Marc Hallin
email: mhallin@ulb.ac.be

Davy Paindaveine

Thomas Verdebout
email: thomas.verdebout@univ-lille3.fr.

"Optimal Rank-based Tests for Common Principal Components"

Keywords: Common Principal Components, rank-based methods, local asymptotic normality, robustness

HAL is

This paper provides optimal testing procedures for the m-sample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudo-Gaussian test proposed in Hallin et al. (2010a) is locally and asymptotically optimal under Gaussian densities. We also show how to compute its local powers and asymptotic relative efficiencies (AREs). A numerical evaluation of those AREs, however, reveals that, while remaining valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogenous m-tuple of elliptical densities, irrespective of any moment assumptions-in elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rank-based tests are not only validity-robust in the sense that they survive arbitrary elliptical population

Introduction.

Principal components-arguably, the oldest and most popular tool of multivariate analysis-were originally introduced by [START_REF] Pearson | On lines and planes of closest fit to system of points in space[END_REF], then rediscovered by [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF], in a one-sample context. Multisample principal component problems only came much later, when [START_REF] Flury | Common principal components in k groups[END_REF] introduced the Common Principal Components (CPC) model. CPC models since then have been used in a number of applications, mainly in a biometric context (see e.g. [START_REF] Airoldi | Age variation in voles (Microtus californicus and Microtus ochrogaster ) and its significance for systematic studies[END_REF], Flury and Riedl (1988)). Under such a model, m ≥ 2 populations of dimension k, with covariance matrices Σ Σ Σ Cov i , i = 1, . . . , m, are assumed to share, with possibly different eigenvalues, the same principal components: namely, these covariance matrices factorize into Σ Σ Σ Cov i = β β βΛ Λ Λ Cov i β β β ′ for some m-tuple of positive diagonal matrices Λ Λ Λ Cov i , i = 1, . . . , m, and some orthogonal matrix β β β-the matrix of common eigenvectors, which does not depend on i and characterizes the common principal components. CPC models later on have been generalized [START_REF] Flury | Two generalizations of the common principal component model[END_REF]) into partial CPC models, in which only a subset of q < k principal components are common to the m populations. More recently, a broader class of models, which includes CPC and partial CPC, but also possible common eigenspaces, has been considered by [START_REF] Boik | Spectral models for covariance matrices[END_REF].

Before considering a statistical analysis based on such model, however, it is natural to check whether the CPC assumption is compatible with the data under study. [START_REF] Flury | Common principal components in k groups[END_REF] therefore developed a Gaussian likelihood ratio test φ (n) N for the null hypothesis H 0 of common principal components. This test is based on the asymptotically chi-square null distribution of -2 log Λ where, denoting by S (n) i , i = 1, . . . , m the empirical covariance matrices computed from m mutually independent samples of k-dimensional independent observations and by β β β the (constrained) maximum likelihood estimator of β β β,

Λ := m i=1 det( β β β ′ S (n) i β β β) det(diag( β β β ′ S (n) i β β β)) ni/2
(1.1) (we write diag(A) for the diagonal matrix having the same diagonal elements as a squared matrix A). Under H 0 , β

β β ′ S (n) i β β β should be nearly diagonal, hence det( β β β ′ S (n) i β β β)
and det(diag( β

β β ′ S (n) i β β β))
approximately equal, in which case Λ is close to one; under the alternative, Λ is closer to zero (hence, -2 log Λ is large), leading to the rejection of the CPC hypothesis. The asymptotically chi-square distribution of -2 log Λ follows from the Optimal Rank-based Tests for CPC 3 classical asymptotic result of [START_REF] Wilks | The large-sample distribution of the likelihood ratio for testing composite hypotheses[END_REF], a result which, however, is valid under Gaussian assumptions only.

It is well known that Gaussian likelihood ratio tests (LRT) for hypotheses involving covariance matrices, are quite sensitive to violations of Gaussian assumptions and to the presence of outliers (on this latter point, see e.g. [START_REF] Boente | A robust approach to common principal components[END_REF]). The test based on (1.1) is no exception to that rule: -2 log Λ under non-Gaussian densities is no longer asymptotically chi-square, but converges in distribution to a weighted sum of independent chi-square variables.

This type of asymptotic behavior is not uncommon, and there exists an extensive literature on how to preserve the chi-square asymptotics of LRT statistics: [START_REF] Muirhead | Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations[END_REF], [START_REF] Browne | The decomposition of multitrait-multimethod matrices[END_REF], [START_REF] Satorra | Scaling corrections for chi-square statistics in covariance structure analysis[END_REF], Bentler and Dungeon (1996), provide adjusted LRTs for various problems. [START_REF] Shapiro | Analysis of covariance structures under elliptical distributions[END_REF] give a necessary and sufficient condition under which such adjusted test statistics remain asymptotically chi-square.

Using this result by Shapiro and Browne, Boik (2002) constructs a test-φ

(n)
Boik , sayfor the null hypothesis of CPC based on a statistic which remains asymptotically chisquare under families of elliptical distributions with finite fourth-order moments and common kurtosis parameter. That is, denoting by κ k (g i ) the kurtosis in the ith elliptical distribution, i = 1, . . . , m (see Section 6 for a precise definition), the validity of this test requires the somewhat stringent assumption of homokurticity κ k (g 1 ) = . . . = κ k (g m ).

In a series of papers, [START_REF] Boente | A robust approach to common principal components[END_REF][START_REF] Boente | Influence functions and outlier detection under the common principal components model: a robust approach[END_REF][START_REF] Boente | Robust tests for the common principal components model[END_REF]) generalize Boik's test by substituting robust scatter matrices for the regular covariance matrices, which reduces the impact of possible outliers. In terms of validity robustness, however, the resulting tests do not improve much on Boik's, as they merely replace the assumption of homokurticity with an assumption of the form ς(g 1 ) = . . . = ς(g m ), where the parameter ς(g) depends on the chosen concept of scatter-a homogeneity assumption that is hardly more natural or realistic than Boik's homokurticity assumption. Hallin et al. (2010a) amend this situation by introducing a pseudo-Gaussian test φ

(n)

HPV the validity of which, unlike that of φ

(n) N and φ (n)
Boik , resists heterokurtic violations of Gaussian assumptions. Under Gaussian densities, this test is asymptotically equivalent to Flury's LRT. However, φ

(n)

HPV still requires finite fourth-order moments, and follows as a robustified version of Flury's LRT φ (n) N , the exact optimality properties of which have not been investigated so far. These issues (certainly, the fourth-order moment assumption) are not dramatic, and most statisticians would feel comfortable using such tests. Unfortunately, it appears from the power analysis in Sections 8.1 and 8.2 below that φ (n) HPV exhibits disturbingly low power against non-Gaussian alternatives. In the two-population case with bivariate t 5 densities (homokurtic case with finite fourth-order moments), the asymptotic relative efficiency of φ

(n)

HPV with respect to the locally optimal procedure is as low as .4286, whereas the normal-score (van der Waerden) test we are proposing here achieves asymptotic relative efficiency .9446-more than twice as much. Under t 4.2 densities, the figures are .1202 and .9303, respectively! It seems, thus, that the robustification of φ

(n) N into φ (n)
HPV is obtained at the expense of efficiency-which, most statisticians will agree, is quite a heavy price.

The objective of this paper is to remedy those validity and efficiency problems by proposing rank-based tests that outperform the available parametric ones on both counts. These rank tests enjoy enhanced validity properties; in particular, they allow for heterokurticity and do not require any moment assumptions at all. In the same time, they exhibit increased efficiency-robustness: see Tables 1 and2. The asymptotic relative efficiencies of the van der Waerden version of our tests with respect to φ (n)

HPV , for instance, are uniformly larger than one-a theoretical finding that is supported by the simulation results of Section 8.2.

Reaching that double objective requires overcoming several technical difficulties. First, the traditional covariance-based concept of common principal components has to be extended in order to cope with the possible absence of second-order moments. In elliptical families, the scatter and shape matrices that characterize the elliptical equidensity contours quite naturally qualify as moment-free substitutes for covariance matrices (with which they coincide, up to a scalar factor, in case second-order moments do exist). Second, based on a parametrization involving those scatter and shape matrices, we establish the local asymptotic normality (LAN) of the model in the vicinity of the CPC hypothesis. This provides us with a clear definition of optimality, and a way of computing local powers and asymptotic relative efficiencies-with, however, the additional difficulty that the limiting local experiments associated with the scatter-or shape-matrix parametrization are not full-rank Gaussian shift experiments but curved ones. Such curved LAN structures were previously studied in [START_REF] Hallin | Optimal rank-based testing for principal components[END_REF]. As a by-product, we establish the exact optimality properties of Flury's φ

Outline of the paper.

The outline of the paper is as follows. Section 3 states the assumptions to be used in the sequel. The parametrization we propose for testing the CPC hypothesis in an elliptical context is described in Section 4. Section 5 provides the uniform local and asymptotic (ULAN) property, on which the construction of optimal tests will be based. Section 6 derives Gaussian and pseudo-Gaussian tests for the CPC hypothesis, and Section 7 introduces optimal rank-based tests for the same. In Section 8, the performances of the proposed tests are investigated both in terms of asymptotic relative efficiencies (Section 8.1) and simulations (Section 8.2). Finally, the Appendix collects technical proofs.

Main assumptions.

For the sake of convenience, we are collecting here the main assumptions to be used in the sequel.

Elliptical symmetry

Denote by (X i1 , . . . , X ini ), i = 1, . . . , m a collection of m mutually independent samples of i.i.d. k-dimensional random vectors with elliptically symmetric densities. More precisely, the n i observations X ij , j = 1, . . . , n i in sample i are independent, with density

f i (x) := c k,fi (det(Σ Σ Σ i )) -1/2 f i ((x -θ θ θ i ) ′ Σ Σ Σ -1 i (x -θ θ θ i )) 1/2 , (3.1)
for some k-dimensional location parameter θ θ θ i , some symmetric and positive definite scatter matrix Σ Σ Σ i and some radial density function

f i : R + 0 → R + ; c k,fi is a normalization constant such that R k f i (x)dx = 1.
Note that, despite the terminology, the radial density f i is not a probability density (over the positive real line), since it does not integrate to one; but fi := µ -1 k-1;fi r k-1 f i (for the sake of simplicity, we write fi instead of fik ), where µ ℓ;f := ∞ 0 r ℓ f (r) dr, is. Define

F := f : f (r) > 0 a.e. and µ k-1;f < ∞ and F 1 := f ∈ F : 1 µ k-1;f 1 0 r k-1 f (r) dr = 1 2 ;
F 1 is a class of standardized radial densities, in the sense that, for any radial density f ∈ F 1 , the probability density f (r) := µ -1 k-1;f r k-1 f (r) is a properly standardized probability density. By "standardized", here, we mean that the corresponding median is one; the median, for a nonvanishing density over R + 0 , indeed, is a scale parameter which does not require any moment conditions.

Summarizing this, we throughout assume that the following assumption holds true.

Assumption (A). The observations X ij , j = 1, . . . , n i , i = 1, . . . , m are mutually independent, with probability densities f i given in (3.1), for some m-tuple of (possibly distinct) radial densities f := (f 1 , . . . , f m ) ∈ (F a ) m , where F a ⊂ F 1 is defined below.

Under Assumption (A), the elliptical distances

d ij (θ θ θ i , Σ Σ Σ i ) := Σ Σ Σ -1/2 i
(X ijθ θ θ i ) , j = 1, . . . , n i , i = 1, . . . , m, have probability density fi , with median one, which identifies the scatter matrices Σ Σ Σ i , i = 1, . . . , m also in the absence of any moments. Under finite second-order moments, however, Σ Σ Σ i is proportional to the covariance matrix Σ Σ Σ Cov i of X ij . Assumption (A) allows for heterogeneity of the m elliptical densities, that is, we may have f i = f i ′ for i = i ′ . Classical examples are the k-variate multinormal distributions, with standardized radial densities f i (r) = φ(r) := exp(-a k r 2 /2), the k-variate Student distributions, with standardized radial densities (for ν ∈ R + 0 degrees of freedom)

f i (r) = f t ν (r) := (1 + a k,ν r 2 /ν) -(k+ν)/2
, and the k-variate power-exponential distributions, with standardized radial densities of the form

f i (r) = f e η (r) := exp(-b k,η r 2η ), η ∈ R + 0 ; the positive constants a k , a k,ν , and b k,η are such that f i ∈ F 1 .
The equidensity contours associated with (3.1) are hyper-ellipsoids centered at θ θ θ i , the shape and orientation of which are determined by the scatter matrix Σ Σ Σ i . The multivariate signs

U ij (θ θ θ i , Σ Σ Σ i ) := Σ Σ Σ -1/2 i (X ij -θ θ θ i )/d ij (θ θ θ i , Σ Σ Σ i ) and standardized radial distances d ij (θ θ θ i , Σ Σ Σ i ) just
defined are X ij 's (within-group) elliptical coordinates associated with those ellipsoids. The observations then decompose into

X ij = θ θ θ i + d ij Σ Σ Σ 1/2 i U ij , where
the U ij 's, j = 1, . . . , n i , i = 1, . . . , m are i.i.d. uniform over the unit sphere in R k , and the d ij 's are i.i.d., independent of the U ij 's, with standardized probability density fi over R + and distribution function Fi . In the sequel, the notation gi and Gi will be used for the same functions computed from a standardized radial density g i (∈ F 1 ).

