Sign Tests for Weak Principal Directions

Transcription

1 Sign Tests for Weak Princial Directions DAVY PAINDAVEINE, JULIEN REMY and THOMAS VERDEBOUT arxiv: v1 [math.st] 21 Dec 2018 Université libre de Bruxelles ECARES and Déartement de Mathématique Avenue F.D. Roosevelt, 50 ECARES, CP114/04 B-1050, Brussels Belgium We consider inference on the first rincial direction of a -variate ellitical distribution. We do so in challenging double asymtotic scenarios for which this direction eventually fails to be identifiable. In order to achieve robustness not only with resect to such weak identifiability but also with resect to heavy tails, we focus on sign-based statistical rocedures, that is, on rocedures that involve the observations only through their direction from the center of the distribution. We actually consider the generic roblem of testing the null hyothesis that the first rincial direction coincides with a given direction of R. We first focus on weak identifiability setus involving single sikes that is, involving sectra for which the smallest eigenvalue has multilicity 1). We show that, irresective of the degree of weak identifiability, such setus offer local alternatives for which the corresonding sequence of statistical exeriments converges in the Le Cam sense. Interestingly, the limiting exeriments deend on the degree of weak identifiability. We exloit this convergence result to build otimal sign tests for the roblem considered. In classical asymtotic scenarios where the sectrum is fixed, these tests are shown to be asymtotically equivalent to the sign-based likelihood ratio tests available in the literature. Unlike the latter, however, the roosed sign tests are robust to arbitrarily weak identifiability. We show that our tests meet the asymtotic level constraint irresective of the structure of the sectrum, hence also in ossibly multi-sike setus. Finally, we fully characterize the non-null asymtotic distributions of the corresonding test statistics under weak identifiability, which allows us to quantify the corresonding local asymtotic owers. Monte Carlo exercises confirm our asymtotic results. MSC 2010 subject classifications: Primary 62F05, 62H25; secondary 62E20. Keywords: Le Cam s asymtotic theory of statistical exeriments, Local asymtotic normality, Princial comonent analysis, Sign tests, Weak identifiability. Corresonding author. 1

2 2 D. Paindaveine, J. Remy and Th. Verdebout 1. Introduction Most classical methods in multivariate statistics are based on Gaussian maximum likelihood estimators of location and scatter, that is, on the samle mean and samle covariance matrix. Irresective of the considered roblem location or scatter roblems, multivariate regression roblems, rincial comonent analysis, canonical correlation analysis, etc.), these methods exhibit oor efficiency roerties when Gaussian assumtions are violated, articularly when the underlying distributions have heavy tails. Moreover, the resulting rocedures are very sensitive to ossible outliers in the data. To imrove on this, many robust rocedures were develoed. In articular, multivariate sign methods, that is, methods that use the observations only through their direction from the center of the distribution, have become increasingly oular. For location roblems, multivariate sign tests were considered in Randles 1989), Möttönen and Oja 1995), Hallin and Paindaveine 2002) and Paindaveine and Verdebout 2016), whereas sign rocedures for scatter or shae matrices were considered in Tyler 1987a), Dümbgen 1998), Hallin and Paindaveine 2006a), Dürre, Vogel and Fried 2015) and Dürre, Fried and Vogel 2017), to cite only a few. PCA techniques based on multivariate signs and on the comanion concet of ranks) were studied in Hallin, Paindaveine and Verdebout 2010), Taskinen, Koch and Oja 2012), Hallin et al. 2013) and Dürre, Tyler and Vogel 2016). Multivariate sign tests were also develoed, e.g., for testing i.i.d.-ness against serial deendence see Paindaveine, 2009), or for testing for multivariate indeendence see Taskinen, Kankainen and Oja, 2003 and Taskinen, Oja and Randles, 2005). Most references above actually focus on satial sign rocedures, that is, on rocedures that are based on the signs U i := X i / X i, i = 1,..., n, obtained by rojecting the -variate observations at hand onto the unit shere S 1 := {x R : x 2 = x x = 1} of R sometimes, the rojection is erformed on standardized observations in order to achieve affine invariance). Since they discard the radii X i, i = 1,..., n, satial sign rocedures can deal with arbitrarily heavy tails and are robust to observations that would be far from the center of the distribution. For more details on satial sign methods, we refer to the monograh Oja 2010). The resent aer considers rincial comonent analysis, or more recisely, inference on rincial comonent directions. Consider the case where the observations X 1,..., X n at hand form a random samle from a -variate centered ellitical distribution, that is, a distribution whose characteristic function is of the form t ext Σt) for some symmetric ositive definite matrix Σ. Assume that the ordered eigenvalues of Σ satisfy λ 1 > λ 2... λ, so that the leading eigenvector θ 1, ossibly unlike the other eigenvectors θ j, j = 2,...,, is identifiable as usual, identifiability is u to an unimortant sign). Throughout, we will then consider the roblem of testing the null hy-

3 Sign tests for weak rincial directions 3 othesis H 0 : θ 1 = against the alternative hyothesis H 1 : θ 1, where is a fixed unit -vector. This testing roblem has attracted much attention in the ast decades, and the textbook rocedure, namely the Anderson 1963) Gaussian likelihood ratio test, has been extended in various directions. To mention only a few, Jolicoeur 1984) considered a small-samle test, whereas Flury 1988) roosed an extension to a larger number of eigenvectors. Tyler 1981, 1983) robustified the Anderson 1963) test to ossible ellitical deartures from multinormality the original likelihood ratio test require Gaussian assumtions). Schwartzman, Mascarenhas and Taylor 2008) considered extensions to the case of Gaussian random matrices, and Hallin, Paindaveine and Verdebout 2010) obtained Le Cam otimal tests for the roblem considered. All asymtotic tests above assume that the eigenvalues λ 1 > λ 2... λ are fixed, so that the eigenvector θ 1 remains asymtotically identifiable. The null asymtotic distribution of the corresonding test statistics, however, may very oorly aroximate their fixed-n distribution when λ 1 /λ 2 is close to one. In the resent work, we therefore consider general asymtotic scenarios that address this issue. More recisely, we allow for scatter values Σ = Σ n for which the corresonding ratio λ n1 /λ n2 > 1) converges to one as n diverges to infinity. In such asymtotic scenarios, the leading eigenvector θ n1 remains identifiable for any n but is no longer identifiable in the limit. One then says that θ n1 is weakly identifiable. The distributional framework considered here formalizes situations that are often encountered in ractice where two samle eigenvalues are close to each other so that inference about the corresonding eigenvectors is a riori difficult. Inference on weakly identified arameters has already been much considered in the literature: see, e.g., Pötscher 2002), Forchini and Hillier 2003), Dufour 2006) and Antoine and Lavergne 2014). Liu and Shao 2003) and Zhu and Zhang 2006) consider asymtotic inference under a total lack of identifiability. Recently, Paindaveine, Remy and Verdebout 2018) considered Gaussian tests on weakly identified eigenvectors. While these tests can handle weak identifiability, they are based on samle covariance matrices, hence cannot deal with heavy tails and are very sensitive to ossible outliers. In the resent work, we tackle the same roblem but develo satial sign tests that not only inherit the robustness of sign rocedures but also can deal with both heavy tails and weak identifiability. Assume that the arent ellitical distributions do not charge the origin of R, so that satial signs are well defined with robability one. Then, suitable satial sign tests are to be determined in the image of the model by the rojection onto the unit shere S 1. Now, if the random -vector X is centered ellitical with scatter matrix Σ, then the corresonding satial sign U := X/ X follows the angular Gaussian distribution with shae matrix V := Σ/trΣ); see Tyler 1987b). Since Σ and V share the same eigenvectors, the induced testing roblem is still the roblem of testing H 0 : θ 1 = against H 1 : θ 1, where θ 1 is the leading eigenvector of V. Also, since eigenvalues of Σ and V are equal

