[END_REF] when

Let us investigate now the situations where possibly

Under H 0 , F 1 = F 2 = F , we simply obtain

If {G, F } is free (G = F on non-null measure set) we necessarily have

which means that we have an infinite number of solutions. Actually this conclusion is not an obstacle to our testing approach. In fact it is enough to arbitrarily fix a value for p 1 , let say p1 in ]δ, 1[, 0 < δ < 1, and investigate p 2 such that p 2 = (p * 2 p1 )/p * 1 which will automatically become the only parameter λ = p 2 under the p1 -setup, not necessarily belonging to [δ, 1] but probably to

Given the above identifiability discussion, the natural extension of our method is to consider a one single parameter fitting approach.

In order to consistently pick the right

with F.1 Empirical levels -Study under H 0

We give here an illustration of the different frameworks studied in our simulations to assess the empirical level of our test. The component weights have systematically been fixed to 50% for illustration purpose, see Figure 9. Table 2 provides the corresponding parameters for each mixture component, also used within the assessment of empirical levels in our global simulation study. Table 2: Parameters of mixture components corresponding to Fig. 9, used for the assessment of empirical levels of the test. Recall that we are under H 0 , meaning that 2) P( 4) P( 2) U(0, 0.6) U(0, 0.2) U(0, 0.9) U(0, 0.4) G 2 P(2.5) P(0.5) P( 5) P( 7) U(0.1, 0.5) U(0.2, 0.4) U(0.1, 1) U(0, 1)

F.2 Empirical powers -Study under H 1

We give here an illustration of the different frameworks studied in our simulations to assess the empirical power of our test. The component weights have systematically been fixed to 50% for illustration purpose, see Figure 10 and corresponding parameters in Table 3.

Table 3: Parameters corresponding to Fig. 10 and Fig. 11, used for the assessment of empirical powers. Case (a) : different means for F 1 and F 2 ; Case (b) : close means between F 1 and F 2 ; Case (c) : same means, different variances for F 1 and F 2 ; Case (d) : same means, close variances between F 1 and F 2 ; Case (e) : same means and variances for F 1 and F 2 , but different distributions. 3.