The derivation of locally and asymptotically optimal tests at a given m-tuple f = (f 1 , . . . , f m ) of radial densities will be based on the uniform local and asymptotic normality (ULAN) of the corresponding model. This ULAN property only holds under some mild regularity conditions on the f i 's. More precisely, ULAN (see Proposition 5.1 below) requires the f i 's to belong to the collection F a of those radial densities f ∈ F 1 which are absolutely continuous, with almost everywhere derivative ḟ such that, letting ϕ f := -ḟ /f and denoting by F the distribution function associated with f , the integrals satisfy the following assumption.

I k (f ) := 1 0 ϕ 2 f ( F -1 (u)) du and J k (f ) := 1 0 ϕ 2 f ( F -1 (u))( F -1 (u))

Assumption (B). For all

i = 1, . . . , m, n i = n (n) i → ∞ as n → ∞.
This assumption is weaker than the assumption usually made in (univariate or multivariate) multisample problems, where it is required that n (n) i /n be bounded away from 0 and 1 for all i as n → ∞. Letting r

(n) i := n (n) i /n, it is easy to check that Assumption (B) is actually equivalent to the Noether conditions max 1 -r (n) i r (n) i , r (n) i 1 -r (n) i = o(n) as n → ∞, for all i.
However, in the derivation of asymptotic distributions under local alternatives, we will need-mainly, for notational comfort-the following classical reinforcement:

Assumption (B ′ ). For all i = 1, . . . , m, r

(n) i → r i ∈ (0, 1), as n → ∞.
For notational simplicity, we henceforth omit superfluous (n) superscripts.

Score functions.

The signed-rank tests considered in Section 6 are based on m-tuples K = (K 1 , . . . , K m ) of score functions, which we assume to satisfy the following regularity conditions.

Assumption (C). For any i = 1, . . . , m, the mapping (from (0, 1) to R) u → K i (u) (C1) is continuous and square-integrable; (C2) can be expressed as the difference of two monotone increasing functions, and (C3) satisfies

1 0 K i (u) du = k. Assumption (C3) is a normalization constraint that is automatically satisfied by the score functions K i (u) = K fi (u) := ϕ fi ( F -1 i (u)) F -1 i (
u) leading to local and asymptotic optimality at m-tuples of radial densities f = (f 1 , . . . , f m ) for which ULAN holds; see Section 5.

For score functions K, K 1 , K 2 satisfying Assumption (C), let (throughout, U stands for a random variable uniformly distributed over (0, 1)),

J k (K 1 , K 2 ) := E[K 1 (U )K 2 (U )].
For simplicity, we write

J k (K) for J k (K, K), J k (K, f ) for E[K(U )K f (U )], etc.
The power score functions K a (u) := k(a + 1)u a (a ≥ 0) provide some traditional score functions satisfying Assumption (C), with J k (K a ) = k 2 (a + 1) 2 /(2a + 1): the Laplace, Wilcoxon and Spearman scores are obtained for a = 0, 1, and 2, respectively. As for score functions of the form K fi , an important particular case is that of van der Waerden or normal scores, obtained for f i = φ. Then, denoting by Ψ k the chi-square distribution function with k degrees of freedom, K φ (u) = Ψ -1 k (u), and J k (φ) = k(k + 2). Similarly, writing G k,ν for the Fisher-Snedecor distribution function with k and ν degrees of freedom, Student densities

f i = f t ν yield K f t ν (u) = k(k + ν)G -1 k,ν (u) ν + kG -1 k,ν (u) and J k (f t ν ) = k(k + 2)(k + ν) k + ν + 2 .

Parametrization of m-sample elliptical models.

A natural notation for the joint distribution of the n-tuple (X ′ 11 , . . . , X ′ mnm ) ′ under Assumption (A), parameter values θ θ θ 1 , . . . , θ θ θ m , Σ Σ Σ 1 , . . . , Σ Σ Σ m , and the m-tuple f of radial densities, is P (n) θ θ θ1,...,θ θ θm;Σ Σ Σ1,...,Σ Σ Σm;f . Such parametrization, however, is not well adapted to the present context, due to the fact that eigenvectors and eigenvalues are complicated functions of the scatter matrices. As in Hallin et al. (2010b), a parametrization based on eigenvectors and eigenvalues, which we now describe, will prove much more adequate.

Scatter, scale, and shape.

Since the eigenvectors β β β (1)

, . . . , β β β (m) of Σ Σ Σ 1 , . . . , Σ Σ Σ m are scale-free functions of the Σ Σ Σ i 's, it is appropriate to first decompose each Σ Σ Σ i into a product Σ Σ Σ i = σ 2 i V i
, where σ i is a scalar global scale parameter and V i a shape matrix (see Hallin and Paindaveine (2006) Hallin, Paindaveine and Verdebout for details) for sample i. [START_REF] Paindaveine | A canonical definition of shape[END_REF] has shown the advantages of doing so by defining σ 2 i as (det Σ Σ Σ i ) 1/k . This definition, which we adopt here, implies that the eigenvalues λ V ij of the shape matrices V i are such that k j=1 λ V ij = 1 for all i = 1, . . . , m; clearly, V i and Σ Σ Σ i share the same eigenvectors.

Shape eigenvalues and eigenvectors.

Shape matrices in turn factorize into

V i = β β β (i)′ Λ Λ Λ V i β β β (i) , with Λ Λ Λ V i := diag(λ V i1 , . . . , λ V ik ) (throughout diag(B 1 , B 2 , . . . , B m ) stands for the block-diagonal matrix with diagonal blocks B 1 , B 2 , . . . , B m ). Even in case the λ V ij '
s are all distinct, this factorization, due to possible permutations of eigenvalues and the columns of β β β (i) , is not unique, and it is usually imposed, without any loss of generality, that the diagonal elements of Λ Λ Λ V i are ranked in decreasing order of magnitude, which provides each eigenvalue λ V ij and the corresponding eigenvector β β β (i) j with a well-defined label j. That way of labeling eigenvalues and eigenvectors is used in the statement of ULAN in Section 5 below. The same labeling however is no longer adequate when describing the null hypothesis H 0 of CPC. The existence of a common β β β indeed induces a matching between the eigenvalues of the various populations, and it would be natural to label them so that λ V i1 , for i = 1, . . . , m be associated with

β β β's first column β β β 1 , λ V i2 , i = 1, . . . , m with β β β 2 , etc. Under such labeling, H 0 would take the simple form β β β (1) = . . . = β β β (m) instead of "there exist (m -1) permutation matrices M Π 2 , . . . , M Π m such that β β β (1) = M Π 2 β β β (2) = . . . = M Π m β β β (m) =: β β β "
. This β β β-induced labeling, however, only exists under H 0 , and, being β β β-dependent, only holds over a neighborhood of β β β; hence, it is local. We therefore adopt the "traditional" labeling in the statement of ULAN, and switch to the local β β β-induced labels when optimal tests are to be derived (these tests, typically, will involve the ordering of eigenvalues induced by some adequate estimator β β β). Establishing ULAN for a parametrization involving eigenvector matrices β β β (i) and eigenvalues Λ Λ Λ V i requires a differentiable correspondence between the V i 's and the corresponding (β β β (i) , Λ Λ Λ V i )'s. Therefore, we need the following assumption. Assumption (D). For all i = 1, . . . , m, the scatter Σ Σ Σ i (equivalently, the shape

V i ) has k distinct eigenvalues: λ Σ Σ Σ i1 > . . . > λ Σ Σ Σ ik .
While ULAN indeed requires Assumption (D), the tests we will propose, as we will show, remain (asymptotically) valid under the weaker

Assumption (D ′ ). For any 1 ≤ j = j ′ ≤ k, there exists i ∈ {1, . . . , m} such that λ Σ Σ Σ ij = λ Σ Σ Σ ij ′ .
Now, under the null and Assumption (D ′ ), the matrix β β β := (β β β 1 , . . . , β β β k ) of common eigenvectors is identified up to an arbitrary permutation of its columns (we still forget about the irrelevant sign changes of the β β β j 's). However, it is easy to fix an ordering, hence to make the β β β j 's-hence also the corresponding λ Σ Σ Σ ij 's-(individually) identifiable.

For instance, one can require that λ Σ Σ Σ 11 ≥ λ Σ Σ Σ 12 ≥ . . . ≥ λ Σ Σ Σ 1k (> 0), and that, for any sequence of the form

λ Σ Σ Σ 1j = λ Σ Σ Σ 1,j+1 = . . . = λ Σ Σ Σ 1,j+ℓ , one has λ Σ Σ Σ 2j ≥ λ Σ Σ Σ 2,j+1 ≥ . . . ≥ λ Σ Σ Σ 2,j+ℓ .
Recursively, if further ties occur among those λ Σ Σ Σ 2,j 's, the ranking can be based on the way the λ Σ Σ Σ 3,j 's are ordered, etc. Clearly, Assumption (D ′ ) ensures that this correctly defines a unique ordering of the common principal directions and corresponding eigenvalues. Note that the largest eigenspace common to Σ Σ Σ 1 , . . . , Σ Σ Σ m (equivalently, to V 1 , . . . , V m ) then has dimension less than or equal to one.

Parameter space.

The parametrization we are adopting in the sequel is similar to that considered in the one-sample case by [START_REF] Hallin | Optimal rank-based testing for principal components[END_REF]; it is based on the L := mk(k + 2)-dimensional vector (we denote by dvec (A) =: (A 11 , (dv • ec (A)) ′ ) ′ the vector obtained by stacking the diagonal elements of a squared matrix A)

ϑ ϑ ϑ := (ϑ ϑ ϑ ′ I , ϑ ϑ ϑ ′ II , ϑ ϑ ϑ ′ III , ϑ ϑ ϑ ′ IV ) ′ := (θ θ θ ′ 1 , . . . , θ θ θ ′ m , σ 2 1 , . . . , σ 2 m , (dv • ec Λ Λ Λ V 1 ) ′ , . . . , (dv • ec Λ Λ Λ V m ) ′ , (vec β β β (1) ) ′ , . . . , (vec β β β (m) ) ′ ) ′ ,
where θ θ θ i and σ 2 i are the location and scale parameters, Λ Λ Λ V i := diag(λ V i1 , . . . , λ V ik ) and β β β (i) the shape eigenvalue and eigenvector matrices, respectively, in population i, i = 1, . . . , m; the reason why λ V i1 is omitted in the parametrization is that, V i being a shape matrix,

λ V i1 = 1/ m j=2 λ V ij . The parameter space is thus Θ Θ Θ := R mk × (R + 0 ) m × (C k-1 ) m × (vec (SO k )) m , where C k-1
is the open cone of (R + 0 ) k-1 with strictly ordered (from largest to smallest) coordinates, and SO k stands for the class of k × k real orthogonal matrices with determinant one. Note that Assumption (D) is explicitly incorporated in the definition of Θ Θ Θ.

We denote by P

(n) ϑ ϑ ϑ;f or P (n)
ϑ ϑ ϑI ,ϑ ϑ ϑII ,ϑ ϑ ϑIII ,ϑ ϑ ϑIV ;f the joint distribution of the n observations under parameter value ϑ ϑ ϑ and standardized radial densities f = (f 1 , . . . , f m ).

Uniform local asymptotic normality (ULAN).

