SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES AGAINST MONOTONE ROTATIONALLY SYMMETRIC ALTERNATIVES

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1 Submitted to the Annals of Statistics SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES AGAINST MONOTONE ROTATIONALLY SYMMETRIC ALTERNATIVES By Christine Cutting, Davy Paindaveine Thomas Verdebout Université libre de Bruxelles In this supplementary article, we first derive the fixed-p asymptotic non-null distribution of the Rayleigh test statistic provided in Equation 3.4. Then we show that, under FvML distributions, the conditions i-iii of Theorem 5. always hold, that is, that they hold without any constraint on the concentration n nor on the way the dimension p n goes to infinity with n. Throughout, this supplement, n.m resp., S.n.m denotes Equation m of Section n from the main manuscript resp., from this supplement article. Theorem m.n or Lemma S.m.n are used in a similar way.. Asymptotic non-null distribution of the Rayleigh test statistic in the fixed-p case. We focus on the fixed-p case p n p for any n derive the asymptotic distribution of the Rayleigh test statistic R n under sequences of contiguous rotationally symmetric alternatives. More specifically, we consider sequences of alternatives of the form P n θ n, n,f, with n τ n / n, where the sequence τ n converges to some τ 0,. Since R n is invariant when a common rotation is applied to X ni, i,..., n, we will without any loss of generality assume that θ n coincides for any n with the first vector of the canonical basis of R p to avoid introducing an extra notation, we will simply write θ for this constant value of θ n. The null of uniformity will still be denoted as P n 0. Adopting the notation u ni, v ni, S ni introduced in Appendix B of the main manuscript, Research is supported by an A.R.C. contract from the Communauté Française de Belgique, the IAP research network grant nr. P7/06 of the Belgian government Belgian Science Policy, the Crédit de Recherche J.03.6 of the FNRS Fonds National pour la Recherche Scientifique, Communauté Française de Belgique, a grant from the National Bank of Belgium. MSC 00 subject classifications: Primary 6H, 6G0; secondary 6H5 Keywords phrases: Contiguity, directional statistics, high-dimensional statistics, local asymptotic normality, rotationally symmetric distributions, tests of uniformity

2 CHR. CUTTING, D. PAINDAVEINE, AND TH. VERDEBOUT rewrite then the Rayleigh test statistic R n as R n p n X nix nj p n i,j p n where we let A : pi p θθ, Y n : p n u ni θ v ni S ni u nj θ v nj S nj i,j u ni u nj v ni v nj S nis nj Yn p Z naz n, i,j u ni Z n : i p n Under P n 0, the multivariate CLT, along with the identities S.. E [ u ] ni p E[ vni ] see Lemma B.i, provides S.. Clearly, Y n Yn Z n is asymptotically χ D N 0, 0 0 Σ v ni S ni. i E [ S ni S ] ni p I p θθ, with Σ : p I p θθ. under Pn 0. By using the identities ΣAΣAΣ ΣAΣ tr[σa] p, Theorem 9.. in Rao Mitra 97 shows that /pz naz n is asymptotically χ under Pn 0. Since the joint asymptotic normality result in S.. ensures asymptotic independence of Yn /pz naz n, this confirms that R n is asymptotically χ p under P n 0. Let us now turn to the sequence of alternatives P n θ, n,f considered above. In view of S.. Theorem., Le Cam s third lemma directly shows that S..3 Yn Z n D N τ, Σ as n under P n θ n, n,f. Under the same sequence of alternatives, Y n is therefore asymptotically χ τ that is, non-central χ with non-centrality parameter τ p Z naz n is still asymptotically χ. Asymptotic independence between Y n Z n still holds under contiguous alternatives, which shows that R n D χ τ χ, where the χ terms are independent. This establishes the asymptotic result in 3.4.

