Asymptotic non-null distributions of test statistics for redundancy in high-dimensional canonical correlation analysis

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1 Asymtotic non-null distributions of test statistics for redundancy in high-dimensional canonical correlation analysis Ryoya Oda, Hirokazu Yanagihara, and Yasunori Fujikoshi, Deartment of Mathematics, Graduate School of Science, Hiroshima University -3- Kagamiyama, Higashi-Hiroshima, Hiroshima , Jaan Hiroshima University Statistical Science Research Core -3- Kagamiyama, Higashi-Hiroshima, Hiroshima , Jaan Last Modified: March, 07 Abstract In this aer, we derive asymtotic non-null distributions of three test statistics, i.e., the likelihood ratio criterion, the Lawley-Hotelling criterion and the Bartlett-Nanda-Pillai criterion, for redundancy in high-dimensional canonical correlation analysis. Since our setting is that the dimension of one of two observation vectors may be large but does not exceed the samle size, we use the high-dimensional asymtotic framework such that the samle size and the dimension divided by the samle size tend to and some ositive constant being included in [0,, resectively, for evaluating asymtotic distributions. Additionally, we derive asymtotic null distributions of the three test statistics under the high-dimensional asymtotic framework by using the same transformation as when we derive asymtotic non-null distribution. We verified that new aroximations based on derived asymtotic distributions are more accurate than those based on asymtotic distributions evaluated from the classical asymtotic framework, i.e., only the samle size tends to. Introduction Canonical correlation analysis CCA is a statistical method emloyed to investigate the relationshis between a air of q- and -dimensional random vectors, x = x,..., x q and y = y,..., y. Introductions to CCA are rovided in many textbooks for alied statistical analysis see, e.g., Srivastava, 00, cha. 4.7; Timm, 00, cha. 8.7, and it has widesread alications in many fields e.g., Chung et al., 07; Prera et al., 04; Sakar et al., 06; Singh et al., 04; Zheng et al., 04. In an actual data analysis, it is imortant to clarify whether or not some variables are irrelevant for analysis. In CCA, one of methods to solve the roblem is hyothesis testing for redundancy. Since a distribution of a test statistic lays an imortant rule in hyothesis testing, it has been studied by many authors. It is known that the null distribution of the likelihood ratio LR criterion for redundancy is distributed according to the Wilks Lambda distribution e.g., Fujikoshi, 98. Since the Wilks Lambda distribution is Correonding author. m5553@hiroshima-u.ac.j

2 user-unfriendly, it is imortant to derive an asymtotic aroximation of the distribution of the LR criterion. Fujikoshi 98 derived an asymtotic exansion of the null distribution of the LR criterion under the large-samle LS asymtotic framework such that only the samle size aroaches. However, an aroximation based on the asymtotic exansion worsen as the dimension of either of observation vectors increases. To avoid the deterioration of aroximation accuracy, Sakurai 009 and Wakaki and Fujikoshi 0 used the asymtotic exansion of the null distribution of the LR criterion under the asymtotic framework such that the samle size and the dimension of either of observation vectors tend to. On the other hand, as for a non-null distribution of the LR criterion, under the LS asymtotic framework, Suzukawa and Sato 996 obtained an asymtotic distribution of the LR criterion under a local alternative hyothesis. These results have been derived from the fact that the LR criterion is distributed according to the Wilks Lambda distribution under the null hyothesis. However, the asymtotic distribution of the LR criterion has not been derived under the fixed alternative hyothesis because the criterion is not distributed according to the Wilks Lambda distribution under the hyothesis. In CCA, the Lawley-Hotelling LH criterion and the Bartlett-Nanda-Pillai BNP criterion are commonly used in the test for redundancy just like the LR criterion. However, since the two test statistics are not distributed according to the Wilks Lambda distribution even under the null hyothesis, the asymtotic null distributions have not been derived under excet the LS asymtotic framework. Let q be the number of elements of x assumed to be irrelevant variables and q = q q. In this aer, since our setting is that the dimension of one of two observation vectors may be large but does not exceed the samle size, we use the following high-dimensional HD asymtotic framework for evaluating asymtotic distributions: n, + q n c [0,, q : fixed. It should be emhasized that the LS asymtotic framework is included in the HD asymtotic framework as a secial case. An aim of this aer is deriving asymtotic distributions of the three test statistics under a fixed alternative hyothesis by using the HD asymtotic framework. To derive the asymtotic distributions, we transform the test statistics by using the result in Yanagihara et al. 07 because the test statistics are not distributed according to the Wilks Lambda distribution under the fixed alternative hyothesis. In Yanagihara et al. 07, the asymtotic null distribution of the LR criterion by using the HD asymtotic framework, but the asymtotic non-null distribution have not been derived. Additionally, we derive the asymtotic null distributions of the three test statistics under the HD asymtotic framework because those of the LH criterion and the BNP criterion have not been derived. This aer is organized as follows: In section, we introduce three test statistics for redundancy in CCA. In section 3, we resent our key lemmas and main results. In Section 4, we verify the accuracies of aroximations under the HD asymtotic framework by conducting numerical exeriments. Technical details are rovided in the Aendix.