As mentioned in Section 1, we plan to construct tests that are optimal at correctly specified densities, in the sense of Le Cam's asymptotic theory of statistical experiments. In this section, we state the ULAN result (with respect to ϑ ϑ ϑ ∈ Θ Θ Θ, for fixed radial densities f = (f 1 , . . . , f m )) on which optimality will be based. Denote by

ϑ ϑ ϑ (n) := (ϑ ϑ ϑ (n)′ I , ϑ ϑ ϑ (n)′ II , ϑ ϑ ϑ (n)′ III , ϑ ϑ ϑ (n)′ IV ) ′ := (θ θ θ (n)′ 1 , . . . , θ θ θ (n)′ m , σ 2(n) 1 , . . . , σ 2(n) m , (dv • ec Λ Λ Λ V(n) 1
) ′ , . . . , (dv

• ec Λ Λ Λ V(n) m ) ′ , (vec β β β (1),(n) ) ′ , . . . , (vec β β β (m),(n) ) ′ ) ′ a local sequence such that ϑ ϑ ϑ (n) ∈ Θ Θ Θ and ϑ ϑ ϑ (n) -ϑ ϑ ϑ = O(n -1/2 ). Letting r (n) := diag((r (n) 1 ) -1/2 , . . . , (r (n) m ) -1/2 )
Hallin, Paindaveine and Verdebout (see Section 3.2), define

ς ς ς (n) := diag ς ς ς (n) I , ς ς ς (n) II , ς ς ς (n) III , ς ς ς (n) IV := diag r (n) ⊗ I k , r (n) , r (n) ⊗ I k-1 , r (n) ⊗ I k 2 (5.1)
and consider further sequences of the form ϑ ϑ ϑ (n) + n -1/2 ς ς ς (n) τ τ τ (n) , where

τ τ τ (n) = (τ τ τ (n)′ I , τ τ τ (n)′ II , τ τ τ (n)′ III , τ τ τ (n)′ IV ) ′ = (t (n)′ 1 , . . . , t (n)′ m , s (n) 1 , . . . , s (n) m , l (n)′ 1 , . . . , l (n)′ m , (vec b (1),(n) ) ′ , . . . , (vec b (m),(n) ) ′ ) ′ is such that sup n τ τ τ (n)′ τ τ τ (n) < ∞ and ϑ ϑ ϑ (n) + n -1/2 ς ς ς (n) τ τ τ (n) ∈ Θ Θ Θ. Under Assumption (B ′ ),
we also write ς ς ς for lim n→∞ ς ς ς (n) .

Strong restrictions are imposed on τ τ τ (n) = (τ τ τ

(n)′ I , τ τ τ (n)′ II , τ τ τ (n)′ III , τ τ τ (n)′ IV
) ′ in order for the perturbed parameter values ϑ ϑ ϑ (n) + n -1/2 ς ς ς (n) τ τ τ (n) to belong to Θ Θ Θ. In particular, the perturbed orthogonal matrices should remain orthogonal; we refer to [START_REF] Hallin | Optimal rank-based testing for principal components[END_REF] for details.

The statement of ULAN in Proposition 5.1 below still requires some additional notation. Write V ⊗2 for the Kronecker product V ⊗ V. Denoting by e ℓ the ℓth vector of the canonical basis of R k , let

K k := k i,j=1 (e i e ′ j ) ⊗ (e j e ′ i ) be the classical (k 2 × k 2 ) commutation matrix. Define H k as the k × k 2 matrix such that H k vec (A) = dvec (A) for any k × k matrix A. For any k × k diagonal matrix Λ Λ Λ = diag(λ 1 , λ 2 , . . . , λ k ), write M Λ Λ Λ k for the (k -1) × k matrix -λ 1 (λ -1 2 , . . . , λ -1 k ) ′ . . . I k-1 and L β β β (i) , Λ Λ Λ V i k for (L β β β (i) , Λ Λ Λ V i k;12 L β β β (i) , Λ Λ Λ V i k;13 . . . L β β β (i) , Λ Λ Λ V i k;(k-1)k ) ′ , with L β β β (i) , Λ Λ Λ V i k;jh := (λ V ih -λ V ij )(β β β (i) h ⊗ β β β (i) j ). Finally, let G β β β (i) k := (G β β β (i) k;12 G β β β (i) k;13 . . . G β β β (i) k;(k-1)k ), with G β β β (i) k;jh := e j ⊗ β β β (i) h -e h ⊗ β β β (i)
j , and ν ν ν (i) := diag(ν

(i) 12 , ν (i) 13 , . . . , ν (i) (k-1)k ) with ν (i) jh := λ V ij λ V ih /(λ V ij -λ V ih ) 2 .
We then have the following ULAN result.

:= P (n) ϑ ϑ ϑ;f | ϑ ϑ ϑ ∈ Θ Θ Θ is ULAN, with central sequence ∆ ∆ ∆ ϑ ϑ ϑ;f = ∆ ∆ ∆ (n) ϑ ϑ ϑ;f := ∆ ∆ ∆ I(n)′ ϑ ϑ ϑ;f , ∆ ∆ ∆ II(n)′ ϑ ϑ ϑ;f , ∆ ∆ ∆ III(n)′ ϑ ϑ ϑ;f , ∆ ∆ ∆ IV (n)′ ϑ ϑ ϑ;f ′ , ∆ ∆ ∆ I ϑ ϑ ϑ;f =    ∆ ∆ ∆ I,1 ϑ ϑ ϑ;f1 . . . ∆ ∆ ∆ I,m ϑ ϑ ϑ;fm    , ∆ ∆ ∆ II ϑ ϑ ϑ;f =    ∆ II,1 ϑ ϑ ϑ;f1 . . . ∆ II,m ϑ ϑ ϑ;fm    , ∆ ∆ ∆ III ϑ ϑ ϑ;f =    ∆ ∆ ∆ III,1 ϑ ϑ ϑ;f1 . . . ∆ ∆ ∆ III,m ϑ ϑ ϑ;fm   , ∆ ∆ ∆ IV ϑ ϑ ϑ;f =    ∆ ∆ ∆ IV ,1 ϑ ϑ ϑ;f1 . . . ∆ ∆ ∆ IV ,m ϑ ϑ ϑ;fm   ,
where (with

d ij = d ij (θ θ θ i , V i ) and U ij = U ij (θ θ θ i , V i )) ∆ ∆ ∆ I,i ϑ ϑ ϑ;fi := 1 √ n i σ i ni j=1 ϕ fi d ij σ i V -1/2 i U ij , ∆ II,i ϑ ϑ ϑ;fi := 1 2 √ n i σ 2 i ni j=1 ϕ fi d ij σ i d ij σ i -k , ∆ ∆ ∆ III,i ϑ ϑ ϑ;fi := 1 2 √ n i M Λ Λ Λ V i k H k (Λ Λ Λ V i ) -1/2 β β β (i)′ ⊗2 ni j=1 vec ϕ fi d ij σ i d ij σ i U ij U ′ ij , ∆ ∆ ∆ IV ,i ϑ ϑ ϑ;fi := 1 2 √ n i G β β β (i) k L β β β (i) , Λ Λ Λ V i k V ⊗2 i -1/2 ni j=1 vec ϕ fi d ij σ i d ij σ i U ij U ′ ij , i = 1, .
. . , m, and with block-diagonal information matrix

Γ Γ Γ ϑ ϑ ϑ;f := diag(Γ Γ Γ I ϑ ϑ ϑ;f , Γ Γ Γ II ϑ ϑ ϑ;f , Γ Γ Γ III ϑ ϑ ϑ;f , Γ Γ Γ IV ϑ ϑ ϑ;f ), (5.2)
where

Γ Γ Γ I ϑ ϑ ϑ;f = diag(Γ Γ Γ I,1 ϑ ϑ ϑ;f1 , . . . , Γ Γ Γ I,m ϑ ϑ ϑ;fm ), Γ Γ Γ II ϑ ϑ ϑ;f = diag(Γ II,1 ϑ ϑ ϑ;f1 , . . . , Γ II,m ϑ ϑ ϑ;fm ), Γ Γ Γ III ϑ ϑ ϑ;f = diag(Γ Γ Γ III,1 ϑ ϑ ϑ;f1 , . . . , Γ Γ Γ III,m ϑ ϑ ϑ;fm ), and Γ Γ Γ IV ϑ ϑ ϑ;f = diag(Γ Γ Γ IV ,1 ϑ ϑ ϑ;f1 , . . . , Γ Γ Γ IV ,m ϑ ϑ ϑ;fm ), with Γ Γ Γ I,i ϑ ϑ ϑ;fi := I k (f i ) kσ 2 i V -1 i , Γ II,i ϑ ϑ ϑ;fi := J k (f i ) -k 2 4σ 4 i , Γ Γ Γ III,i ϑ ϑ ϑ;fi := J k (f i ) 4k(k + 2) M Λ Λ Λ V i k H k ((Λ Λ Λ V i ) -1 ) ⊗2 [I k 2 + K k ] H ′ k (M Λ Λ Λ V i k ) ′ ,
and

Γ Γ Γ IV ,i ϑ ϑ ϑ;fi := J k (f i ) 4k(k + 2) G β β β (i) k (ν ν ν (i) ) -1 G β β β (i) k ′ .
More precisely, for any ϑ ϑ ϑ

(n) = ϑ ϑ ϑ + O(n -1/2
) and any bounded sequence τ τ τ (n) , we have, under

P (n) ϑ ϑ ϑ (n) ;f , Λ (n) ϑ ϑ ϑ (n) +n -1/2 ς ς ς (n) τ τ τ (n) /ϑ ϑ ϑ (n) ;f := log dP (n) ϑ ϑ ϑ (n) +n -1/2 ς ς ς (n) τ τ τ (n) ;f /dP (n) ϑ ϑ ϑ (n) ;f = (τ τ τ (n) ) ′ ∆ ∆ ∆ (n) ϑ ϑ ϑ (n) ;f - 1 2 (τ τ τ (n) ) ′ Γ Γ Γ ϑ ϑ ϑ;f τ τ τ (n) + o P (1)
and

∆ ∆ ∆ ϑ ϑ ϑ (n) ;f L -→ N (0, Γ Γ Γ ϑ ϑ ϑ;f ), as n → ∞.
Proposition 5.1, which is the multi-sample version of Theorem 2.1 in Hallin et al. (2010b), is the key result for constructing optimal inference procedures for eigenvectors and eigenvalues in multisample elliptical families. However, the standard methods for defining locally and asymptotically optimal tests under ULAN, which are based on the fact that local experiments converge to Gaussian shift experiments, do not apply here. Indeed, the parameter space Θ Θ Θ is a nonlinear manifold of R L (since (vec (SO k )) m is a nonlinear manifold of R mk 2 ). Just as in the one-sample situation, local limiting experiments therefore are curved Gaussian experiments. The problem of constructing optimal tests for diferentiable hypotheses in curved experiments has been considered in [START_REF] Hallin | Optimal rank-based testing for principal components[END_REF], where general results are provided, which we apply in the present situation.

Consider a parameter value ϑ ϑ ϑ 0 satisfying H 0 for some common eigenvector matrix β β β. As explained in Section 4.2,

β β β ′ V i β β β =: Λ Λ Λ V;β β β i , in general, is a reordered version of Λ Λ Λ V i , since the eigenvalues in Λ Λ Λ V
i are ranked in decreasing order of magnitude but not necessarily so in the locally β β β-reordered (we also call it ϑ ϑ ϑ 0 -reordered) Λ Λ Λ V;β β β i . At ϑ ϑ ϑ 0 , the locally Hallin, Paindaveine and Verdebout reordered Λ Λ Λ V;β β β i is a much more natural parameter than the original Λ Λ Λ V i . In that local reparametrization, the null hypothesis H 0 of CPC actually consists of the intersection of the nonlinear manifold Θ Θ Θ and the linear one

C := R mk ×(R + 0 ) m ×(C k-1 ) m ×M(1 m ⊗I k 2 )
, where M(A) stands for the vector space spanned by the columns of A.