4 4 D. Paindaveine, J. Remy and Th. Verdebout u to a common ositive factor, weak) identifiability occurs for the original ellitical roblem if and only if it does for the induced angular Gaussian roblem. These considerations exlain that identifying otimal satial sign tests under weak identifiability for the ellitical roblem should be done in the setu where one observes triangular arrays of satial signs U n1,..., U nn, n = 1, 2,..., randomly drawn from the angular Gaussian distribution with shae matrix V n. Accordingly, we consider the roblem of testing the null hyothesis H 0 : θ n1 = in an angular Gaussian double asymtotic scenario for which λ n1 /λ n2 > 1) may converge to one at an arbitrary rate, which rovides weak identifiability for the leading rincial direction θ n1. We first focus on sequences of single-sike shae matrices characterized by sectra of the form λ n1 > λ n2 =... = λ n ). We will show that, irresective of the rate of convergence of λ n1 /λ n2 to one, there exist suitable local alternatives for which the corresonding sequence of statistical exeriments converges in the Le Cam sense. Interestingly, the limiting exeriment deends on the degree of weak identifiability. This aves the way, in this single-sike setu, to the construction of otimal sign tests for the roblem considered. The resulting tests, however, in rincile meet the asymtotic level constraint in the single-sike setu only. We will show that, fortunately, these tests show the target asymtotic null size irresective of the structure of the sectrum. Finally, we will also fully characterize the non-null asymtotic distributions of the corresonding test statistics under weak identifiability, which will allow us to extensively quantify the corresonding local asymtotic owers. The outline of the aer is as follows. In Section 2, we introduce the notation, describe the sequence of angular Gaussian models to be considered, and derive the aforementioned results on limiting exeriments. This is used to derive a sign test that can deal with weak identifiability and enjoys nice Le Cam otimality roerties. However, since this test involves nuisance arameters, it is an infeasible statistical rocedure. In Section 3, we therefore derive a feasible version of this test. We show that, under the null hyothesis, hence also under sequences of contiguous hyotheses, the infeasible and feasible tests are asymtotically equivalent, so that the latter inherits the otimality roerties of the former. We also show that, in classical asymtotic scenarios where one stays away from weak identifiability, the roosed test is asymtotically equivalent to the likelihood ratio test from Tyler 1987b). In Section 4, we investigate the asymtotic roerties of the roosed sign test. We first show that this test asymtotically achieves the target null size even when the underlying scatter matrix does not have a single-sike structure. Then, for any degree of weak identifiability, we characterize the asymtotic distribution of the roosed test statistic under suitable local alternatives. In Section 5, we show through Monte Carlo exercises that, unlike its cometitors, the roosed sign test can deal with both heavy tails and weak identifiability. Finally, we rovide a wra u and

5 Sign tests for weak rincial directions 5 final comments in Section 6. All roofs are collected in a technical aendix. 2. Limits of angular Gaussian exeriments Consider a triangular array of observations X ni, i = 1,..., n, n = 1, 2,..., such that for any n, the random -vectors X n1,..., X nn form a random samle from a centered ellitical distribution that does not charge the origin of R. For any n, the leading eigenvector θ n1 of the corresonding scatter matrix Σ n is assumed to be well identified, u to a sign. In this general setu, we aim at designing suitable sign tests for the roblem of testing the null hyothesis H 0 : θ n1 = against the alternative hyothesis H 1 : θ n1, where is a fixed unit -vector. As exlained in the Introduction, such tests should be determined in the image of the model by the rojection of the model onto the unit shere S 1. For any n, the rojected observations U n1,..., U nn, where U ni = X ni / X ni is the satial sign of X ni, form a random samle from the -variate angular Gaussian distribution with shae matrix V n = Σ n /trσ n ) shae matrices throughout are normalized to have trace ), so that the density of U n1 with resect to the surface area measure on S 1 is u S 1 ) Γ 2 ) 2π /2 det V n ) 1/2 u V 1 n u) /2, 1) where Γ ) is the Euler Gamma function; see Tyler 1987b). We will denote the corresonding hyothesis as P n) V n. In this angular Gaussian framework, the aforementioned ellitical testing roblem induces the roblem of testing the null hyothesis H 0 : θ n1 = against the alternative hyothesis H 1 : θ n1, where θ n1 is the leading eigenvector of V n recall that V n and Σ n share the same eigenvectors). Throughout, λ nj and θ nj will refer to the ordered) eigenvalues and eigenvectors of V n. Recall that weak identifiability of θ n1 in the original ellitical roblem occurs if and only if λ n1 /λ n2 > 1) 1. The goal of this section is to determine otimal sign tests, under ossibly weak identifiability of θ n1, in the articular case of single-sike sectra, that is, in situations where the smallest eigenvalue has multilicity 1. Accordingly, we will consider sequences of null shae matrices of the form := 1 δ ) nξ I + δ n ξ θ 0, 2) where ξ > 0 is a locality arameter and δ n is a bounded ositive sequence that may be o1); throughout, we tacitly assume that ξ and δ n are chosen so that is ositive definite. It is straigthforward to check that indeed has a single sike: the largest eigenvalue is λ n1 = 1 + 1)δ n ξ/, with multilicity one and corresonding eigenvector, whereas the remaining eigenvalues are λ n2 =... = λ n = 1 δ n ξ/, j = 2,...,,

6 6 D. Paindaveine, J. Remy and Th. Verdebout with an eigensace that is the orthogonal comlement to. In this setu, weak identifiability of course occurs if and only if δ n is o1). To discuss otimality issues, we consider local alternatives associated with erturbations + ν n τ n of, where ν n ) is a ositive sequence and τ n ) is a bounded sequence in R so that + ν n τ n S 1 for any n. It is easy to show that the latter condition entails that ν n and τ n must satisfy θ 0τ n = ν n 2 τ n 2 3) for any n. The resulting sequence of alternatives is then associated with the single-sike shae matrices V 1n := 1 δ nξ ) I + δ n ξ + ν n τ n ) + ν n τ n ). 4) Our construction of otimal sign tests requires studying the asymtotic behavior, under P n), of the log-likelihood ratios Λ n := logdp n) V 1n /dp n) ). To do so, let γ n := δ n ξ + 1)δ n ξ and S n V) := 1 n n V 1/2 U ni U ni V 1/2 V 1/2 U ni 2 5) Note that the sequence γ n ) is Oδ n ). We then have the following result. Theorem 2.1. Fix S 1. Let ν n ) and τ n ) be such that + ν n τ n S 1 for any n. Let be as in 2). Then, as n, under P n), we have the following: i) if δ n 1, then, for ν n = 1/ nγ n ), Λ n = τ n i) 1 2 τ nγ i) τ n + o P 1) 6) as n, where and where Γ i) := + 1)ξ) + 2) ξ) I θ 0), i) := + 1)ξ ξ I θ 0) n S n ) 1 I ) θ0 is asymtotically normal with mean zero and covariance matrix Γ i) ; ii) if δ n is o1) with nδ n, then, for ν n = 1/ nγ n ), Λ n = τ n ii) 1 2 τ nγ ii) τ n + o P 1) 7)