3 Test for redundancy hyothesis In this section, we introduce three test statistics for redundancy in CCA. Let z = x, y be a q + -dimensional random vector distributed according to q + -variate normal distribution with E[z] = µ = µ x, µ y, Cov[z] = Σ = Σ xx where µ x and µ y are q- and -dimension mean vectors of x and y, Σ xx and Σ yy are q q and covariance matrices of x and y, and Σ xy is the q covariance matrix of x and y. We artition x as x = x, x, x : q, x : q and exress µ x, Σ xx and Σ xy corresonding to the division of x as follows: µ x = µ, µ, Σ xx = Σ Σ Σ y, Σ xy = Σ Σ Σ y. We are interested in whether or not the elements of x are irrelevant variables in CCA. Fujikoshi Σ yx Σ xy Σ yy 985 defined that x is irrelevant if the following null hyothesis is true: H 0 : trσ xx Σ xy Σ yy Σ yx = trσ Σ yσ yy Σ y. In addition, we note that the fixed alternative hyothesis is as follows: H : trσ xx Σ xy Σ yy Σ yx trσ Σ yσ yy Σ y. The left-hand side and the right-hand side of the equation or are the sum of squares of the canonical correlation coefficients between x and y, and the sum of squares of the canonical correlation coefficients between x and y, resectively. Let O q, be a q matrix of zeros. In articular, we note that is equivalent to H 0 : Σ y = O q,, 3 where Σ ab c = Σ ab Σ ac Σ cc Σ cb. Let z,..., z n be n indeendent random vectors from z, and let z be the samle mean of z,..., z n given by z = n n i= z i and S be the usual unbiased estimator of Σ given by S = n n i= z i zz i z, divided in the same way as we divided Σ, as follows: S = S xx S yx S xy S yy S S S y = S S S y. S y S y S yy Suose that z,..., z n N q+ µ, Σ. Then, the three test statistics, i.e., the LR, LH and BNP criteria are exressed as follows: log S yy G = LR S yy x T G = tr S yy S yy x Syy x G = LH, tr S yy S yy x Syy G = BNP where S ab c = S ab S ac S cc S cb., 3