3 SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES 3. Universality of the asymptotic non-null distribution of the Rayleigh test in the FvML case. In the rest of this supplement, we prove that the conditions i- iii of Theorem 5. universally hold under FvML distributions, in the sense that, if F n is FvML for any n, then the theorem does not require any condition on the dependence of p n n on n but for the fact that p n as n. The proof is quite lengthy requires several preliminary results. Lemma S... Write e l : e l;p, : E[X θ l ], where X follows a p-dimensional FvML distribution with concentration > 0 location θ S. Then e r, e p r, e pp 3 r p e 4 p p p 3 r p p, where we let r : r p, : I p /I p as in the main manuscript, I ν sts here for the order-ν modified Bessel function of the first kind. Proof of Lemma S... Using integration by parts in the representation result S..4 I ν z z/ ν π Γν see, e.g., 0.3. in Olver et al. 00 provides s s ν expzs ds z ν s ν expzs ds s ν expzs ds S..5 z π Γν 3 I νz ν z/ ν, which readily leads to e / p / πγ I p s s p 3 exps ds Γ p I p Γ I p I p I p Turning to e, S..4 above yields s s ν expzs ds s s ν expzs ds π Γν I νz z/ ν π Γν 3 I νz z/ ν

4 4 CHR. CUTTING, D. PAINDAVEINE, AND TH. VERDEBOUT Hence, e / p / πγ I p / p / πγ I p as was to be showed. The results for s s p 3 exps ds [ π Γ I p / p Γ p I p /Γ I p p I p, I p π Γ p I p ] / p/ e l / p / π Γ I p s s p 3 exps ds, l 3, 4, follow similarly by using the expressions obtained by plugging S..4-S..5 into s 3 s ν expzs ds s s ν expzs ds s s ν expzs ds s 4 s ν expzs ds s ν expzs ds along with the well-known recurrence relation s ν expzs ds s ν expzs ds, S..6 I ν z I ν z ν z I νz; see 0.9. in Olver et al. 00. Note that closed form expressions for f l : f l;p, E[ X θ l/ ], where X still follows a p-dimensional FvML distribution with concentration > 0 location θ, can be obtained much more directly than in Lemma S.., as S..4 readily yields S..7 f l / p / πγ I p s pl 3 exps ds Γ pl I pl / l Γ I p The main result of this supplementary article is the following theorem, that implies that, under FvML distributions, Theorem 5. does not impose any condition on the way p n should go to infinity as a function of n.

5 SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES 5 Theorem S... Let us still write e l : e l;p, : E[X θ l ] f l : f l;p, E[ X θ l/ ], where X follows a p-dimensional FvML distribution with concentration > 0 location θ. Further let ẽ l : ẽ l;p, : E[X θ e l ]. Then there exist a positive integer p 0 a real constant C such that i pẽ f C, for any p p 0 any > 0. ii ẽ4 ẽ C, iii f 4 f C, The proof requires both following lemmas on the modified Bessel functions ratio R ν z : I ν z/i ν z we adopt the same notation as in Hornika Grün 03. Lemma S... Fix ν > 0 z > 0, let G α,β t t/α t β. Then i R ν z G ν,ν z ii R ν z G ν/,ν3/ z iii R ν z G ν/,ν/ z iv R ν z G ν,ν z, v R ν z G ν,ν z, vi R ν z G ν/, ν/ν3/ z. These bounds, that have been obtained in Amos 974 i-v Simpson Spector 984 vi, are actually sufficient to establish Theorem S..i iii. To prove Theorem S..ii, however, we will need the following reinforcement of the bounds in Lemma S..ii-iii an appropriate control of the resulting approximation error; see Paindaveine 06 for a proof. Lemma S..3. Fix ν > 0 z 0, let a ν z : ν 3 ν 4 z ν 3 ν 7, b νz : ν 5 ν z z ν 3 ν 7, z c ν z : ν 3 ν 5 z ν 3 ν 7 z, d νz : ν ν 3 z ν ν 3 z e ν z : ν ν 5 z ν ν 3 z Then i L ν z R ν z U ν z, with z L ν z : bν a ν zν zν 3 cν zz U ν z : ii there exists ν 0 > 0 such that z d ν zν dν zν eν zz ; ν 7 z 7 for any ν ν 0 any z > 0. z 3 U ν z L ν z 3ν 3 8