7 Sign tests for weak rincial directions 7 as n, where and where Γ ii) := + 2 I θ 0) 8) ii) := I θ 0) n S n ) 1 I ) θ0 9) is asymtotically normal with mean zero and covariance matrix Γ θ0 ; iii) if δ n = 1/ n, then, for ν n = 1/ nγ n ) or equivalently ν n 1/ξ), where Λ n = τ nυ n) + 1 2ξ τ nυ n) τ n Υ n) τ n ) 4ξ 2 τ n 4) + o P 1), 10) := n S n ) 1 I ) is such that vecυ n) ) is asymtotically normal with mean zero and covariance matrix / + 2))I 2 + K + J ) J ; iv) if nδ n 0, then, even for ν n 1/ξ, we have Λ n = o P 1). Theorem 2.1 shows that the asymtotic behavior of Λ n crucially deends on the sequence δ n ) and identifies four different regimes. In the classical regime i) where δ n 1, standard erturbations with ν n 1/ n rovide a sequence of exeriments that is locally asymtotically normal LAN), with central sequence i) and Fisher information Γ i). In such a LAN setu, the locally and asymtotically maximin test φ n) for H 0 : θ n1 = against H 1 : θ n1 rejects the null hyothesis at asymtotic level α when T n ) := i) ) Γ i) ) i) = n + 2) I θ 0)S n ) 2 > χ 2 1,1 α, 11) where A stands for the Moore-Penrose inverse of A and where χ 2 l,β denotes the order-β quantile of the chi-square distribution with l degrees of freedom. In regime ii), erturbations with ν n 1/ nδ n ) that is, erturbations that are more severe than in the standard regime i) make the sequence of exeriments LAN, here with the central sequence ii) in 9) and the Fisher information matrix Γ ii) in 8). Since ii) ) Γ ii) ) ii) = T n ), the test φ n) is still locally asymtotically maximin in regime ii). The situation in regime iii) is quite different. While the sequence of exeriments there is not LAN nor locally asymtotically mixed normal, it still converges in the Le Cam sense. It is easy to check that, in this regime, τ nυ n) + 1 2ξ τ nυ n) τ n

8 8 D. Paindaveine, J. Remy and Th. Verdebout is asymtotically normal with mean zero and variance τ n ξ 2 τ n 4), so that the Le Cam first lemma entails that the sequences of hyotheses P n) and P n) V 1n are mutually contiguous in regime iii), too. As we will show in the next section, the test φ n) shows non-trivial asymtotic owers under these contiguous alternatives. Finally, in regime iv), Theorem 2.1 shows that no test can distinguish between the null hyothesis and the alternatives associated with ν n 1, which are the most severe ones that can be considered. 3. The roosed sign test The test φ n) from the revious section enjoys nice otimality roerties but it is unfortunately infeasible: its test statistic T n ) indeed involves the oulation shae matrix, which is of course unknown in ractice. Another issue is that the test φ n) will in rincile meet the asymtotic level constraint only under the single-sike shae structure in 2). In this section, we therefore construct a version φ n) of φ n) that is feasible and that can coe with more general shae structures. To do so, let ) be a sequence of shae matrices associated with the null hyothesis. In other words, we assume that, for any n, the shae admits the sectral decomosition = λ n1 θ 0 + λ nj θ nj θ nj, 12) where the θ nj s form an orthonormal basis of the orthogonal comlement to and where λ n1 > λ n2... λ n. In articular, the 1 smallest eigenvalues here do not need to be equal, so that the number of sikes may be arbitrary. With this notation, it is clear that estimating requires estimating the eigenvalues λ n1,..., λ n and the eigenvectors θ n2,..., θ n. To do so, we consider the Tyler 1987a) M-estimator, that is defined as the shae matrix satisfying S n ˆV n ) = 1/)I see 5)) and decomose it into ˆV n = j=2 ˆλ njˆθ njˆθ nj. 13) j=1 The eigenvalues ˆλ n1,..., ˆλ n of Tyler s M-estimator rovide estimates of the eigenvalues in 12). Later asymtotic results, however, will require that the estimators of the eigenvectors θ nj are orthogonal to the null value of the first eigenvector θ n1, a constraint

9 Sign tests for weak rincial directions 9 that the eigenvectors ˆθ nj dot not meet in general. To correct for this, we will rather use the estimators θ nj, j = 2,..., resulting from a Gram-Schmidt orthogonalization of, ˆθ n2,..., ˆθ n. In other words, θ nj is defined recursively through θ nj := I θ 0 j 1 θ k=2 nk θ nk)ˆθ nj I θ 0 j 1 θ k=2 nk θ, j = 2,...,, 14) nk)ˆθ nj with summation over an emty collection of indices being equal to zero. In the rest of the aer, Ṽ 0n will denote the estimator of obtained by substituting in 12) the Tyler eigenvalues ˆλ nj and eigenvectors θ nj for the λ nj s and θ nj s. Note that since Tyler s M-estimator is normalized to have trace, the estimator Ṽ0n also has trace, hence is a shae matrix. Quite nicely, relacing with Ṽ0n in the test statistic T n ) of φ n) has no asymtotic imact in robability under the null hyothesis. More recisely, we have the following result. Theorem 3.1. Let ) be a sequence of null shae matrices as in 12). Then, under P n), T n Ṽ0n) = T n ) + o P 1) as n. Based on this result, the sign test we roose in this aer is the test φ n) that rejects the null hyothesis H 0 : θ n1 = at asymtotic level α whenever T n Ṽ0n) = n + 2) I θ 0)S n Ṽ0n) 2 > χ 2 1,1 α. 15) Unlike φ n), this new sign test is a feasible statistical rocedure. An alternative test for the same roblem is the likelihood ratio test, φ n) Tyl say, that rejects the null hyothesis at asymtotic level α whenever L n := n ˆλn1 θ ˆV n + ˆλ n1 θ ˆV 0 n 2 ) > χ 2 1,1 α, 16) where ˆV n stands for Tyler s M-estimator defined above; see Tyler 1987b). As shown by the following result, the roosed sign test φ n), in classical asymtotic scenarios where one stays away from weak identifiability, is asymtotically equivalent to φ n) Tyl under the null hyothesis. Theorem 3.2. Let ) be a sequence of null shae matrices as in 12). Assume that there exists η > 0 such that both corresonding leading eigenvalues satisfy λ n1 /λ n2 1+η for n large enough. Then, under P n), T n Ṽ0n) = L n + o P 1) as n. From contiguity, this result also entails that φ n) and φ n) Tyl are asymtotically equivalent, hence exhibit the same asymtotic owers under the local alternatives considered in