4 3 Main results In this section, we derive asymtotic non-null distributions of the three test statistics within the HD asymtotic framework. First of all, we resent key lemmas which are required for deriving our main results. Under the HD asymtotic framework, it is a serious roblem that increasing the dimension of S yy and S yy x with an increase in the dimension. However, we can avoid the roblem by using the following lemma. Lemma 3. Suose that V is a q matrix and W is a non-singular matrix. Then, W + V V W = I q + V W V, 4 trv V W = trv W V, 5 trw W + V V = trv W V I q + V W V. 6 The roof of the lemma is omitted because it is easy to obtain from elementary linear algebra. By alying this lemma to our roblem, we can evaluate the asymtotic behaviors of the three test statistics from two random matrices of which dimensions do not increase with an increase in the dimension. Yanagihara et al. 07 derived the asymtotic null distribution of the LR criterion under the HD asymtotic framework by using the following lemma as for the roof, see Yanagihara et al. 07. Lemma 3. Let E, A, and B be mutually indeendent random matrices, which are defined by E N n O n,, I I n, A N n q q O n,q q, I q q I n, and B N n q O n,q, Σ xx I n, where E and B are indeendent, and let B denote the n q matrix defined by B = B, B. Then, n S yy x = Σ / yy xe I n P EΣ / yy x, n S yy = Σ / yy xaγ + E I n P AΓ + EΣ / yy x, where P = BB B B, P = B B B B and Γ = Σ / yy x Σ y Σ /. It is easy to see from 3 that Γ = O,q under H 0. Therefore, Γ is called non-centrality arameter matrix, and it is known that rankγ 0 under H. However, since it is clearly that Γ O,q under H, we can not aly the equations in Lemma 3. to the three test statistics. Therefore, we have to give another exressions of S yy and S yy x in order to aly the equations in Lemma 3. to the three test statistics. The following lemma is what the result in Lemma 3. is extended to the roof is given in Aendix A: Lemma 3.3 Let U, U and T be mutually indeendent random matrices, which are defined by U N n q O n q,, I I n q, U N q O q,, I I q, 4

5 and T = t ij q i,j= is a q q lower triangular matrix of which elements are mutually indeendent random variables, which are defined by t ii χ i and t ij N0, i > j. Also, let J K be a singular value decomosition of Γ defined in Lemma 3., where J is a -th orthogonal matrix, K is a q -th orthogonal matrix. Then, n S yy x = Σ / yy xu U Σ / yy x, 7 n S yy = Σ / yy xu U + JT + U T + U J Σ / yy x. 8 By using Lemma 3. and Lemma 3.3, we can derive the following another exressions of T G the roof is given in Aendix B. Lemma 3.4 Suose that > q. Let D = T + U, where T, and U are defined in Lemma 3.3, and Ũ is a q q random matrix distributed according to W q n q, I q, where D and Ũ are mutually indeendent. Then, log I q + Ũ DD T G = trũ DD trdd DD + Ũ G = LR G = LH G = BNP. 9 Next, we evaluate the asymtotic behaviors of T G by using Lemma 3.4. following theorem the roof is given in Aendix C: The result is the Theorem 3. Suose that > q. When H is true, the asymtotic non-null distributions of T G as n, + q /n c [0, is given by τ G T G η G d N0,, where η G and τg are standardizing constants given by q log + log I + ΓΓ n q η G = trγγ q + n q n q q + tr ΓΓ I + ΓΓ Here, ξ G are constants given by n q ξ LR τg n q = 3 ξ LH G = LR G = LH 4 n q ξ BNP G = BNP G = LR G = LH G = BNP ξ LR = n q tr ΓΓ I + ΓΓ + tr I + ΓΓ q, ξ LH = n q tr ΓΓ + tr ΓΓ + q, ξ BNP = n q tr ΓΓ I + ΓΓ 4 + tr ΓΓ I + ΓΓ 4 + tr I + ΓΓ 4 q.., 5