Proposition 3.2 in Hallin et al. (2010b) on locally and asympotically optimal tests for differentiable hypotheses in curved ULAN experiments imply that, in the present context, a most stringent (at

ϑ ϑ ϑ 0 = (ϑ ϑ ϑ ′ I , ϑ ϑ ϑ ′ II , ϑ ϑ ϑ ′ III , 1 ′ m ⊗(vec (β β β)) ′ ) ′
) test for H 0 can be based on the quadratic form provided by the "classical" most stringent test for the (linear) null hypothesis consisting of the intersection of C and the tangent space to Θ Θ Θ at ϑ ϑ ϑ 0 . That intersection, still in the vicinity of ϑ ϑ ϑ 0 , reduces to

                                   ϑ ϑ ϑ I + n -1/2 ς ς ς (n) I τ τ τ (n) I ϑ ϑ ϑ II + n -1/2 ς ς ς (n) II τ τ τ (n) II ϑ ϑ ϑ III + n -1/2 ς ς ς (n) III τ τ τ (n) III vec (β β β + n -1/2 (r (n) 1 ) -1/2 b (1),(n) ) . . . vec (β β β + n -1/2 (r (n) m ) -1/2 b (m),(n) )             such that β β β ′ b (i),(n) + (b (i),(n) ) ′ β β β = 0, i = 1, . . . , m and (r (n) 1 ) -1/2 b (1),(n) = . . . = (r (n) m ) -1/2 b (m),(n)                        .
Solving this system leads to

ς ς ς (n) IV τ τ τ (n) IV = (r (n) 1 ) -1/2 (vec b (1),(n) ) ′ , . . . , (r (n) m ) -1/2 (vec b (m),(n) ) ′ ′ ∈ M(Ψ Ψ Ψ), with Ψ Ψ Ψ := 1 m ⊗       I k -β β β 1 β β β ′ 1 -β β β 2 β β β ′ 1 . . . -β β β k β β β ′ 1 -β β β 1 β β β ′ 2 I k -β β β 2 β β β ′ 2 . . . . . . . . . . . . . . . -β β β k β β β ′ k-1 -β β β 1 β β β ′ k . . . -β β β k-1 β β β ′ k I k -β β β k β β β ′ k      
, where β β β ℓ denotes β β β's ℓth column. Hence, the null hypothesis of CPC, locally at ϑ ϑ ϑ 0 , takes the form ς ς ς (n) τ τ τ (n) ∈ M(Υ Υ Υ), where

Υ Υ Υ := diag Υ Υ Υ I , Υ Υ Υ II , Υ Υ Υ III , Υ Υ Υ IV := diag I mk , I m , I m(k-1) , Ψ Ψ Ψ .
It then follows from Hallin et al. (2010b, Section 4.1) that, for given f, a locally and asymptotically most stringent test φ

(n) f , say, rejects H 0 for large values of Q (n) θ ϑ ϑ,f
, where

(throughout, we denote by A -the Moore-Penrose inverse of A)

Q (n) ϑ ϑ ϑ,f := (∆ ∆ ∆ ϑ ϑ ϑ,f ) ′ Γ Γ Γ - ϑ ϑ ϑ,f -(ς ς ς (n) ) -1 Υ Υ Υ Υ Υ Υ ′ (ς ς ς (n) ) -1 Γ Γ Γ ϑ ϑ ϑ,f (ς ς ς (n) ) -1 Υ Υ Υ -Υ Υ Υ ′ (ς ς ς (n) ) -1 ∆ ∆ ∆ ϑ ϑ ϑ,f (5.3)
= (∆ ∆ ∆ IV ϑ ϑ ϑ,f ) ′ (Γ Γ Γ IV ϑ ϑ ϑ,f ) --(ς ς ς (n) IV ) -1 Υ Υ Υ IV Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 Γ Γ Γ IV ϑ ϑ ϑ,f (ς ς ς (n) IV ) -1 Υ Υ Υ IV -Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 ∆ ∆ ∆ IV ϑ ϑ ϑ,f ,
and θ ϑ ϑ := θ ϑ ϑ (n) denotes a sequence of estimators satisfying the following Assumption (E)

with K reducing to {f}.

Optimal Rank-based Tests for CPC

13

Assumption (E). We say that a sequence of estimators ϑ ϑ ϑ (n) of ϑ ϑ ϑ, n ∈ N, satisfies Assumption (E) for some given collection K of m-tuples of standardized radial densities if, as n → ∞ as in Assumption (B), ϑ ϑ ϑ (n) is (E1) constrained:

P (n)
ϑ ϑ ϑ0;g ϑ ϑ ϑ (n) ∈ H 0 = 1 for all n, ϑ ϑ ϑ 0 ∈ H 0 , and g ∈ K;

(E2) n 1/2 (ς ς ς (n) ) -1 -consistent: for all ϑ ϑ ϑ 0 ∈ H 0 , n 1/2 ς ς ς (n) -1 (ϑ ϑ ϑ (n) -ϑ ϑ ϑ 0 ) = O P (1), as n → ∞, under P (n)
ϑ ϑ ϑ0;g , for all g ∈ K; (E3) locally asymptotically discrete: for all ϑ ϑ ϑ 0 ∈ H 0 and all c > 0, there exists M = M (c) > 0 such that the number of possible values of ϑ ϑ ϑ (n) in balls of the form

{t ∈ R L : n 1/2 ς ς ς (n) -1 (t -ϑ ϑ ϑ 0 ) ≤ c} is bounded by M , uniformly in n.
Assumption (E3) is a theoretical assumption that has no impact in practice (see pages 125 or 188 of Le Cam and Yang (2000) for a discussion). Any estimator satisfying (E1) and (E2) can be turned into an estimator also satisfying (E3) by discretization (see, e.g., Hallin et al. (2006)), a fact we will no further emphasize in the notation by tacitly assuming, in the statement of asymptotic results, that any ϑ ϑ ϑ (n) , when necessary, has been adequately discretized.

The sequences of tests φ

(n) f
associated with the m-tuple f achieve local asymptotic optimality at f. Moreover, they are of a purely parametric nature since, in general, they are valid at f only-that is, they achieve the correct nominal asymptotic level under correctly specified f only, even when based on an estimator ϑ ϑ ϑ (n) satisfying Assumption (E) under a broad collection K of densities. An exception is the Gaussian test φ (n) N associated with an m-tuple of Gaussian radial densities which, with a Gaussian MLE ϑ ϑ ϑ (n) , remains valid under any m-tuple g = (g 1 , . . . , g m ) such that g i has Gaussian kurtosis (that is, in the notation of Section 6 below, κ k (g i ) = 0) for all i = 1, . . . , m (this, of course, requires finite fourth-order moments). Clearly, this is somewhat unsatisfactory in practice, and there is a need to define alternative optimal tests, that remain valid under much broader conditions. The next two sections are devoted to the construction of such tests.

Gaussian and pseudo-Gaussian tests.

In this section, we construct a pseudo-Gaussian version φ

(n) † N of the Gaussian test φ (n) N ,
that is, a test that shares the optimality properties of φ (n) N in the multinormal case, while remaining valid under a much broader class of densities-namely, the class of all (possibly heterokurtic) m-tuples of elliptic densities with finite fourth-order moments. Our construction is based on a general method proposed by Hallin and Paindaveine (2008a), which exploits the ULAN structure of the experiment. Finally, we show that this pseudo-Gaussian test φ

(F 4 1 ) m := g = (g 1 , . . . , g m ) ∈ (F 1 ) m : E k (g i ) := 1 0 ( G-1 ik (u)) 4 du < ∞, i = 1, . . . , m ,
where r → Gik (r) := (µ k-1;gi ) -1 r 0 s k-1 g i (s) ds stands for the distribution function, under P

(n) ϑ ϑ ϑ;g , of the d ij (ϑ ϑ ϑ)'s, j = 1, . . . , n i . Then, writing D k (g i ) := 1 0 ( G-1 ik (u)) 2 du, κ k (g i ) := k k + 2 × E k (g i ) D 2 k (g i ) -1,
for any g ∈ (F 4 1 ) m , is a measure of kurtosis in the ith elliptic population under P

(n) ϑ ϑ ϑ;g ; see, e.g., page 54 of [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis[END_REF].

If g i is Gaussian, E k (g i ) = k(k + 2)/a 2 k and D k (g i ) = k/a k , so that κ k (g i ) = 0.
Since the optimal Gaussian test φ N clearly requires controlling the asymptotic behavior of ∆ ∆ ∆ IV ϑ ϑ ϑ;φ also away from the Gaussian case. This is made possible by the following result. Lemma 6.1. Assume that (A), (B), and (D ′ ) hold. Fix any ϑ ϑ ϑ 0 ∈ H 0 (the eigenvalues are the ϑ ϑ ϑ 0 -ordered eigenvalues) and g ∈ (F 4 1 ) m . Then, (i) under P (n) ϑ ϑ ϑ0;g , ∆ ∆ ∆ IV ϑ ϑ ϑ0;φ is asymptotically normal, with mean zero and covariance matrix Γ Γ Γ g,IV ϑ ϑ ϑ0;φ := diag Γ Γ Γ g,IV ,1 ϑ ϑ ϑ0;φ , . . . , Γ Γ Γ g,IV ,m ϑ ϑ ϑ0;φ

, where Γ Γ Γ g,IV ,i ϑ ϑ ϑ0;φ :=

a 2 k E k (gi) 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ ;
(ii) reinforcing (D ′ ) into (D) and defining Γ Γ Γ g,IV ϑ ϑ ϑ0;φ,g := diag Γ Γ Γ g,IV ,1 ϑ ϑ ϑ0;φ,g , . . . , Γ Γ Γ g,IV ,m ϑ ϑ ϑ0;φ,g , with Γ Γ Γ g,IV ,i ϑ ϑ ϑ0;φ,g :=

a k D k (gi) 4k G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ , we have that ∆ ∆ ∆ IV ϑ ϑ ϑ0+n -1/2 ς ς ς (n) τ τ τ (n) ;φ -∆ ∆ ∆ IV ϑ ϑ ϑ0;φ + Γ Γ Γ g,IV ϑ ϑ ϑ0;φ,g τ τ τ (n) IV is o P (1) as n → ∞, under P (n) ϑ ϑ ϑ0;g ; (iii) still with (D ′ ) reinforced into (D), ∆ ∆ ∆ IV ϑ ϑ ϑ0;φ -Γ Γ Γ g,IV ϑ ϑ ϑ0;φ,g τ τ τ (n)
IV is asymptotically normal, with mean zero and covariance matrix Γ Γ Γ g,IV ϑ ϑ ϑ0;φ under P (n)

ϑ ϑ ϑ0+n -1/2 ς ς ς (n) τ τ τ (n) ;g for any g ∈ (F 4 a ) m := (F 4 1 ) m ∩ (F a ) m .
Point (i) of this Lemma directly follows from the multivariate central limit theorem. Note that Assumption (D ′ ) is sufficient for asymptotic normality since the common value β β β of the eigenvector matrix is well identified under Assumption (D ′ ). However, points (ii) and (iii) require ULAN and therefore Assumption (D); they directly follow from Lemma 4.

Q N (n) ϑ ϑ ϑ0,g := ∆ ∆ ∆ IV ϑ ϑ ϑ0;φ ′ (Γ Γ Γ g,IV ϑ ϑ ϑ0;φ ) ⊥ ∆ ∆ ∆ IV ϑ ϑ ϑ0;φ , (6.1)
with

(Γ Γ Γ g,IV ϑ ϑ ϑ;φ ) ⊥ := (Γ Γ Γ g,IV ϑ ϑ ϑ;φ ) --(Γ Γ Γ g,IV ϑ ϑ ϑ;φ ) -Γ Γ Γ g,IV ϑ ϑ ϑ;φ,g (ς ς ς (n) IV ) -1 Υ Υ Υ IV Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 Γ Γ Γ g,IV ϑ ϑ ϑ;φ,g (Γ Γ Γ g,IV ϑ ϑ ϑ;φ ) -Γ Γ Γ g,IV ϑ ϑ ϑ;φ,g (ς ς ς (n) IV ) -1 Υ Υ Υ IV -Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 Γ Γ Γ g,IV ϑ ϑ ϑ;φ,g (Γ Γ Γ g,IV ϑ ϑ ϑ;φ ) -, where Γ Γ Γ g,IV
ϑ ϑ ϑ;φ and Γ Γ Γ g,IV ϑ ϑ ϑ;φ,g are defined in Lemma 6.1, and eigenvalues have been ϑ ϑ ϑ 0 -reordered as explained in Sections 3 and 4. Now, using the fact that Γ Γ Γ g,IV ,i ϑ ϑ ϑ;φ

-= k(k + 2) a 2 k E k (g i ) G β β β (i) k ν ν ν (i) (G β β β (i) k ) ′ , the quadratic form Q N (n)
ϑ ϑ ϑ0,g after some algebra rewrites

Q N (n) ϑ ϑ ϑ0,g = m i=1 1≤j<j ′ ≤k n i 1 + κ k (g i ) (β β β ′ j S (n) φ,i β β β j ′ ) 2 (6.2) - m i,i ′ =1 1≤j<j ′ ≤k n i n i ′ n 1 (1 + κ k (g i ))(1 + κ k (g i ′ )) ν jj ′ (g) (ν (i) jj ′ ν (i ′ ) jj ′ ) 1/2 (β β β ′ j S (n) φ,i β β β j ′ ) (β β β ′ j S (n) φ,i ′ β β β j ′ ),
where