10 10 D. Paindaveine, J. Remy and Th. Verdebout Theorem 2.1i). There is no guarantee, however, that this asymtotic equivalence extends to the weak identifiability situations considered in Theorem 2.1ii)-iv). To investigate the validity of φ n) under such non-standard asymtotic scenarios, we now thoroughly study the null and non-null asymtotic roerties of φ n). 4. Asymtotic roerties of the roosed test We first focus on the null hyothesis. Under the single-sike null hyotheses associated n) with the shae matrices in 2), the test statistic T n Ṽ0n) of the feasible test φ is asymtotically chi-square with 1 degrees of freedom, which easily follows from the asymtotic equivalence result in Theorem 3.1 and from the fact that Theorem 2.1 imlies that T n Ṽ0n) is asymtotically chi-square with 1 degrees of freedom under such sequences. Since the latter theorem focuses on single-sike shae matrices, there is no guarantee, however, that this extends to more general shae matrices. The following result shows that the null asymtotic distribution of T n Ṽ0n) remains asymtotically chi-square with 1 degrees of freedom for arbitrary sequences of null shae matrices. Theorem 4.1. Let ) be a sequence of null shae matrices as in 12). Then, under P n), T n Ṽ0n) is asymtotically chi-square with 1 degrees of freedom. We stress that this result does not only allow for general, multi-sike, null shae matrices, but also for weakly identifiable eigenvectors θ n1. In articular, it alies in each of the four asymtotic scenarios considered in Theorem 2.1. Since there is no guarantee that the asymtotic equivalence in Theorem 3.2 holds under weak identifiability, it is unclear whether or not the Tyler test φ n) Tyl is, like φ n), robust to weak identifiability. As we will show through simulations in the next section, it actually turns out that the Tyler test severely fails to be robust in this sense. Now, the nice robustness roerties above are not sufficient, on their own, to justify resorting to the roosed sign test, as it might be the case that such robustness is obtained at the exense of ower. To see whether or not this is the case, we turn to the investigation of the non-null asymtotic roerties of φ n). We have the following result. Theorem 4.2. Fix S 1. Let δ n ) be a ositive sequence that is O1) and take ν n = 1/ nγ n ). Let τ n ) be a sequence converging to τ and such that + ν n τ n S 1 for any n. Then, under P n) V 1n, with V 1n as in 4), we have the following as n : i) if δ n 1, then T n Ṽ0n) is asymtotically chi-square with 1 degrees of freedom

11 Sign tests for weak rincial directions 11 and non-centrality arameter + 1)ξ) + 2) ξ) τ 2 ; ii) if δ n is o1) with nδ n, then T n Ṽ0n) is asymtotically chi-square with 1 degrees of freedom and non-centrality arameter + 2 τ 2 ; iii) if δ n = 1/ n, then T n Ṽ0n) is asymtotically chi-square with 1 degrees of freedom and non-centrality arameter + 2 τ ξ 2 τ 2) 1 1 4ξ 2 τ 2) ; iv) if δ n = 1/ n, then, under P n) V n1 with V n1 based on δ n 1/ξ, T n Ṽ0n) remains asymtotically chi-square with 1 degrees of freedom. From contiguity, Theorem 3.1 imlies that the test φ n) enjoys the same asymtotic behavior as φ n) under the contiguous alternatives of Theorem 2.1i)-ii), hence inherits the Le Cam otimality roerties of the latter test in the corresonding regimes. Consequently, the local asymtotic owers associated with the non-null results in Theorem 4.2i)-ii) are the maximal ones that can be achieved. In regime iii), Theorem 4.2 entails that the test φ n) is rate-otimal, in the sense that it shows non-trivial asymtotic owers against the corresonding contiguous alternatives the non-standard nature of the limiting exeriment in this regime does not allow to state stronger otimality roerties, though). Finally, this test is otimal in regime iv), but trivially so since, as already mentioned, Theorem 2.1iv) imlies that the trivial α-test is also otimal in this regime. We erformed the following simulation to check the validity of Theorem 4.2. For any combination of l {0, 1, 2, 3, 4} and w {0, 1, 2}, we generated M = 2,500 mutually indeendent random samles X l,w) 1,..., X n l,w) of size n = 200,000 from the six-variate = 6) multinormal distribution with mean zero and covariance matrix Σ l,w) n := 1 n w/6 ) I + n w/6 θ l) n1 θl) n1, 17) with := 1, 0,..., 0) R and θ l) n1 := cos α nl, sin α nl, 0,..., 0) R, where α nl := 2 arcsinν n l 1)/2) is based on ν n := 1/ nγ n ); here, γ n is as in 5), with the values δ n = n w/6 and ξ = 1 that are induced by 17). The value w = 0 yields the standard asymtotic scenario in Theorem 4.2i), while w = 1, 2 are associated with weak identifiability situations covered by Theorem 4.2ii). The value l = 0 corresonds to the null

12 12 D. Paindaveine, J. Remy and Th. Verdebout hyothesis H 0 : θ n1 =, whereas l = 1, 2, 3, 4 rovide increasingly severe alternatives of the form θ l) n1 = + ν n τ nl, where τ nl is some -vector with norm l 1; this allows to obtain the corresonding asymtotic local owers from Theorem 4.2i)-ii). For each samle, we erformed the roosed sign test for H 0 : θ n1 = in 15) at nominal level 5%. Clearly, the resulting rejection frequencies, that are rovided in Figure 1, are in agreement with the theoretical asymtotic owers comuted from Theorem 4.2, but maybe for the case w = 2. Note, however, that at any finite samle size, emirical ower curves will eventually converge to flat ower curves at the nominal level α for weak enough identifiability this follows from Theorem 4.2iv)), which exlains this small deviation observed for w = 2. The sign nature of the roosed test makes it suerfluous to also consider non-gaussian ellitical distributions here. w=0 w=1 w=2 Rejection frequencies l l l Figure 1. Rejection frequencies, under various null and non-null distributions, of the roosed sign test in 15) erformed at asymtotic level α = 5%; the value w = 0 corresonds to the classical asymtotic scenario where θ n1 remains asymtotically identifiable, whereas w = 1, 2 rovide asymtotic scenarios involving weak identifiability the lighter the color, the weaker the identifiability). Random samles were drawn from six-variate multinormal distributions; see Section 4 for details. 5. Finite-samle comarisons with cometing tests The objective of this section is to comare the roosed sign test to some cometitors in terms of emirical size under the null hyothesis. We did so by erforming the following Monte Carlo exercise. For any w {0, 1, 2, 3}, we generated M = 5,000 mutually indeendent random samles X w) 1,..., X w) n of size n = 400 from the six-variate = 6) Gaussian distribution with mean zero and covariance matrix ) Σ w) n := 1 n w/4 I + n w/4 θ 0,