6 We can also derive asymtotic non-null distributions of the three test statistics under the LS asymtotic framework. Although the non-null distributions can be derived by a different way from the method for roving Theorem 3., the result corresonds with the one in Theorem 3. when /n 0 and q /n 0. Therefore, the roof is omitted. Corollary 3. When H is true, the asymtotic non-null distributions of T G as n is given by ν G T G θ G d N0,, where θ G and ν G are standardizing constants given by log I + ΓΓ θ G = trγγ tr I + ΓΓ G = LR G = LH G = BNP, νg = n 4tr ΓΓ I + ΓΓ n 4tr ΓΓ + 4tr ΓΓ n 4tr ΓΓ I + ΓΓ 4 + 4tr ΓΓ I + ΓΓ 4 G = LR G = LH G = BNP Asymtotic null distributions of the three test statistics under the HD asymtotic framework can be obtained as the following theorem in the same way as the roof of Theorem 3. the roof is given in Aendix D. Theorem 3. When H 0 is true, the asymtotic null distributions of T G as n, +q /n c [0, is given by τ G,0 T G η G,0 d q χ q q : fixed N0,,. where η G,0 and τ G,0 are standardizing constants given by q log G = LR n q q η G,0 = G = LH n q q G = BNP τg,0 = n q q G = LR n q 3 q G = LH 3 n q q G = BNP,. It is easy to see that q / χ q q d N0, as. Hence, we suggest that q / χ q,α q is used as the threshold value for the null distributions of TG with significance level α, where χ q,α is the uer 00α % oint of the χ -distribution with q degrees of freedom. By using this aroximation, we can test regardless of the size of. 6

7 4 Numerical Studies In this section, we comare the accuracies of aroximations based on derived asymtotic distributions under the HD asymtotic framework with those based on asymtotic distributions under the LS asymtotic framework. The accuracies are evaluated from normal or chi-square aroximation. First, we comared aroximations under H 0. Let α G, and α G, be uer robabilities when asymtotic distributions under the HD and the LS asymtotic frameworks are used resectively, which are defined by α G, = P nt G > n χ q,α q + nη G,0, α G, = P nt G > χ q,α. q τg,0 We comared α G, with α G, when α = Tables -5 show the α G, and α G, which were evaluated by Monte Carlo simulations with 0,000 iterations. Note that c = /n and c = q /n in Tables. From Tables and, the aroximations under the LS asymtotic framework erformed a little well but lose to those under the HD asymtotic framework even when at least one of and q is small. From Tables 3-5, the aroximations under the HD asymtotic framework erformed well than those under the LS asymtotic framework. Table : q = 7, q = 3 n α LR, α LR, α LH, α LH, α BNP, α BNP,

8 Table : = 7, q = 3 n q α LR, α LR, α LH, α LH, α BNP, α BNP, Table 3: q = 7, q = 3 c n α LR, α LR, α LH, α LH, α BNP, α BNP,

9 Table 4: = 7, q = 3 c n q α LR, α LR, α LH, α LH, α BNP, α BNP, Table 5: q = 3 c, c n, q α LR, α LR, α LH, α LH, α BNP, α BNP, Next, we comared aroximations under H. It should be ket in mind that sometimes some elements of Γ defined in Lemma 3. become infinite and other times those are finite as at least one of and q tends to. Therefore, we treat the following two non-centrality matrices. Structure at least of elements of Γ tend to Λ, O q, q Γ = q, Λ = diag q, 0,..., 0. Λ, O,q < q Structure all elements of Γ are bounded Γ = Λ, O q, q q Λ, O,q < q, Λ = diag + q +, 0,..., 0. + q 9

10 It is clearly that the non-zero element of Γ tends to in Structure, and all elements of Γ are bounded in Structure as at least one of and q tends to. Let α G,3 and α G,4 be uer robabilities when asymtotic distributions under the HD and the LS asymtotic frameworks are used resectively, which are defined by α G,3 = P T G > z τ α + η G, α G,4 = P T G > z G ν α + θ G, G where z α is the uer 00α % oint of the standard normal distribution. We comared α G,3 with α G,4 when α = Tables 6-5 show the α G,3 and α G,4 which were evaluated by Monte Carlo simulations with 0,000 iterations. Note that c = /n and c = q /n in Tables. From Tables 6 and 7, the aroximations under the LS asymtotic framework erformed a little well but lose to those under the HD asymtotic framework even when at least one of and q is small. The reason is that the HD asymtotic framework includes in the LS asymtotic framework. From Tables 8-5, the aroximations under the HD asymtotic framework erformed well, but the aroximations under the LS asymtotic framework are oor. Table 6: Structure : q = 7, q = 3 n α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP,