S (n) φ,i := k σ 2 D k (g i ) β β β(Λ Λ Λ V;β β β i ) -1/2 β β β ′ 1 n i ni j=1 (X ij -θ θ θ i )(X ij -θ θ θ i ) ′ β β β(Λ Λ Λ V;β β β i ) -1/2 β β β ′ and diag(ν 12 (g), . . . , ν (k-1)k (g)) := m i=1 r (n) i 1 + κ k (g i ) (ν ν ν (i) ) -1 -1 =: ν ν ν(g).
In order to obtain a genuine test statistic (that is, a random variable that does not depend anymore on ϑ ϑ ϑ 0 nor g) which nevertheless, under any P (n) ϑ ϑ ϑ0;g (with ϑ ϑ ϑ 0 ∈ H 0 and g ∈ (F 4 1 ) m ) and contiguous alternatives, is asymptotically equivalent to Q

N (n)
ϑ ϑ ϑ0,g , it is sufficient to (a) replace ϑ ϑ ϑ 0 in (6.2) with some estimator ϑ ϑ ϑ (n) satisfying Assumption (E) for the class K = (F 4 1 ) m , and (b) replace the coefficients D k (g i ) and the kurtoses κ k (g i ) with consistent (still under P

(n)

ϑ ϑ ϑ0;g , ϑ ϑ ϑ 0 ∈ H 0 , g ∈ (F 4 1 ) m ) estimators D(n) i and κ(n) i , respectively. In this pseudo-Gaussian context, a natural estimator for ϑ ϑ ϑ 0 is ϑ ϑ ϑ (n) N := X′ 1 , . . . , X′ m , σ2 1 , . . . , σ2 m , (6.3)
(dv

• ec Λ Λ Λ 1 ) ′ / k j=1 ( λ1j ) 1/k , . . . , (dv • ec Λ Λ Λ m ) ′ / k j=1 ( λmj ) 1/k , 1 ′ m ⊗ (vec β β β) ′ ′ ,

Hallin, Paindaveine and Verdebout

where Xi := n -1 i ni j=1 X ij , Λ Λ Λ i = diag( λi1 , . . . , λik ), i = 1, . . . , m, and β β β are the maximum likelihood estimators of the corresponding parameters in the CPC model (see [START_REF] Flury | Asymptotic theory for common principal components analysis[END_REF], and σ2

i denotes the empirical median of the

d 2 ij ( Xi , β β β Λ Λ Λ i β β β ′ / k j=1 ( λij ) 1/k
)'s, j = 1, . . . , n i . Note that the estimators β β β and Λ Λ Λ i , resulting from the Flury and Gautschi (1986) algorithm, do not provide consistent estimators of β β β and the ϑ ϑ ϑ 0 -reordered eigenvalues matrices Λ Λ Λ V;β β β i , respectively, because of the possibly different ordering of eigenvalues (and also because the determinant of Λ Λ Λ i is not equal to one in general). However, [START_REF] Flury | Asymptotic theory for common principal components analysis[END_REF] shows that they are root-n consistent for

N Π β β β = N Π β β β (1) = N Π M Π2 2 β β β (2) = . . . = N Π M Πm m β β β (m)
and the corresponding reordered version of the Λ Λ Λ V;β β β i 's for some global permutation matrix N Π . Now, since both the null hypothesis and the test statistic

Q N (n) ϑ ϑ ϑ0,g
are invariant with respect to such global permutations, no reordering of the eigenvalues is needed here. Note that D k (g i ) is consistently estimated by

k k j=1 ( λij ) 1/k /σ 2 i , i = 1, . . . , m. Finally, an obvious choice for κ(n) i is then κ(n) i := k k + 2 × n -1 i ni j=1 d 4 ij ( Xi , β β β Λ Λ Λ i β β β ′ ) n -1 i ni j=1 d 2 ij ( Xi , β β β Λ Λ Λ i β β β ′ ) 2 -1. Letting S (n) i := n i -1 ni j=1 (X ij -Xi )(X ij -Xi ) ′ , this leads to the test statistic Q (n) † N := m i=1 1≤j<j ′ ≤k n i 1 + κ(n) i ( λij λij ′ ) -1 ( β β β ′ j S (n) i β β β j ′ ) 2 (6.4) - m i,i ′ =1 1≤j<j ′ ≤k n i n i ′ n ( λij λij ′ ) -1/2 ( λi ′ j λi ′ j ′ ) -1/2 (1 + κ(n) i )(1 + κ(n) i ′ ) νjj ′ (ν (i) jj ′ ν(i ′ ) jj ′ ) 1/2 ( β β β ′ j S (n) i β β β j ′ ) ( β β β ′ j S (n) i ′ β β β j ′ ),
where we write ν(i) jj ′ and νjj ′ , respectively, for the ν exceeds the α-upper quantile of the chi-square distribution with (m -1)k(k -1)/2 degrees of freedom.

To investigate the asymptotic behavior of this pseudo-Gaussian test under local alternatives, we consider perturbations ϑ ϑ ϑ 0 + n -1/2 ς ς ς (n) τ τ τ (n) such that, letting Assumption (B ′ ) hold and putting ς ς ςτ τ τ := lim n→∞ ς ς ς (n) τ τ τ (n) , with

ς ς ς (n) IV τ τ τ (n) IV = ((r (n) 1 ) -1/2 (vec b (1),(n) ) ′ , . . . , (r (n) m ) -1/2 (vec b (m),(n) ) ′ ) ′ and ς ς ς IV τ τ τ IV = (r -1/2 1 (vec b (1) ) ′ , . . . , r -1/2 m (vec b (m) ) ′ ) ′ ,
we still have, for all i = 1, . . . , m, β β β ′ b (i) + (b (i) ) ′ β β β = 0 (where β β β is the common value, under ϑ ϑ ϑ 0 , of the m eigenvector matrices). Assume furthermore that the corresponding perturbed value of ϑ ϑ ϑ 0 does not belong to H 0 anymore (does not belong to the linear manifold C). Letting, for any such ϑ ϑ ϑ 0 and any g ∈ (F 4 1 ) m ,

C N ϑ ϑ ϑ0;g := diag 1 1 + κ k (g 1 ) (ν ν ν (1) ) -1 , . . . , 1 1 + κ k (g m ) (ν ν ν (m) ) -1
and D

N (n) ϑ ϑ ϑ0;g := C N ϑ ϑ ϑ0;g -C N ϑ ϑ ϑ0;g ((r (n) ) -1 1 m 1 ′ m (r (n) ) -1 ) ⊗ ν ν ν(g) C N ϑ ϑ ϑ0;g , standard algebra yields l N ϑ ϑ ϑ0,τ τ τ ;g := lim n→∞ (τ τ τ (n) IV ) ′ (I m ⊗ G β β β k ) D N (n) ϑ ϑ ϑ0;g (I m ⊗ G β β β k ) ′ (τ τ τ (n) IV ) (6.5) = m i,i ′ =1 (vec b (i) ) ′ G β β β k δ ii ′ T N (i,i ′ ) g -(r i r i ′ ) 1/2 T N (i,i ′ ) g ν ν ν(g)T N (i,i ′ ) g G β β β k ′ (vec b (i ′ ) ),
where T

N (i,i ′ ) g := (1 + κ k (g i ))(1 + κ k (g i ′ )) -1/2 (ν ν ν (i) ) -1/2 (ν ν ν (i ′ )
) -1/2 ; r (n) was defined on Page 10. The following result then summarizes the asymptotic properties of

Q (n) † N and φ (n) † N .
Proposition 6.1. Assume that (A), (B), and (D ′ ) hold. Then, (i)

Q (n) † N is asymptotically chi-square with (m -1)k(k -1)/2 degrees of freedom under ϑ ϑ ϑ∈H0 g∈(F 4 1 ) m {P (n)
ϑ ϑ ϑ;g }, and (provided that (D ′ ) is reinforced into (D) and (B) into (B ′ )) asymptotically noncentral chi-square, still with (m -1)k(k -1)/2 degrees of freedom, but with noncentrality parameter l N ϑ ϑ ϑ,τ τ τ ;g under P (n) ϑ ϑ ϑ+n -1/2 ς ς ς (n) τ τ τ (n) ;g , ϑ ϑ ϑ ∈ H 0 , ς ς ςτ τ τ := lim n→∞ ς ς ς (n) τ τ τ (n) as described above, and g ∈ (F 4 a ) m ; (ii) φ Even though the pseudo-Gaussian test φ (n) † N of the previous section is valid under a broad class of densities, it still requires finite fourth-order moments, and may be poorly Hallin, Paindaveine and Verdebout robust, since it is based on empirical covariance matrices. In this section, we show how ranks (actually, a multivariate generalization of signed ranks) allow us to improve on the performances of pseudo-Gaussian tests both in terms of validity and efficiency.

A general result by [START_REF] Hallin | Semiparametric efficiency, distributionfreeness, and invariance[END_REF] implies that, in adaptive semiparametric models for which fixed-f submodels are ULAN and fixed-ϑ ϑ ϑ submodels are generated by a group G ϑ ϑ ϑ of transformations (acting on the observation space), invariant versions of central sequences exist under very general assumptions. In the present case, the ULAN structure of fixed-f submodels is established in Section 5. As for the fixed-ϑ ϑ ϑ submodels, consider the group G ϑ ϑ ϑ ,• of continuous monotone radial transformations G h of the form

X → G h (X 11 , . . . , X mnm ) := (θ θ θ 1 +h 1 (d 11 (θ θ θ 1 ,β β β (1) Λ Λ Λ V 1 β β β (1)′ ))β β β (1) (Λ Λ Λ V 1 ) 1/2 β β β (1)′ U 11 (θ θ θ 1 ,β β β (1) Λ Λ Λ V 1 β β β (1)′ ), . . . , θ θ θ m +h m (d mnm (θ θ θ m ,β β β (m) Λ Λ Λ V m β β β (m)′ ))β β β (m) (Λ Λ Λ V m ) 1/2 β β β (m)′ U mnm (θ θ θ m ,β β β (m) Λ Λ Λ V m β β β (m)′ )),
where for all i = 1, . . . , m, h i : R + → R + is continuous, monotone increasing, and such that h i (0) = 0 and lim r→∞ h i (r) = ∞. Letting σ 2 := (σ 2 1 , . . . , σ 2 m ) ′ , this group is a generating group for the submodel σ 2 f P (n) ϑ ϑ ϑ I ,σ 2 ,ϑ ϑ ϑ III ,ϑ ϑ ϑ IV ;f (a nonparametric family). The invariance principle suggests basing inference on statistics that are measurable with respect to the corresponding maximal invariant, namely the vectors (U 11 , . . . , U mnm ) (a multivariate generalization of signs) along with the vector (R 11 , . . . , R mnm ) of ranks, where

U ij = U ij (θ θ θ i , β β β (i) Λ Λ Λ V i β β β (i)′ ), and R ij = R ij (θ θ θ i , β β β (i) Λ Λ Λ V i β β β (i)′ ) denotes the rank of d ij (θ θ θ i , β β β (i) Λ Λ Λ V i β β β (i)′ ) among d i1 (θ θ θ i , β β β (i) Λ Λ Λ V i β β β (i)′ ), . . . , d ini (θ θ θ i , β β β (i) Λ Λ Λ V i β β β (i)′
). Such invariant statistics of course are distribution-free under σ 2 f P (n) ϑ ϑ ϑ I ,σ 2 ,ϑ ϑ ϑ III ,ϑ ϑ ϑ IV ;f . The existence of central sequences that are measurable with respect to the multivariate signs U ij and the ranks R ij (recall that central sequences are always defined up to o P (1) quantities) is established by the asymptotic representation result of Lemma 7.1(i) below.

Denoting by K an m-tuple of score functions satisfying Assumption (C), consider the random vectors

∆ ∆ ∆ IV ϑ ϑ ϑ;K := (( ∆ ∆ ∆ IV ,1 ϑ ϑ ϑ;K1 ) ′ , . . . , ( ∆ ∆ ∆ IV ,m ϑ ϑ ϑ;Km ) ′ ) ′ , with ∆ ∆ ∆ IV ,i ϑ ϑ ϑ;Ki := 1 2 √ n i G β β β (i) k L β β β (i) , Λ Λ Λ V i k V ⊗2 i -1/2 ni j=1 K i R ij n i + 1 vec U ij U ′ ij . (7.1)
In order to describe the asymptotic behavior of ∆ ∆ ∆ IV ϑ ϑ ϑ;K , similarly define

∆ ∆ ∆ IV ϑ ϑ ϑ;K;g := ((∆ ∆ ∆ IV ,1 ϑ ϑ ϑ;K1;g1 ) ′ , . . . , (∆ ∆ ∆ IV ,m ϑ ϑ ϑ;Km;gm ) ′ ) ′ , with ∆ ∆ ∆ IV ,i ϑ ϑ ϑ;Ki;gi := 1 2 √ n i G β β β (i) k L β β β (i) , Λ Λ Λ V i k V ⊗2 i -1/2 ni j=1 K i Gik d ij σ vec U ij U ′ ij .
We then have the following result.