13 Sign tests for weak rincial directions 13 with = 1, 0,..., 0) R. As in the simulation conducted at the end of Section 4, the value w = 0 is associated with classical situations where θ n1 remains asymtotically identifiable whereas the values w = 1, 2, 3 rovide situations where identifiability of θ n1 is weaker and weaker. In each relication, we erformed four tests for H 0 : θ n1 = at nominal level 5% : the classical Gaussian likelihood ratio test from Anderson 1963), the Tyler 1987b) test in 16), the Gaussian test from Paindaveine, Remy and Verdebout 2018), and the roosed sign test in 15). The same exercise was reeated with random samles drawn from multivariate t distributions with 2, 4 and 6 degrees of freedom, in each case with mean zero and scatter matrix Σ w) n. The resulting null rejection frequencies are reorted in Figure 2. Clearly, the Anderson 1963) test meets the nominal level constraint only in the Gaussian, well identified, case. As exected, the Tyler 1987b) test, which is a sign test, can deal with heavy tails, but the results make it clear that this test strongly overrejects the null hyothesis under weak identifiability. The oosite holds for the Paindaveine, Remy and Verdebout 2018) test, that resists weak identifiability situations in the Gaussian case but cannot deal with heavy tails. In line with the theoretical results of the revious sections, the roosed sign test resists both heavy tails and weak identifiability. Since this test is the only one achieving this double robustness, it makes little sense to comare finite-samle owers of this test with those of its cometitors; such a comarison would only bring information on the finite-samle owers of the roosed sign test, that were already investigated in Section Wra u and final comments In this work, we considered hyothesis testing for rincial directions in challenging, weak identifiability, scenarios. Under elliticity assumtions, we roosed a sign test that, unlike its cometitors, meets the asymtotic level constraint both under heavy tails and under weak identifiability. By resorting to Le Cam s asymtotic theory of statistical exeriments, we also roved that this test enjoys strong otimality roerties in the class of satial sign tests all otimality statements below are relative to this class of tests). In articular, it is locally asymtotically otimal in classical situations where θ n1 remains asymtotically identifiable. It follows from our results that the likelihood ratio test from Tyler 1987b) satisfies the same otimality roerty. Our sign test, however, shows strong advantages over the Tyler test: not only does our test meet the level constraint under any weak identifiability situation, but it also remains locally asymtotically otimal in all cases, but for the case δ n 1/ n for which our test is still rate-otimal. The following comments are in order. First, our sign test is not only robust to heavy tails and weak identifiability but also to some) deartures from elliticity. More recisely,

14 14 D. Paindaveine, J. Remy and Th. Verdebout Gaussian Anderson Tyler PVR Gaussian PVR Sign t t 6 t w=0 w=1 w=2 w=3 w=0 w=1 w=2 w=3 w=0 w=1 w=2 w=3 w=0 w=1 w=2 w=3 Figure 2. Null rejection frequencies, under six-variate Gaussian, t 6, t 4 and t 2 densities, of four tests for the null hyothesis H 0 : θ n1 = 1, 0,..., 0) R 6 ), all erformed at asymtotic level α = 5%. The tests considered are the Anderson 1963) test, the Tyler 1987b) test, the Gaussian test from Paindaveine, Remy and Verdebout 2018), and the roosed sign test; w = 0 corresonds to the classical asymtotic scenario where θ n1 remains asymtotically identifiable, whereas w = 1, 2, 3 rovide asymtotic scenarios involving weaker and weaker identifiability; see Section 5 for details.

15 Sign tests for weak rincial directions 15 it should be clear that our sign test only assumes that the satial signs U ni = X ni / X ni follow an angular Gaussian distribution. As a consequence, the satial signs do not need be indeendent of the radii X ni, which imlies in articular that our sign test can deal with some skewed distributions. More recisely, the roosed test only require that the observations X ni, i = 1,..., n, form a random samle from a distribution with ellitical directions that does not charge the origin of R ); see Randles 2000). Second, it has been throughout assumed that the arent ellitical distribution was centered. This was mainly for the sake of readability, as our results can easily be extended to the unsecified location case. More recisely, an unsecified-location version of the roosed test can simly be obtained by relacing the satial signs U ni = X ni / X ni in our sign test with centered versions U ni ˆµ n ) = X ni ˆµ n )/ X ni ˆµ n, where ˆµ n is an arbitrary root-n consistent estimator of the center of the underlying ellitical distribution. A natural choice, that would be root-n consistent even in the large class of distributions with ellitical directions, is the affine-equivariant median from Hettmanserger and Randles 2002). Due to the orthogonality between location and scatter arameters under elliticity see Hallin and Paindaveine, 2006b), all asymtotic results of this aer naturally extend to the resulting unsecified-location sign test. While this work rovides an overall good rocedure to test for rincial directions under weak identifiability, it also oens ersectives for future research. As mentioned above, the otimality of the roosed test is relative to the class of satial sign tests. Restricting to sign tests of course is a guarantee for excellent robustness roerties, yet it might be so that, if some slightly lower robustness is also accetable, then higher asymtotic efficiency could be achieved. In articular, it should be ossible to develo signed rank tests that rovide a nice trade-off between efficiency and robustness. It is exected that such tests can deal with heavy tails and are robust to weak identifiability, while uniformly dominating, in terms of asymtotic relative efficiencies, arametric Gaussian tests in classical cases where the leading rincial direction remains asymtotically identifiable; see Paindaveine 2006). Aendix A: Proof of Theorem 2.1 We start with some reliminary lemmas. Lemma A.1. Consider the shae matrices and V n1 in 2) and 4). Then, i) and V n1 share the same determinant; ii) for any real number a, the ath matrix owers

16 16 D. Paindaveine, J. Remy and Th. Verdebout of and V n1 are given by V0n a = 1 δ ) nξ ai θ 0) )δ nξ = 1 δ ) nξ ai + λ a,n θ 0 and V a 1n = = 1 δ nξ 1 δ nξ ) aθ0 θ 0 ) ai + ν n τ n ) + ν n τ n ) ) + λ a,n + ν n τ n ) + ν n τ n ) ) ai + λ a,n + ν n τ n ) + ν n τ n ), where we let λ a,n := 1 + 1)δ n ξ/) a 1 δ n ξ/) a. Proof of Lemma A.1. i) Letting θ n := + ν n τ n, r := 1 δ n ξ/ and s := 1 + 1)δ n ξ/, rewrite and V n1 as = ri θ 0) + s θ 0 and V 1n = ri θ n θ n) + sθ n θ n. 18) Both these matrices have eigenvalues r with multilicity 1 and s with multilicity one, hence have determinant r 1 s. ii) The result directly follows from the sectral decomositions in 18). Lemma A.2. If δ n 1, then lim n δ n ξ/γ n = + 1)ξ, where γ n is defined in 5). If δ n = o1), then lim n δ n ξ/γ n =. Proof of Lemma A.2. Since δ n ξ/γ n = + 1)δ n ξ, the result follows. To state the next result, we first introduce some notation. Denoting as e l the lth vector of the canonical basis of R and by A B the Kronecker roduct between the matrices A and B, we let K := i,j=1 e ie j ) e je i ) stand for the 2 2 commutation matrix and define J := i,j=1 e ie j ) e ie j ) = vec I )vec I ). Further let T n V) := 1 n n vec U ni V)U niv) ) vec U ni V)U niv) )) with U ni V) := V 1/2 U ni / V 1/2 U ni ; note that with this notation, S n V) in 5) rewrites S n V) := 1 n U ni V)U n niv). We then have the following result.