11 Table 7: Structure : = 7, q = 3 n q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP, Table 8: Structure : q = 7, q = 3 c n α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP,

12 Table 9: Structure : = 7, q = 3 c n q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP, Table 0: Structure : q = 3 c, c n, q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP,

13 Table : Structure : q = 7, q = 3 n α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP, Table : Structure : = 7, q = 3 n q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP,

14 Table 3: Structure : q = 7, q = 3 c n α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP, Table 4: Structure : = 7, q = 3 c n q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP,

15 Table 5: Structure : q = 3 c, c n, q α LR,3 α LR,4 α LH,3 α LH,4 α BNP,3 α BNP, Aendix A Proof of Lemma 3.3 At first, we show the 7 and 8. Considering the conditional distribution of B given B in Lemma 3., B is given by B and A defined in Lemma 3., as follows: This imlies that B = B Σ Σ + AΣ /. A = B B Σ Σ Σ /. Notice that P B = B, P B = B and I n P A = O n,q, where P = BB B B and B = B, B. Hence, we have P P A = I n P A I n P A = I n P A, A. where P = B B B B. From A., by using Γ and E defined in Lemma 3., we can calculate as follows: AΓ + E I n P AΓ + E = AΓ + E I n P + P P AΓ + E = AΓ + E I n P AΓ + E + AΓ + E P P AΓ + E = E I n P E + AΓ + E P P AΓ + E. 5

16 Since P P is an idemotent matrix, the following equation is derived: AΓ + E P P AΓ + E = AΓ + E P P AΓ + E = P P AΓ + P P E P P AΓ + P P E = P P I n P AΓ + P P E P P I n P AΓ + P P E P = I n P AΓ + E P I n P AΓ + E. Let Q Q and QQ be sectral decomositions of P P and I n P, resectively, where Q is an n q matrix satisfying Q Q = I q, and Q is an n n q matrix satisfying Q Q = I n q. Then, AΓ + E I n P AΓ + E is exressed as follows: AΓ + E I n P AΓ + E = E QQ E + Q I n P AΓ + Q E Q I n P AΓ + Q E. Let J K be a singular decomosition of Γ, where J is a -th orthogonal matrix and K is a q -th orthogonal matrix. Then, the another exression of AΓ + E I n P AΓ + E is given as AΓ + E I n P AΓ + E = E QQ E + Q I n P AK J + Q E Q I n P AK J + Q E = E QQ E + J Q I n P AK + Q EJ Q I n P AK + Q EJ J. A. Here, without loss of generality, we can relace AK with A. Then, A. is exressed as follows: AΓ + E I n P AΓ + E = E QQ E + J Q I n P A + Q EJ Q I n P A + Q EJ J. It is easy to see that A I n P A is distributed according to W q, I q because A and P are mutually indeendent. Hence, we aly the Bartlett decomosition see, e.g., Fujikoshi et al., 00, cha..3 to A I n P A as follows: A I n P A = T T, A.3 where T = t ij q i,j= is a q q lower triangular matrix of which elements are mutually indeendent random variables, which are defined by t ii χ i and t ij N0, i > j, and T, E and B are mutually indeendent since A, E, and B are mutually indeendent. Let HLG be a singular value decomosition of T, where L is a diagonal matrix of which diagonal elements are the singular values of T, and H and G are q -th orthogonal matrices. The following singular value decomosition of Q I n P A is also used: Q I n P A = CLH = RT, 6