Lemma 7.1. Assume that (A), (B), (C), and (D ′ ) hold. Fix any ϑ ϑ ϑ 0 ∈ H 0 (the eigenvalues are the ϑ ϑ ϑ 0 -reordered eigenvalues) and g ∈ (F 1 ) m . Then,

(i) ∆ ∆ ∆ IV ϑ ϑ ϑ0;K = ∆ ∆ ∆ IV ϑ ϑ ϑ0;K;g + o L 2 (1), under P (n) ϑ ϑ ϑ0;g , as n → ∞; (ii) under P (n) ϑ ϑ ϑ0;g , ∆ ∆ ∆ IV
ϑ ϑ ϑ0;K;g is asymptotically normal with mean zero and covariance matrix Γ Γ Γ IV ϑ ϑ ϑ0;K := diag(Γ Γ Γ IV ,1 ϑ ϑ ϑ0;K1 , . . . , Γ Γ Γ IV ,m ϑ ϑ ϑ0;Km ), with Γ Γ Γ IV ,i ϑ ϑ ϑ0;Ki :

= J k (Ki) 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ ; (iii) reinforcing (D ′ ) into (D) and defining Γ Γ Γ IV ϑ ϑ ϑ0;K,g := diag(Γ Γ Γ IV ,1 ϑ ϑ ϑ0;K1,g1 , . . . , Γ Γ Γ IV ,m ϑ ϑ ϑ0;Km,gm ), with Γ Γ Γ IV ,i ϑ ϑ ϑ0;Ki,gi := J k (Ki,gi) 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ , and assuming moreover that g ∈ (F a ) m , ∆ ∆ ∆ IV ϑ ϑ ϑ0;K;g -Γ Γ Γ IV ϑ ϑ ϑ0;K,g τ τ τ (n)
IV is asymptotically normal with mean zero and covariance ma-

trix Γ Γ Γ IV ϑ ϑ ϑ0;K under P (n) ϑ ϑ ϑ0+n -1/2 ς ς ς (n) τ τ τ (n) ;g .
An immediate corollary of the asymptotic representation result in Part (i) of this lemma is that ∆ ∆ ∆ IV ϑ ϑ ϑ;f := ∆ ∆ ∆ IV ϑ ϑ ϑ;K f , with K f := (K f1 , . . . , K fm ), constitutes a signed-rank version of the eigenvector part ∆ ∆ ∆ IV ϑ ϑ ϑ;f of the f-central sequence; Parts (ii) and (iii) provide the asymptotic distribution of ∆ ∆ ∆ IV ϑ ϑ ϑ;K f , under the null and local alternatives. In order to construct a test statistic based on ∆ ∆ ∆ IV ϑ ϑ ϑ;K , we also need to know how it is affected (asymptotically, under the null hypothesis and contiguous alternatives) by the substitution, for ϑ ϑ ϑ, of an estimator ϑ ϑ ϑ (n) satisfying Assumption (E). This important step is taken care of by the asymptotic linearity result of Lemma 7.2. This lemma uses the local reordering of eigenvalues described in the previous sections. Lemma 7.2. Assume that (A), (B), (C), and (D ′ ) hold, and let ϑ ϑ ϑ (n) be an estimator satisfying Assumption (E). Fix ϑ ϑ ϑ 0 ∈ H 0 (with common value β β β of the eigenvector matrices and the corresponding reordering of eigenvalues). Then, for all g ∈ (F a ) m ,

∆ ∆ ∆ IV θ ϑ ϑ;K -∆ ∆ ∆ IV ϑ ϑ ϑ0;K + Γ Γ Γ IV ϑ ϑ ϑ0;K,g (ς ς ς (n) IV ) -1 n 1/2 1 m ⊗ vec ( β β β -β β β) = o P (1)
as n → ∞, under P

ϑ ϑ ϑ0;g .

See the appendix for the proof. Finally, the construction of the rank-based tests of Section 7.2 requires consistent estimation of the cross-information quantities J k (K i , g i ), i = 1, . . . , m. The following method, which is inspired by a local maximum likelihood argument, heavily relies on the asymptotic linearity result of Lemma 7.2, and was first proposed, in a different context, by Hallin et al. (2006). Fix i ∈ {1, . . . , m} and g ∈ (F a ) m , and let ϑ ϑ ϑ (n) satisfy Assumption (E). Denote by β β β the estimator of the common eigenvector matrix in ϑ ϑ ϑ (n) ; that is, assume that ϑ ϑ ϑ (n) =: ( θ ϑ ϑ

′ I , θ ϑ ϑ ′ II , θ ϑ ϑ ′ III , 1 ′ m ⊗ (vec β β β) ′ ) ′ . Define, for any ρ ≥ 0, vec ( β β β(ρ)) := vec ( β β β) + n -1/2 i ρ k(k + 2)G β β β k ν ν ν (i) (G β β β k ) ′ ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki . (7.2)
Consider the (almost surely) piecewise continuous quadratic form

ρ → h (n) i (ρ) := ( ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ,Ki ) ′ (Γ Γ Γ IV ,i ϑ ϑ ϑ (n) ;Ki ) -∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) (ρ);Ki ,
where ϑ ϑ ϑ (n) (ρ) is simply obtained from ϑ ϑ ϑ (n) by replacing β

β β with β β β(ρ), that is, ϑ ϑ ϑ (n) (ρ) := ( θ ϑ ϑ ′ I , θ ϑ ϑ ′ II , θ ϑ ϑ ′ III , 1 ′ m ⊗ (vec β β β(ρ)) ′ ) ′ .
Then, Lemma 7.2, the consistency of ϑ ϑ ϑ (n) , and the definition of β β β(ρ) in (7.2) imply that

h (n) i (ρ) = ( ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki ) ′ (Γ Γ Γ IV ,i ϑ ϑ ϑ (n) ;Ki ) -∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki -Γ Γ Γ IV ,i ϑ ϑ ϑ (n) ;Ki,gi n 1/2 i vec ( β β β(ρ) -β β β) + o P (1) = ( ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki ) ′ (Γ Γ Γ IV ,i ϑ ϑ ϑ (n) ;Ki ) -I k 2 -ρk(k + 2)Γ Γ Γ IV ,i ϑ ϑ ϑ (n) ;Ki,gi G β β β k ν ν ν (i) (G β β β k ) ′ ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki + o P (1) (7.3)
as n → ∞, under P

(n) ϑ ϑ ϑ0;g . Now, note that k(k+2)G β β β k ν ν ν (i) (G β β β k ) ′ is the Moore-Penrose general- ized inverse of 1 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ . Hence, recalling that Γ Γ Γ IV ,i ϑ ϑ ϑ0;Ki := J k (Ki) 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ and Γ Γ Γ IV ,i ϑ ϑ ϑ0;K,g := J k (Ki,gi) 4k(k+2) G β β β k (ν ν ν (i) ) -1 (G β β β k ) ′ , (7.
3) can be rewritten as

h (n) i (ρ) = 1 -J k (K i , g i )ρ h (n) i (0) + o P (1), (7.4)
still as n → ∞, under P

(n) ϑ ϑ ϑ0;g . Since h

(n) i (0) > 0, an intuitively appealing estimator for (J k (K i , g i )) -1 , in view of (7.4), is given by ρ := inf{ρ > 0 : h (n) i (ρ) < 0}. By proceeding along the same lines as in Hallin et al. (2006), it is easily shown that Ĵk (K i , g i ) := ρ-1 is, after adequate discretization (which still has no impact in fixed-n i practice), a consistent estimator of J k (K i , g i ) under P (n) ϑ ϑ ϑ0;g .

Optimal rank-based tests.

Motivated by the form of the pseudo-Gaussian statistic in (6.1), consider the signed-rank statistic

Q (n) ϑ ϑ ϑ0;K,g := ∆ ∆ ∆ IV ϑ ϑ ϑ0;K ′ (Γ Γ Γ IV ϑ ϑ ϑ0;K,g ) ⊥ ∆ ∆ ∆ IV ϑ ϑ ϑ0;K , (7.5)
with

(Γ Γ Γ IV ϑ ϑ ϑ;K,g ) ⊥ := (Γ Γ Γ IV ϑ ϑ ϑ;K ) --(Γ Γ Γ IV ϑ ϑ ϑ;K ) -Γ Γ Γ IV ϑ ϑ ϑ;K,g (ς ς ς (n) IV ) -1 Υ Υ Υ IV × Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 Γ Γ Γ IV ϑ ϑ ϑ;K,g (Γ Γ Γ IV ϑ ϑ ϑ;K ) -Γ Γ Γ IV ϑ ϑ ϑ;K,g (ς ς ς (n) IV ) -1 Υ Υ Υ IV -Υ Υ Υ IV ′ (ς ς ς (n) IV ) -1 Γ Γ Γ IV ϑ ϑ ϑ;K,g (Γ Γ Γ IV ϑ ϑ ϑ;K ) -,
where Γ Γ Γ IV ϑ ϑ ϑ;K and Γ Γ Γ IV ϑ ϑ ϑ;K,g are defined in Lemma 7.1 (this includes the ϑ ϑ ϑ 0 -reordering of eigenvalues). Now, by using the facts that (G

β β β (i) k ) ′ G β β β (i) k = 2I k(k-1)/2 and Γ Γ Γ IV ,i ϑ ϑ ϑ0;K -= k(k+2) J k (K1) G β β β (i) k ν ν ν (i) (G β β β (i) k ) ′ , standard algebra yields Q (n) ϑ ϑ ϑ0;K,g = k(k + 2) m i=1 1≤j<j ′ ≤k n i J k (K i ) (β β β ′ j S K;i β β β j ′ ) 2 (7.6) - m i,i ′ =1 1≤j<j ′ ≤k n i n i ′ n J k (K i , g i )J k (K i ′ , g i ′ ) J k (K i )J k (K i ′ ) ν jj ′ (K, g) (ν (i) jj ′ ν (i ′ ) jj ′ ) 1/2 (β β β ′ j S K;i β β β j ′ ) (β β β ′ j S K;i ′ β β β j ′ ) ,
where

S K;i := 1 n i ni j=1 K i R ij (ϑ ϑ ϑ 0 ) n + 1 U ij (ϑ ϑ ϑ 0 )U ′ ij (ϑ ϑ ϑ 0 ) and diag(ν 12 (K, g), . . . , ν (k-1)k (K, g)) := m i=1 r (n) i J 2 k (K i , g i ) J k (K i ) (ν ν ν (i) ) -1 -1 =: ν ν ν(K, g).
As for the pseudo-Gaussian tests of Section 6, obtaining a genuine test statistic requires replacing in (7.6) the parameter value ϑ ϑ ϑ 0 with an estimator ϑ ϑ ϑ (n) satisfying Assumption (E)-here, with K = (F a ) m -and replacing the cross-information quantities J k (K i , g i ) with consistent (under P (n) ϑ ϑ ϑ0;g , g ∈ (F a ) m ) estimates. The estimates Ĵk (K i , g i ) defined at the end of Section 7.1 can be used for that purpose. As for ϑ ϑ ϑ (n) , many choices are possible. Still avoiding moment assumptions, we propose the following one. Let θ θ θ

(n) i and V (n)
i , i = 1, . . . , m be the location and shape estimators associated with the affineequivariant multivariate median proposed by [START_REF] Hettmansperger | A practical affine equivariant multivariate median[END_REF], which are implicitly defined by

1 n i ni j=1 U ij (θ θ θ (n) i , V (n) i ) = 0 and 1 n i ni j=1 U ij (θ θ θ (n) i , V (n) i )U ′ ij (θ θ θ (n) i , V (n)
i ) = 1 k I k , with det(V (n) i ) = 1, i = 1, .
. . , m. Under H 0 , the eigenvalue matrices Λ Λ Λ V i = diag(λ V i1 , . . . , λ V ik ), i = 1, . . . , m and the matrix β β β = (β β β 1 , . . . , β β β k ) of common eigenvectors then can be estimated consistently by using the plug-in method as in [START_REF] Boente | A robust approach to common principal components[END_REF]. More precisely, the resulting estimates Λ Λ Λ V i , i = 1, . . . , m and β β β are obtained by solving the ML-type equations

β β β ′ j m i=1 n i λ V ij -λ V il λ V ij λ V il V (n) i β β β l = 0, j = l = 1, . . . , k, (7.7)
β β β ′ j V (n) i β β β j = λ V ij , i = 1, . . . , m, j = 1, . . . , k, β β β ′ j β β β l = δ jl , j, l = 1, . . . , k,
where δ jl is the usual Kronecker symbol. As in the pseudo-Gaussian context, the resulting estimators are root-n consistent up to a global permutation (see the comments below (6.3)). Now, the scale parameters σ 2 i , i = 1, . . . , m do not appear in (7.6), so that the resulting "estimators" for ϑ ϑ ϑ can be chosen as if they were specified:

ϑ ϑ ϑ (n) := θ θ θ ′ 1 , . . . , θ θ θ ′ m , σ 2 1 , . . . , σ 2 m , (dv • ec ( Λ Λ Λ V 1 )) ′ , . . . , (dv • ec ( Λ Λ Λ V m )) ′ , 1 ′ m ⊗ (vec β β β) ′ ′ . (7.8)
It can be checked that, after appropriate discretization, ϑ ϑ ϑ (n) in (7.8) satisfies Assumption (E) with K = (F 1 ) m (hence without requiring any moment condition), so that it can

Hallin, Paindaveine and Verdebout be used advantageously in our signed-rank tests. Summing up, the signed-rank statistic we propose is

Q (n) K := k(k + 2) m i=1 1≤j<j ′ ≤k n i J (K i ) ( β β β ′ j Ŝ K;i β β β j ′ ) 2 - m i,i ′ =1 1≤j<j ′ ≤k n i n i ′ n Ĵk (K i , g i ) Ĵk (K i ′ , g i ′ ) J (K i )J (K i ′ ) νjj ′ (K, g) (ν (i) jj ′ ν(i ′ ) jj ′ ) 1/2 ( β β β ′ j Ŝ K;i β β β j ′ ) ( β β β ′ j Ŝ K;i ′ β β β j ′ ) , where Ŝ K;i := 1 n i ni j=1 K i R ij (ϑ ϑ ϑ (n) ) n + 1 U ij (ϑ ϑ ϑ (n) )U ′ ij (ϑ ϑ ϑ (n) ) and diag(ν 12 (K, g), . . . , ν(k-1)k (K, g)) := m i=1 r (n) i Ĵ 2 k (K i , g i ) J (K i ) (ν ν ν (i) ) -1 -1 =: ν ν ν(K, g),
all parameters being estimated via the chosen estimator ϑ ϑ ϑ (n) (given in (7.8), for instance).

The resulting test φ

K rejects the null of CPC, at asymptotic level α, as soon as Q (n) K exceeds the α-upper quantile of the chi-square distribution with (m-1)k(k-1)/2 degrees of freedom. Consider perturbations τ τ τ (n) as described in Proposition 6.1 above. Letting

C ϑ ϑ ϑ0;K,g := diag J 2 k (K 1 , g 1 ) J k (K 1 ) (ν ν ν (1) ) -1 , . . . , J 2 k (K m , g m ) J k (K m ) (ν ν ν (m) ) -1 and
D (n) ϑ ϑ ϑ0;K,g := C ϑ ϑ ϑ0;K,g -C ϑ ϑ ϑ0;K,g ((r (n) ) -1 1 m 1 ′ m (r (n) ) -1 ) ⊗ ν ν ν(K, g) C ϑ ϑ ϑ0;K,g ,
the quantities characterizing the asymptotic distribution of Q (n) K under the corresponding local alternatives (see Part (i) of Theorem 7.1 below) are

l ϑ ϑ ϑ0,τ τ τ ;K,g := lim n→∞ (τ τ τ (n) IV ) ′ (I m ⊗ G β β β k ) D (n) ϑ ϑ ϑ0;K,g (I m ⊗ G β β β k ) ′ (τ τ τ (n) IV ) (7.9) = m i,i ′ =1 (vec b (i) ) ′ G β β β k δ ii ′ T (i,i ′ ) K,g -(r i r i ′ ) 1/2 T (i,i ′ ) K,g ν ν ν(K, g)T (i,i ′ ) K,g G β β β k ′ (vec b (i ′ ) ), where
T (i,i ′ ) K,g := J k (K i , g i )J k (K i ′ , g i ′ ) J (K i ) J (K i ′ ) 1/2 (ν ν ν (i) ) -1/2 (ν ν ν (i ′ ) ) -1/2 .
We are now ready to state the main result of this paper. ϑ ϑ ϑ;g }, and (provided that (B) and (D ′ ) are reinforced into (B ′ ) and (D), respectively) asymptotically noncentral chi-square, still with (m -1)k(k -1)/2 degrees of freedom, but with noncentrality parameter l ϑ ϑ ϑ,τ τ τ ;K,g /k(k + 2) under P The signed-rank test φ

(n) ϑ ϑ ϑ+n -1/2 ς ς ς (n) τ τ τ (n) ;g , ϑ ϑ ϑ ∈ H 0 , ς ς ςτ τ τ := lim n→∞ ς ς ς (n) τ τ τ (n) as in Proposition 6.1 and g ∈ (F a ) m ; (ii) φ (n) K has asymptotic level α under ϑ ϑ ϑ∈H0 g∈(Fa) m {P (n) ϑ ϑ ϑ;g }; (iii) reinforcing (D ′ ) into (D), φ (n) K f , K f := (K f1 , . . . , K fm ),
K is asymptotically invariant with respect to continuous monotone radial transformatons in the sense that it is asymptotically equivalent to a random variable which is invariant under such transformations. ϑ ϑ ϑ+n -1/2 ς ς ς (n) τ τ τ (n) ;g , with ϑ ϑ ϑ ∈ H 0 , ς ς ςτ τ τ := lim n→∞ ς ς ς (n) τ τ τ (n) as described in Proposition 6.1, and g ∈ (F 4 a ) m , is

ARE k,g (φ (n) K /φ (n) † N ) = l ϑ ϑ ϑ,τ τ τ ;K,g /k(k + 2)l N ϑ ϑ ϑ,τ τ τ ;g , (8.1)
where l N ϑ ϑ ϑ,τ τ τ ;g and l ϑ ϑ ϑ,τ τ τ ;K,g are defined in (6.5) and (7.9), respectively.

Note that, if g = (g 1 , . . . , g 1 ) (homogeneous elliptical densities) and if the same score function-namely, K 1 -is used for the m rankings, (8.1) simplifies into ARE k,g (φ 3) of the van der Waerden tests with respect to the optimal parametric (or the optimal rank-based) ones under various bivariate Student t ν and Gaussian population densities g = (g 1 , g 2 ), with r 2 = 1r 1 = 120/220. with bivariate standard Gaussian densities (ε ε ε 1;1j1 and ε ε ε 1;2j2 : Gaussian case), bivariate Gaussian (ε ε ε 2;1j1 ) and t 5 (ε ε ε 2;2j2 ) (non-Gaussian heterokurtic case with finite fourth-order moments), bivariate standard t 1 densities (ε ε ε 3;1j1 and ε ε ε 3;2j2 : non-Gaussian homokurtic case with infinite fourth-order moments) and bivariate standard t 5 (ε ε ε 4;1j1 ) and t 1 (ε ε ε 4;2j2 ) (non-Gaussian heterokurtic case with infinite fourth-order moments), respectively. Each replication of the ε ε ε ℓ;1j1 's was transformed into Clearly, the scatter matrices of X ℓ;1j1 and X ℓ;2j2;0 have common eigenvectors β β β, with distinct eigenvalue matrices Λ Λ Λ 1 and Λ Λ Λ 2 , while the eigenvectors of X ℓ;2j2;ξ , ξ = 1, 2, 3 3. Rejection frequencies (out of N = 1, 000 replications), under the null (ξ = 0) and three alternatives (ξ = 1, 2, 3; see Section 8.2 for details), of the Flury test (φ Flury ), the pseudo-Gaussian tests φ † N (= φHPV), the signed-rank van der Waerden (φ vdW -Gaussian scores in both samples), homogeneous tν -score (φ t 5 ,t 5 and φ t 1 ,t 1 -identical Student scores in both samples), heterogeneous tν -score test (φ t 1 ,t 5 -t 1 -scores in sample one and t 5 -scores in sample two) and heterogeneous Gaussian and t 5 -scores (φ vdW,t 5 -Gaussian scores in sample one, t 5 -scores in sample two). Sample sizes are n 1 = 100 and n 2 = 120. In the last column, we give the AREs with respect to the optimal (for the densities under reference) rank-based test.

(n) K1 /φ (n) † N ) = (1 + κ k (g 1 ))J 2 k (K 1 , g 1 )/k(k + 2)J k (K 1 ); ( 8
Hallin, Paindaveine and Verdebout as n → ∞, still under P (n) ϑ ϑ ϑ0;g . It directly follows from (9.1), (9.2) and the fact that

L β β β, Λ Λ Λ V i k (V i -1/2 ) ⊗2 J k = 0 that ∆ ∆ ∆ IV ,i ϑ ϑ ϑ (n) ;Ki -∆ ∆ ∆ IV ,i ϑ ϑ ϑ;Ki = J k (K, g 1 ) 4k(k + 2) G β β β k L β β β, Λ Λ Λ V i k V ⊗2 i -1 I k 2 +K k n 1/2
i vec ( Vi -V i )+o P (1) (9.3)

Next, following the same argument as in the proof of Lemma 4.2 in Hallin, Paindaveine, Verdebout (2010b), we have that

n 1/2 i vec ( Vi -V i ) = (L β β β, Λ Λ Λ V i k ) ′ (G β β β k ) ′ n 1/2 i vec ( β β β -β β β) + β β β ⊗2 H ′ k n 1/2 i dvec( Λ Λ Λ V i -Λ Λ Λ V i ) + o P (1)
(9.4) as n → ∞ under P (n) ϑ ϑ ϑ0;g . The result follows by plugging (9.4) into (9.3), then using the fact that (L

β β β, Λ Λ Λ V i k ) ′ V ⊗2 i -1 I k 2 + K k β β β ⊗2 H ′ k = 0.
Proof of Proposition 7.1 Simple algebra yields, for ϑ ϑ ϑ 0 ∈ H 0 , Γ Γ Γ IV ϑ ϑ ϑ0,K,g (ς ς ς (n) ) -1 Υ Υ Υ IV = 2 Γ Γ Γ IV ϑ ϑ ϑ0,K,g (ς ς ς (n) ) -1 Υ Υ Υ, with Υ Υ Υ := 1 m ⊗ I k 2 . This implies that

Q (n)
ϑ ϑ ϑ0;K,g = ( ∆ ∆ ∆ IV ϑ ϑ ϑ0;K ) ′ (Γ Γ Γ IV ϑ ϑ ϑ0,K,g ) ⊥ ∆ ∆ ∆ IV ϑ ϑ ϑ0;K , where (Γ Γ Γ IV ϑ ϑ ϑ0,K,g ) ⊥ = (Γ Γ Γ IV ϑ ϑ ϑ0,K ) --(Γ Γ Γ IV ϑ ϑ ϑ0,K ) -Γ Γ Γ IV ϑ ϑ ϑ0,K,g (ς ς ς (n) ) -1 Υ Υ Υ (9.5) Υ Υ Υ ′ (ς ς ς (n) ) -1 Γ Γ Γ IV ϑ ϑ ϑ0,K,g (Γ Γ Γ IV ϑ ϑ ϑ0,K ) -Γ Γ Γ IV ϑ ϑ ϑ0,K,g (ς ς ς (n) ) -1 Υ Υ Υ -Υ Υ Υ ′ (ς ς ς (n) ) -1 Γ Γ Γ IV ϑ ϑ ϑ0,K,g (Γ Γ Γ IV ϑ ϑ ϑ0,K ) -.

Using Slutzky's Lemma jointly with Lemma 7.2, we obtain that

Q (n) K -Q (n) ϑ ϑ ϑ0;K,g is o P (1)
under P ϑ ϑ ϑ0,g for ϑ ϑ ϑ 0 ∈ H 0 and g ∈ (F a ) m iff (denoting by β β β the common eigenvector matrix under ϑ ϑ ϑ 0 ) (Γ Γ Γ IV ϑ ϑ ϑ0,K,g ) ⊥ Γ Γ Γ IV ϑ ϑ ϑ0,K,g (ς ς ς (n) ) -1 Υ Υ Υn 1/2 vec ( β β ββ β β) = 0.