17 Sign tests for weak rincial directions 17 Lemma A.3. and ii) as n. Fix an arbitrary sequence of shae matrices V n ). Then, under P n) V n, i) T n V n ) = 1 + 2) I + K + J ) + o P 1) n vec Sn V n ) 1 I ) D 1 N 0, + 2) I + K + J ) 1 ) 2 J Proof of Lemma A.3. For any n, U n1 V n ),..., U nn V n ) form a random samle from the uniform distribution on S 1. Therefore, the result follows from i) the weak law of large numbers and from ii) the central limit theorem, by using in both cases Lemma A.2 in Paindaveine and Verdebout 2016). Proof of Theorem 2.1. First note that the quantity γ n in 5) satisfies γ n = 1 δ ) nξ λ 1,n, 19) where λ 1,n was defined in Lemma A.1. Part ii) of this lemma therefore yields ) 1 ) 11 U niv0n 1 U ni = + λ 1,n U ni ) 2 = γn U ni ) 2) 20) and, similarly, 1 δ nξ 1 δ nξ U nivn1 1 U ni = 1 δ ) nξ 11 γn U ni + ν n τ n )) 2). Recalling the angular density in 1) and using Lemma A.1i), we then obtain Λ n = n { } logu 2 niv1n 1 U ni) logu niv0n 1 U ni) = 2 n { } log1 γ n U ni + ν n τ n )) 2 ) log1 γ n U ni ) 2 ), which, by writing γ n U ni +ν n τ n )) 2 = γ n U ni ) 2 +2γ n ν n U ni )U ni τ n )+γ n νnu 2 ni τ n ) 2, yields Λ n = n log 1 2γ nν n U ni )U ni τ n ) + γ n νnu 2 ni τ n ) 2 ) 2 1 γ n U ni ) 2 = n 2U log 1 ni γ n ν )U ni τ n ) + ν n U ni τ n ) 2 ) n 2 1 γ n U ni ) 2 =: n log ) 1 γ n ν n R ni. 21) 2

18 18 D. Paindaveine, J. Remy and Th. Verdebout A Taylor exansion yields Λ n = γn ν n 2 n =: L n1 + L n2 + L n3, ) γ 2 R ni + n νn 2 4 n R 2 ni ) γ 3 + n νn 3 6 n Rni 3 ) 1 γ n ν n H ni ) 3 for some H ni between 0 and R ni. Note that, in all cases i)-iv) considered in the theorem, we have that ν n γ n = O1/ n), ν n = O1) and that there exists η 0, 1) such that γ n < 1 η recall that, in case i), ξ is chosen in such a way that is ositive definite). Consequently, the boundedness of the sequence τ n ), along with the fact that U ni = 1 almost surely, ensures that there exists a ositive constant C such that L n3 C/ n almost surely, so that L n3 is o P 1) all stochastic convergences in this roof are as n under P n) ). Now, using 20), rewrite L n1 as L n1 = γ nν n n 2U ni )U ni τ n ) + ν n U ni τ n ) γ n U ni ) 2 n τ = γ n ν nu ni U ni θ ν ) nτ n n 1 γ n U ni ) 2 = γ n ν n 1 δ ) nξ 1 n τ nu ni U ni θ ν ) nτ n U ni V 1 0n U ni Note that, using 3), we have = 2 nγ n ν n δ n ξ τ nv 1/2 0n S n )V 1/2 0n θ ν ) nτ n. τ n θ ν ) nτ n = 1 δ ) nξ τ n θ ν ) nτ n + δn ξτ n )θ 0 θ ν ) nτ n = δ n ξτ n ) ν nθ 0τ n ) = 1 2 ν nδ n ξ τ n ν nθ 0τ n ),

19 Sign tests for weak rincial directions 19 which yields L n1 = 2 nγ n ν n δ n ξ τ nv 1/2 0n Sn ) 1 I 1/2 ) θ ν ) nτ n + nγ nν n δ n ξ τ n θ ν nτ n ) = 2 nγ n ν n δ n ξ τ nv 1/2 0n Sn ) 1 I 1/2 ) θ ν ) nτ n where, from Lemma A.1, we have V 1/2 0n = Turning to L n2, L n2 = γ2 nν 2 n 4 = γ2 nν 2 n )δ nξ nγ nν 2 nδ n ξ 2 δ n ξ) τ n ν nθ 0τ n ), 22) ) 1/2θ0 and V 1/2 0n τ n = n {2U ni )U ni τ n ) + ν n U ni τ n ) 2 } 2 1 γ n U ni ) 2 ) 2 1 δ nξ { n 4 ) vecu ni U ni ) vecu ni U ni )) 1 γ n U ni ) 2 ) 2 τ n τ n ) ) 1/2τ n + λ 1/2,n θ 0τ n ). n +4ν n τ n τ n ) vecu ni U ni ) vecu ni U ni )) 1 γ n U ni ) 2 ) 2 τ n ) n +νnτ 2 n τ n ) vecu ni U ni ) vecu ni U ni )) } 1 γ n U ni ) 2 ) 2 τ n τ n ). Using 20) again, Lemma A.3i), and the fact that v v) K = K 1 v v) = v v) 23)

20 20 D. Paindaveine, J. Remy and Th. Verdebout for any -vector v, this yields L n2 = nγ2 nνn 2 1 δ nξ 4 = n 2 γ 2 nν 2 n 4 + 2) δ n ξ) 2 ) 2 { 4V 1/2 0n ) V 1/2 0n )) T n )V 1/2 0n τ n ) V 1/2 0n τ n )) +4ν n V 1/2 0n τ n ) V 1/2 0n τ n )) T n )V 1/2 0n τ n ) V 1/2 0n )) } +ν 2 nv 1/2 0n τ n ) V 1/2 0n τ n )) T n )V 1/2 0n τ n ) V 1/2 0n τ n )) { 4V 1/2 0n ) V 1/2 0n )) 2I + J )V 1/2 0n τ n ) V 1/2 0n τ n )) +4ν n V 1/2 0n τ n ) V 1/2 0n τ n )) 2I + J )V 1/2 0n τ n ) V 1/2 0n )) 24) } +ν 2 nv 1/2 0n τ n ) V 1/2 0n τ n )) 2I + J )V 1/2 0n τ n ) V 1/2 0n τ n )) + o P 1). We can now consider the cases i) iv). We start with cases i) ii) and let ζ be equal to one if case i) is considered and to zero if case ii) is. Since ν n γ n = 1/ n in both cases, we have L n1 = 2 n δ n ξ τ nv 1/2 0n Sn ) 1 I 1/2 ) θ ν ) nτ n δ n ξ 2γ n δ n ξ) τ n ν nθ ) 0τ n = 2 n ζξ τ nv 1/2 0n Sn ) 1 I 1/2 ) θ ν ) nτ n + ζ 1)ξ τ n 2 + o P 1), 2 ζξ) where the last equality follows from Lemma A.3ii), Lemma A.2, and the fact that ν n =