6 6 CHR. CUTTING, D. PAINDAVEINE, AND TH. VERDEBOUT We prove Theorem S..i-iii separately. Throughout, we still write r for the key quantity R p I p /I p. Proof of Theorem S..i. From Lemma S.., we obtain S..8 pẽ f pẽ e p r r p r pg a rg b r p r, where we let g a x G, x g b x G, both g a r g b r, which can be achieved by using Lemma S... Starting with g a r, Lemma S..iii readily yields that g a r G, G,. As for g b r, Lemma S..ii entails g b r G, G, p Plugging into S..8 provides p x. We need to control p p p p p pẽ f p p 3 4 r Using Lemma S..ii again then yields pẽ f p p 3 p 4 p 3 4 G p, p 8 p p 3 p p 4 9 4, for any p any > 0, which establishes the result. Proof of Theorem S..iii. From S..7, we obtain f 4 f p I p I p p I p 3I p I p I p,

7 SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES 7 for any p any > 0. By using S..6, this provides f 4 3f I p I p p/i p I p r p r pr r note that Lemma S..iv yields r G p, p /p. Lemma S..i then entails f 4 3f pg, p G, p which proves the result. p p p p p p p p p p Proof of Theorem S..ii. Plugging the expressions of e l, l,, 3, 4, from Lemma S.. in, S..9 ẽ n4 ẽ n e 4 4e 3 e 6e e 4e4 e4 e e e 4 4e 3 e 6e e 3e4 e e e e 4 yields after tedious computations ẽ n4 ẽ n p p 4 3x p p 4 x p 4 x x 3 : hr 3, We need to show that hr is bounded in p, for p large enough, which will be done on the basis of the factorization S..0 where we let f a x : f b x : where g a x G hr p p 4 3f a rf b r g a rg b r, p p p3 p p p3, p p p3 p p p3 x g b x G considered in the proof of Theorem S..i., We start with g a r, which, in view of Lemma S..i, satisfies g a r G, p p3 x, p p3 x, x are the functions already G p, p p p p Cp,

8 8 CHR. CUTTING, D. PAINDAVEINE, AND TH. VERDEBOUT where C sts for a positive real constant that may change from line to line in the rest of the proof. Turning to g b r, Lemma S..vi yields g b r G, G p, p /4 p 4p p Cp 3 p 3 3 p p 4 4 p 4 p 4 Now, by applying Lemma S..v, we obtain f a r r G p p p, p Finally, using the notation results from Lemma S..3, we obtain Cp f b r p p p3 p p p3 U p r U p L p C3 p p 7 7 C3 p p 7 7, for p large enough any > 0. p p3 r Plugging in S..0 the bounds just obtained on g a r, g b r, f a r f b r entails ẽ n4 ẽ n 3 hr C p p 4 3 p p3 3 p p 3 p p 7 7 C p p 3 3 p p 7 7 C, for p large enough any > 0, as was to be proved. References. Amos, D. E Computation of modified Bessel functions their ratios. Mathematics of Computation Hornika, K. Grün, B. 03. Amos-type bounds for modified Bessel function ratios. J. Math. Anal. Appl Olver, F. W. J., Lozier, D. W., Boisvert, R. F. Clark, C. W. 00. NIST Hbook of Mathematical Functions. Cambridge Univ. Press. Paindaveine, D. 06. Generalized Amos-type bounds for ratios of modified Bessel functions. Manuscript in preparation.

9 SUPPLEMENT TO TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES 9 Rao, C. R. Mitra, S. K. 97. Generalized Inverses of Matrices its Applications. J. Wiley, New York. Simpson, H. C. Spector, S. J Some monotonicity results for ratios of modified Bessel functions. Quart. Appl. Math Université libre de Bruxelles Département de Mathématique Bld du Triomphe, Campus Plaine, CP0 B-050, Brussels Belgium Christine.Cutting@ulb.ac.be tverdebo@ulb.ac.be URL: Université libre de Bruxelles ECARES Département de Mathématique Avenue F.D. Roosevelt, 50 ECARES, CP4/04 B-050, Brussels Belgium dpaindav@ulb.ac.be URL: dpaindav