In view of (9.5), however, this follows trivially from the fact that A(A ′ A) -A ′ A = A, a standard properties of Moore-Penrose inverses. Now, since Γ Γ Γ IV ϑ ϑ ϑ0;K (Γ Γ Γ IV ϑ ϑ ϑ0;K,g ) ⊥ is idempotent with trace (m -1)k(k -1)/2, it follows from Theorem 9.2.1 in Rao and Mitra (1971) that Q (n) K is asymptotically chi-square with (m -1)k(k -1)/2 degrees of freedom under P (n) ϑ ϑ ϑ0;g , ϑ ϑ ϑ 0 ∈ H 0 , and asymptotically noncentral chi-square, still with (m -1)k(k -1)/2 degrees of freedom, but with noncentrality parameter lim n→∞ (τ τ τ IV ) (n)′ Γ Γ Γ IV ϑ ϑ ϑ0;K,g (Γ Γ Γ IV ϑ ϑ ϑ0;K,g ) ⊥ Γ Γ Γ IV ϑ ϑ ϑ0;K,g (τ τ τ IV ) (n) (9.6) under P

(n) ϑ ϑ ϑ0+n -1/2 τ τ τ (n) ;g . Evaluation of the limit in (9.6) yields the desired result.

and

  Hallin et al.'s φ (n) HPV . Finally, following the method used in Hallin et al. (2010b) for the one-sample case, we construct rankbased versions of the optimal test statistics associated with various radial densities, and compute the corresponding local powers and asymptotic relative efficiencies.

Proposition 5 . 1 .

 51 Let Assumptions (A) (with f = (f 1 , . . . , f m ) ∈ (F a ) m ), (B) and (D) hold. Then, the family P (n) f

  HPV proposed, on heuristic grounds, in Hallin et al. (2010a), the optimality properties of which thus follow.Let (F 4 1 ) m denote the collection of m-tuples of standardized radial densities yieldingHallin, Paindaveine and Verdebout finite fourth-order moments in each population:

N

  of Section 5 is based on a quadratic form in the eigenvector part ∆ ∆ ∆ IV ϑ ϑ ϑ;φ of the Gaussian central sequence, defining a pseudo-Gaussian version of φ (n)

  2 in Hallin et al. (2010b); the proof is therefore omitted. Transposed to the present context, and temporarily assuming that the actual g ∈ (F 4 1 ) m is known, the pseudo-Gaussian test of Hallin and Paindaveine (2008a) is rejecting the null hypothesis of CPC for large values of

  jj ′ and ν jj ′ (g) values computed from the λij and κ(n) i estimators. The resulting pseudo-Gaussian test φ (n) † N rejects the null hypothesis of CPC, at asymptotic level α, as soon as Q (n) † N

  (n) † N has asymptotic level α under ϑ ϑ ϑ∈H0 g∈(F 4 1 ) m {P (n) ϑ ϑ ϑ;g }; (iii) letting (D ′ ) be reinforced into (D), φ (n) † N is locally and asymptotically most stringent, at asymptotic level α, for ϑ ϑ ϑ∈H0 g∈(F 4 1 ) m {P (n) ϑ ϑ ϑ;g } against alternatives of the form ϑ ϑ ϑ / ∈H0 {P (n) ϑ ϑ ϑ;φ }. One can easily check that φ (n) † N actually coincides with the test φ (n) HPV proposed in Hallin et al. (2010a); theorem 6.1 therefore clarifies the asymptotic optimality properties of the latter.

7 .

 7 Optimal rank-based tests.

7. 1 .

 1 A rank-based central sequence for eigenvectors.

Proposition 7 . 1 .

 71 Assume that (A), (B), (C), and (D ′ ) hold, and let ϑ ϑ ϑ (n) be an estimator satisfying Assumption (E) with K = (F a ) m . Then,(i) Q (n)K is asymptotically chi-square with (m -1)k(k -1)/2 degrees of freedom under ϑ ϑ ϑ∈H0 g∈(Fa) m {P (n)

  is locally and asymptotically most stringent, at asymptotic level α, for ϑ ϑ ϑ∈H0 g∈(Fa) m {P (n) ϑ ϑ ϑ;g } against alternatives of the form ϑ ϑ ϑ / ∈H0 {P (n) ϑ ϑ ϑ;f } with f := (f 1 , . . . , f m ).

8 .

 8 Power comparison and simulations.

8. 1 .Proposition 8 . 1 .

 181 Asymptotic relative efficiencies.The asymptotic relative efficiencies (AREs) of the signed-rank test φ (n) K with respect to the pseudo-Gaussian test φ (n) † N (equivalently, with respect to φ (n) HPV ) directly follow as ratios of noncentrality parameters under local alternatives (see Propositions 6.1 and 7.1). Assume that (A), (B ′ ), (C), and (D) hold. Then, the asymptotic relative efficiency of φ (n) K with respect to φ (n) † N , when testing P (n) ϑ ϑ ϑ;g against P (n)

X 2 10 8 , 2 2

 282 ℓ;1j1 = β β βΛ Λ Λ 1/ε ε ε ℓ;1j1 , ℓ = 1, 2, 3, 4, j 1 = 1, . . . , n 1 ,(8.5)whereβ β β = cos(π/6)sin(π/6) sin(π/6) cos(π/6) and Λ Λ Λ 1 = 16 0while each replication of the ε ε ε ℓ;2j2 's was transformed intoX ℓ;2j2;ξ = β β βB ξ Λ Λ Λ 1/ε ε ε ℓ;2j2 , ℓ = 1, 2, 3, 4, j 2 = 1, . . . , n 2 , ξ = 0, 1

  2 du are finite. The quantities I k (f i ) and J k (f i ) play the roles of radial Fisher information for location and shape/scale, respectively, in sample i, i = 1, . . . , m (seeHallin and Paindaveine 2006).

	3.2. Asymptotic behavior of sample sizes.
	Actually, we throughout consider triangular arrays of observations, of the form
	(X (n) 11 , . . . , X (n) 1n (n) 1	, X (n) 21 , . . . , X (n) 2n (n) 2	, . . . , X (n) m1 , . . . , X (n) mn m (n)	),
	indexed by the total sample size n :=	m i=1 n	(n) i , where the sequences n	(n) i

Table 1 .

 1 Asymptotic relative efficiencies(8.3) of the pseudo-Gaussian tests with respect to the optimal parametric (or the optimal rank-based) ones under various bivariate Student t ν and Gaussian population densities g = (g 1 , g 2 ), with r 2 = 1r 1 = 120/220.

	.2)

Table 2 .

 2 Asymptotic relative efficiencies(8.

  Hallin, Paindaveine and Verdebout

				ξ
	underlying densities g	test	0	1	2	3	ARE k,g ( • /φ opt )
	N , N	.039 N (= φHPV) .041 φ Flury φ † φ vdW .031	.178 .177 .148	.504 .494 .412	.780 .722 .633	1.0000 1.0000 1.0000
		φ t 5 ,t 5	.035	.147	.426	.626	.9446
		φ t 1 ,t 1	.036	.110	.350	.517	.7407
		φ t 1 ,t 5	.043	.146	.414	.595	.8213
		φ vdW,t 5	.039	.149	.433	.631	.9740
	t 1 , t 1	.698 N (= φHPV) .025 φ Flury φ † φ vdW .036	.705 .045 .077	.704 .037 .198	.716 .037 .335	.0000 .0000 .7407
		φ t 5 ,t 5	.041	.088	.261	.416	.8972
		φ t 1 ,t 1	.035	.123	.295	.460	1.0000
		φ t 1 ,t 5	.043	.111	.282	.436	.9505
		φ vdW,t 5	.035	.081	.235	.369	.8045
	t 1 , t 5	.478 N (= φHPV) .230 φ Flury φ † φ vdW .034	.517 .274 .093	.543 .277 .278	.519 .282 .414	.0000 .0000 .8091
		φ t 5 ,t 5	.041	.116	.315	.481	.9348
		φ t 1 ,t 1	.051	.115	.309	.487	.9571
		φ t 1 ,t 5	.059	.141	.345	.528	1.0000
		φ vdW,t 5	.035	.103	.291	.439	.8243
	N , t 5	.124 N (= φHPV) .068 φ Flury φ † φ vdW .040	.236 .156 .139	.480 .374 .377	.707 .536 .566	.0000 .5918 .9706
		φ t 5 ,t 5	.049	.146	.400	.600	.9725
		φ t 1 ,t 1	.049	.141	.353	.514	.8142
		φ t 1 ,t 5	.067	.156	.365	.570	.8556
		φ vdW,t 5	.039	.144	.401	.595	1.0000
	Table

Research supported by the Sonderforschungsbereich "Statistical modelling of nonlinear dynamic processes" (SFB 823) of the Deutsche Forschungsgemeinschaft. Marc Hallin and Davy Paindaveine are also members of ECORE, the association between CORE and ECARES. supported by an Action de Recherche Concertée, Communauté française de Belgique.

‡ Research supported by a BQR (Bonus Qualité Recherche) fund, Université Lille III, France.

Hallin, Paindaveine and Verdebout respectively). The Chernoff-Savage property of [START_REF] Paindaveine | A Chernoff-Savage result for shape. On the non-admissibility of pseudo-Gaussian methods[END_REF] therefore holds: denoting by φ (n) vdW the van der Waerden rank test (based on the Gaussian scores K 1 = . . . = K m := Ψ -1 k , where Ψ -1 k stands for the quantile function of the chi-square distribution with k degrees of freedom), we have that ARE k,g (φ

for all homogeneous g ∈ (F 4 a ) m , with equality in the Gaussian case only. In the bivariate two-population case (m = k = 2) with b (1) = 0 (no perturbation on the eigenvectors of the first population), the ARE under g = (g 1 , g 2 ) of φ

with respect to the optimal parametric test φ

) where denoting by K g1 and K g2 the score functions associated with g 1 and g 2 respectively, K g = (K g1 , K g2 ) and ν 12 (K g , g) naturally stands for the ν 12 (K, g) quantity computed from K g scores. Under the same setting, the ARE of the van der Waerden test φ

takes the form ARE 2,g (φ

12

12 ) -1 J 2 (g 2 )ν 12 (K g , g)

. (8.4)

These AREs do not depend on the value β β β of the common eigenvectors under the null, nor on the perturbation b (2) . Tables 1 and2 provide numerical values of (8.3) and (8.4), respectively, with r 2 = 1r 1 = 120/220 (the sampling scheme considered in the simulations of Section 8.2), for various choices of bivariate Student t ν and Gaussian population densities g = (g 1 , g 2 ). Note that the ARE of the pseudo-Gaussian tests with respect to van der Waerden ones can be as low as .13 under homokurtic bivariate t 4.2 populations, which demonstrates the severe lack of efficiency robustness of the pseudo-Gaussian tests.

Monte-Carlo study

In this section, we concentrate on comparing Flury's traditional Gaussian LRT (φ

Flury ) for the null hypothesis of CPC with the pseudo-Gaussian test φ (n) † N of Section 6 and the signed-rank tests of Section 7. First, we generated N = 1, 000 independent replications of three pairs (m = 2) of mutually independent samples (with respective sizes n 1 = 100 and n 2 = 120) of bivariate (k = 2) random vectors

Optimal Rank-based Tests for CPC differ from those of X ℓ;1j1 , thus characterizing increasingly distant alternatives to the null hypothesis of CPC.

Rejection frequencies (based on the asymptotic chi-square critical values, at nominal 5% level) are reported in Table 3, the inspection of which reveals several well expected facts:

(i) φ

where Ŝ

as n → ∞ under P

(n) ϑ ϑ ϑ0;g . From Lemma A1 in Hallin, Oja and Paindaveine (2006) and Lemma 4.4 in Kreiss (1987), we have that, for ϑ ϑ ϑ (n) satisfying (E),

K has asymptotic level α directly follows from the asymptotic null distribution obtained in (i) and the classical Helly-Bray theorem.

(iii) Optimality is a consequence of the asymptotic equivalence of Q K f and Q ϑ ϑ ϑ0,f described in (5.3)

Proof of Proposition 6.1 (i) It follows from Theorem 4.1 in Hallin, Paindaveine and Verdebout (2010a) that Q

is asymptotically noncentral chi-square, still with (m -1)k(k -1)/2 degrees of freedom, but with noncentrality parameter lim n→∞ (τ τ τ IV ) (n)′ Γ Γ Γ g,IV ϑ ϑ ϑ0;φ,g (Γ Γ Γ g,IV ϑ ϑ ϑ0;φ ) ⊥ Γ Γ Γ g,IV ϑ ϑ ϑ0;φ,g (τ τ τ IV ) (n) (9.7) under P

(n) ϑ ϑ ϑ0+n -1/2 τ τ τ (n) ;g with g ∈ (F 4 a ) m . Evaluation of the limit in (9.7) yields the result. (ii) The fact that φ (n) † N has asymptotic level α directly follows from the asymptotic null distribution in (i) and the classical Helly-Bray theorem.

(iii) Optimality is a consequence of the asymptotic equivalence under g = (φ, . . . , φ) of Q N (n) ϑ ϑ ϑ0,g and Q ϑ ϑ ϑ0,φ described in (5.3).