21 Sign tests for weak rincial directions 21 o1). So, using 23), Lemma A.3 again, and the fact that θ 0τ n = o1), we obtain L n1 = 2 n ζξ 1 ζξ ) 1/2 1 + ζ 1)ξ ) 1/2 τ n Sn ) 1 I + ζ 1)ξ ) θ0 τ n 2 + o P 1) 2 ζξ) = + ζ 1)ξ)1/2 n τ ζξ) 1/2 n Sn ) 1 I + ζ 1)ξ ) θ0 τ n 2 + o P 1) 2 ζξ) = + ζ 1)ξ)1/2 n τ ζξ) ni 1/2 θ 0) S n ) 1 I ) θ0 Now, in cases i) ii), since ν n = 1/ nγ n ) = o1), 24) becomes L n2 = + ζ 1)ξ τ 2 ζξ) ni θ 0)τ n + o P 1). 25) 2 + 2) ζξ) 2 V1/2 0n ) V 1/2 0n )) 2I + J )V 1/2 0n τ n ) V 1/2 0n τ n )) + o P 1), which, by using 23), readily yields L n2 = = 2 + 2) ζξ) )δ nξ ) 1 δ nξ + ζ 1)ξ { 2θ + 2) ζξ) 0τ n ) 2 + τ n 2} + o P 1) ) ) 2I + J )τ n τ n ) + o P 1) = + ζ 1)ξ + 2) ζξ) τ ni θ 0)τ n + o P 1). 26) The results in 6) and 7) then follow from 25)-26), whereas the asymtotic normality results for i) and ii) are direct corollaries of Lemma A.3ii). We turn to case iii), for which δ n = 1/ n and ν n = 1/ nγ n ) = 1/ξ + o1). By using Lemma A.2, 22) here becomes L n1 = ξ/ n) τ nv 1/2 0n Υn) V 1/2 0n θ ν ) δ n ξ nτ n 2γ n ξ/ n)) τ n ν nθ ) 0τ n = τ nv 1/2 0n Υn) V 1/2 0n θ ξ τ ) 1 n 2 τ n ξ τ 2 n 2) + o P 1). Since = I + o1), this yields L n1 = τ nυ n) + 1 2ξ τ nυ n) τ n 1 2 τ n ξ 2 τ n 4 + o P 1). 27)

22 22 D. Paindaveine, J. Remy and Th. Verdebout Turning to L n2, 24) rovides 2 { L n2 = 4 + 2) + o1)) 2 4 ) 2I + J )τ n τ n ) } +4ξ 1 τ n τ n ) 2I + J )τ n ) + ξ 2 τ n τ n ) 2I + J )τ n τ n ) + o P 1) = 1 { 8θ 4 + 2) 0τ n ) τ n ξ 1 θ 0τ n ) τ n 2 + 3ξ 2 τ n 4} + o P 1). = τ n )ξ 2 τ n 4 + o P 1). 28) Hence, from 27) 28), we conclude that Λ n = τ nυ n) + 1 2ξ τ nυ n) τ n 2 + 2) τ n )ξ 2 τ n 4 + o P 1), as was to be shown. Again, the asymtotic normality result for vecυ n) ) easily follows from Lemma A.3ii). Finally, we consider case iv), under which δ n = o1/ n) and ν n = O1). It directly follows from 22) and 24) that L n1 and L n2 are then o P 1), so that Λ n also is. Aendix B: Proofs of Theorems The roofs of this section require the following reliminary result. Lemma B.1. Let ) be a sequence of null shae matrices as in 12) and denote Tyler s M-estimator of scatter as ˆV n. Then, we have the following under P n) as n : i) letting G := I I 2 + K J ), 1/2 G V 1/2 ) 0n n vec ˆVn ) = n vec S n ) 1 I ) + op 1); ii) n ˆV n ) is O P 1). Proof of Lemma B.1. Part i) of the lemma follows from 3.7) 3.8) in Tyler 1987a) and Lemma A.3i), whereas Part ii) follows from Part i) and Lemma A.3ii). In the roofs of this section, all stochastic convergences will be as n under P n).

23 Sign tests for weak rincial directions 23 Proof of Theorem 3.1. Standard roerties of the vec oerator rovide where 1 n n vec = 1 n 1 = n Ṽ 1/2 0n n n U ni U niṽ 1/2 0n U niṽ 1 0n U ni U ni U ) niṽ 1/2 Ṽ 1/2 0n 0n U ni V 1 0n U ni U niṽ 1 0n U ni U ni V 1 0n U ni U niṽ 1 0n U ni)u ni V 1 0n U ni) vecṽ 1/2 0n U ni U niṽ 1/2 0n ) vecṽ 1/2 0n U ni U niṽ 1/2 0n )vecu ni U ) ni )) n Ṽ 1 vec U niṽ 1 0n U ni)u ni V 1 0n U 0n ni) ) V 1 0n = M 1n + M 2n ) n vec Ṽ 1 0n V 1 0n ), M 1n := 1 n n vecṽ 1/2 0n U ni U niṽ 1/2 0n )vecu ni U ni )) U ni V 1 0n U ni) 2 = Ṽ 1/2 0n V 1/2 0n ) Ṽ 1/2 0n V 1/2 0n )) T n ) V 1/2 0n ) V1/2 0n and M 2n := 1 n = 1 n n { 1 U niṽ 1 0n U ni } 1 vec Ṽ 1/2 0n U ni U niṽ 1/2 0n )vecu ni U ni )) U ni V 1 0n U ni U ni V 1 0n U ni n U ni Ṽ 1 0n V 1 0n )U ni U niṽ 1 0n U ni)u ni V 1 0n U ni) 2 vecṽ 1/2 0n U ni U niṽ 1/2 0n )vecu ni U ni)). The squared Frobenius norm M 2n 2 F = tr[m 2nM 2n] is M 2n 2 F = 1 n 2 n i,j=1 U ni Ṽ 1 0n V 1 0n )U ni)u nj Ṽ 1 0n V 1 0n )U nj) U niṽ 1 0n U ni)u ni V 1 0n U ni) 2 U njṽ 1 0n U nj)u nj V 1 0n U nj) 2 U niu nj ) 2 U niṽ 1 0n U nj) 2, so that, denoting as ρa) the sectral radius of A, the Cauchy-Schwarz inequality yields M 2n 2 F 1 n 2 n i,j=1 U ni Ṽ 1 0n V 1 0n )U ni U nj Ṽ 1 0n V 1 0n )U nj U ni V 1 0n U ni) 2 U nj V 1 0n U nj) 2 su u S 1u Ṽ 1 0n V 1 0n )u)2 inf u S 1u V0n 1 = λ 4 Ṽ 1 u)4 n1 ρ 0n )) 2, V 1 0n

24 24 D. Paindaveine, J. Remy and Th. Verdebout which is o P 1) note indeed that Lemma B.1ii) imlies that n Ṽ0n ), hence also n Ṽ 1 0n V 1 0n ), is O P1)). Consequently, we have roved that 1 n Ṽ 1/2 0n U ni U vec niṽ 1/2 0n n U niṽ 1 0n U Ṽ 1/2 0n U ni U ) niṽ 1/2 0n ni U ni V 1 0n U ni = Ṽ 1/2 0n V 1/2 0n ) Ṽ 1/2 0n V 1/2 0n )) T n ) V 1/2 0n ) Ṽ 1 V1/2 0n n vec 0n V 1 0n ) + op 1). Still using the fact that n Ṽ 1 0n V 1 0n ) is O P1), we then obtain from Lemma A.3i) and the continuous maing theorem that 1 n n vec Ṽ 1/2 0n U ni U niṽ 1/2 0n U niṽ 1 0n U ni U ni U ) niṽ 1/2 Ṽ 1/2 0n 0n U ni V 1 0n U ni 1 = Ṽ 1/2 0n V 1/2 0n + 2) ) Ṽ 1/2 0n V 1/2 0n )) I + K + J ) 29) V 1/2 0n ) Ṽ 1 V1/2 0n n vec 0n ) V 1 0n + op 1). Now, since Ṽ 1/2 0n V 1/2 0n is O P 1/ n), Lemma A.3ii) rovides 1 n n vec = n Ṽ 1/2 0n Therefore, 29) 30) rovide U ni U niṽ 1/2 0n U ni V 1 0n U ni [ Ṽ 1/2 0n V 1/2 0n U ni U ) ni V 1/2 V 1/2 0n 0n U ni V 1 0n U ni Ṽ 1/2 0n V 1/2 ) 0n I 2]vec S ) ) n [ Ṽ 1/2 = 0n V 1/2 0n Ṽ 1/2 0n V 1/2 ) 0n I 2] veci ) + o P 1). 30) n vec Sn Ṽ0n) S n ) ) = 1 n 1 = + 2) n + n vec Ṽ 1/2 0n U ni U niṽ 1/2 0n U niṽ 1 0n U ni U ni U ) ni V 1 V 1/2 0n U ni V 1 0n U ni Ṽ 1/2 0n V 1/2 0n Ṽ 1/2 0n V 1/2 ) 0n I + K + J ) V 1/2 0n ) Ṽ 1 V1/2 0n n vec 0n ) V 1 0n [ Ṽ 1/2 0n V 1/2 0n Ṽ 1/2 0n V 1/2 ) 0n I 2] veci ) + o P 1). Hence, using the fact that is an eigenvector of any matrix ower of Ṽ0n and along with the identity I θ 0) = 0, we then obtain θ 0 I θ 0) ) n vec S n Ṽ0n) S n ) ) = o P 1), 0n

25 Sign tests for weak rincial directions 25 which finally roves that T n Ṽ0n) T n ) = n + 2) θ 0 I θ 0) ) vec S n Ṽ0n) S n ) ) 2. is o P 1). The roof of Theorem 3.2 still requires the following result. Lemma B.2. Let ) be a sequence of null shae matrices as in 12) and assume that there exists η > 0 such that both corresonding leading eigenvalues satisfy λ n1 /λ n2 1+η for n large enough. Then, as n under P n), 1 ˆλ n1 j=2 ˆλ 1 nj ˆθ njˆθ nj = 1 λ n1 j=2 λ 1 nj θ nj θ nj + o P 1), where the ˆλ nj s and ˆθ nj s refer to the sectral decomosition of Tyler s M-estimator of scatter as ˆV n in 13). 1 Proof of Lemma B.2. It follows from Lemma B.1ii) that ˆV n V0n 1 is o P1) as n under P n). Since λ n1 /λ n2 stays away from one, we also have that ˆθ n1 θ n1 is o P 1) under the same sequence of hyotheses. Consequently, I ˆθ n1ˆθ 1 n1) ˆV n I θ 0)V0n 1 = j=2 ˆλ 1 nj ˆθ njˆθ nj j=2 λ 1 nj θ nj θ nj is o P 1) as n under P n). The result then follows from the fact that Lemma B.1ii) also imlies that ˆλ n1 λ n1 is o P 1) as n under P n). Proof of Theorem 3.2. Since Ṽ0n = ˆλ n1, we have that L n = n ˆλn1 θ = n + 2 = = n j=1 + 2)ˆλ n1 n + 2)ˆλ n1 1 1 ˆV n + ˆλ n1 θ ˆV 0 n 2 ) ˆλn1ˆλ 1 nj θ 0ˆθ nj ) ˆλ ˆλ n1 nj θ 0ˆθ nj ) 2 2θ 0ˆθ nj ) 2) j=2 j=2 ˆλ 1 { nj ˆλnj ˆλ n1 )θ 0ˆθ nj ) } 2 ˆλ 1 nj {ˆθ nj ˆV n Ṽ0n) } 2.

26 26 D. Paindaveine, J. Remy and Th. Verdebout Since Lemma B.1ii) entails that n ˆV n Ṽ0n) is O P 1) and since both Ṽ0n = ˆλ n1 and = λ n1 are orthogonal to θ nj, j = 2,..., n, Lemma B.2 then rovides L n = = n + 2)λ n1 n + 2)λ n1 j=2 j=2 λ 1 nj θ nj ˆV n Ṽ0n) ) 2 + o P 1) λ 1 nj θ nj ˆV n ) ) 2 + o P 1), which, letting G := I 2 1/ + 2))I 2 + K J ), rewrites L n = + 2 = + 2 { + 2) θ 0 θ nj) } 2 n vec ˆV n ) + o P 1) λ n1 λ nj j=2 j=2 { θ 0 θ 1/2 nj)g V 1/2 ) } 2 0n n vec ˆVn ) + op 1). By using Lemma B.1i), we therefore conclude that L n = + 2) j=2 { θ 0 θ nj) n vec S n ) 1 I ) } 2 + op 1) = n + 2) vec S n ) 1 I )) θ0 θ 0) I θ 0) ) vec S n ) 1 I ) + op 1) = T n ) + o P 1), which, in view of Theorem 3.1, establishes the result. Aendix C: Proofs of Theorems Proof of Theorem 4.1. It directly follows from Lemma A.3 that, under P n), W n := θ 0 I θ 0) ) n vec S n ) ) = θ 0 I θ 0) ) n vec S n ) 1 I ) is asymtotically normal with mean zero and covariance matrix I θ 0. Since I θ 0 is idemotent with rank 1, this imlies that, under the same sequence of hyotheses, T n ) = W ni θ 0)W n is asymtotically chi-square with 1 degrees of freedom. The result then follows from Theorem 3.1.

27 Sign tests for weak rincial directions 27 Proof of Theorem 4.2. The results in i) ii) follow from a routine alication of the Le Cam third lemma. For iii), the mutual contiguity between P n) and P n) V 1n which follows by alying the Le Cam first lemma to Theorem 2.1iii) enables the use of the same Le Cam third lemma. Using the notation introduced in the roof of Theorem 4.1, the central limit theorem yields that, under P n), ) τ nυ n) W n + 1 2ξ τ nυ n) τ n is asymtotically Gaussian with mean zero and covariance matrix I θ ξ τ 2 )τ I 2 θ 0) ξ τ 2 )I 2 θ 0)τ +2 τ n ξ τ 2 n 2) ). The Le cam third lemma therefore ensures that, under the sequence of local alternatives considered in Part iii) of the theorem, W n is asymtotically normal with mean ξ τ 2 )I 2 θ 0)τ and covariance matrix I θ 0. It follows that T n ) = W ni θ 0)W n is asymtotically chi-square with 1 degrees of freedom and non-centrality arameter + 2 τ ξ 2 τ 2) 1 1 4ξ 2 τ 2). From contiguity, Theorem 3.1 imlies that the same holds for T n Ṽ0n), which establishes the result. Finally, Part iv) of the result directly follows from Theorem 2.1iv). Acknowledgement Davy Paindaveine s research is suorted by a research fellowshi from the Francqui Foundation and by the Program of Concerted Research Actions ARC) of the Université libre de Bruxelles. Thomas Verdebout s research is suorted by the Crédit de Recherche J of the FNRS Fonds National our la Recherche Scientifique), Communauté Française de Belgique, and by the aforementioned ARC rogram of the Université libre de Bruxelles. References Anderson, T. W. 1963). Asymtotic theory for rincial comonent analysis. Ann. Math. Statist