17 where C is a q -th orthogonal matrix and R = CG because of Q I n P A Q I n P A = A P P A = A I n P A A I n P A = A I n P A. By using those equations, the following equation can be derived: J Q I n P A + Q EJ = JT + R Q EJ T + R Q EJJ. Q I n P A + Q EJ Hence, by relacing U and U with Q E and R Q EJ, resectively, 7 and 8 can be derived. Next, we show that R Q EJ, Q E and T are mutually indeendent. Let Q Q be the sectral decomosition of I n P, where Q is an n n q matrix satisfying Q Q = I n q. Then, we can see that T is a function of Q A from A.3. J It is easy to see that Q A is distributed according to N n q q O n q,q, I q I n q because Q and A are mutually indeendent. In the same way as Q A, it is easy to see that R Q EJ is distributed according to N q O q,, I I q because T, E, and B are mutually indeendent. We can also see that Q E is distributed according to N n q O n q,, I I n q because Q and E are mutually indeendent. Since R Q EJ and B are indeendent and it is obtained that Q Q = O n q,q from the fact that I n P P P = O n,n, the following exectations can be calculated: [ E vecr Q EJvecQ E ] [ [ ]] = E E vecr Q EJvecQ E B [ ] = E J R Q vecevece I Q = O q,n q, [ E vecr Q EJvecQ A ] = O q,n q q, [ E vecq EvecQ A ] = O n q,n q q. Thus, from the roerty of the multivariate normal distribution, R Q EJ, Q E and T are mutually indeendent. B Proof of Lemma 3.4 From Lemma 3.3, it is clear that U U W n q, I, and U, U and T are mutually indeendent. Let Ũ = DD / DU U D DD /. Then, by using the roerties of Wishart distribution, we can see that Ũ W q n q, I q and Ũ and D are indeendent. Thus, from the equations 4, 5 and 6 in Lemma 3., we can derive the following exressions 7

18 of T G : T LR = log U U + JT + U T + U J U U = log U U + T + U T + U U U = log I + U U T + U T + U = log I q + T + U U U T + U = log I q + Ũ DD, T LH = tr [ U U + T + U T + U U U U U ] = tr DU U D = trũ DD, T BNP = tr [ U U U U + T + U T + U ] = tru U U U + D D = tr [ DU U D I q + DU U D ] = trdd DD + Ũ. Hence, the result in Lemma 3.4 can be derived. C Proof of Theorem 3. From the assumtion > q, we resent = Λ, O q, q, where Λ = diagλ,..., λ q λ λ... λ q 0 of which the diagonal elements of Λ are the singular values of Γ. C. Case of the LR criterion At first, we rove the case of the LR criterion. From the equation 9 in Lemma 3.4, T LR can be exressed as follows: T LR = log I q + Ũ DD, where Ũ and D = T + U are random matrices defined in Lemma 3.4 and Lemma 3.3. Let U = U, U, where U is a q q random matrix, and U is a q q random matrix. Then, DD is calculated DD = T Λ + U, U T Λ + U, U = F F + U U, C. where F = T Λ + U, and T, U, U, and Ũ are mutually indeendent. Here, it should be emhasized that we do not assume the secific Landau asymtotic order to the diagonal elements of Λ. However, by using Φ = I q + Λ, we can see that ΛΦ = O because of ΛΦ = trλ Φ = q i= λ i + λ i q <. 8

19 From these exression, we can derive the following matrix form so that we can exand matrices under the HD asymtotic framework: T LR = log I q + Ũ DD = log I q + Ũ F F + Ũ U U = log I q + Ũ U U + log I q + Ũ + U U F F = log I q + Ũ U U + log Φ + Φ F Ũ + U U F Φ + log Φ. C. Since v ii = i / t ii i d N0, n q, we have t ii = i = + Thus, the following equation is derived: / + n q i v ii v ii + O n. T = I q + V + V + O n /, C.3 where V = diagv,..., v q q and V = T diagt,..., t q q. By using C.3 and the definition of F, F Φ can be exanded as Suose that V 3 = F Φ = T ΛΦ + U Φ = ΛΦ + ΛΦ V + V ΛΦ + U Φ + O n /. C.4 n q Ũ n q I q, V 4 = U q I q. q By using V 3 and V 4, we can exand Ũ + U U as Ũ + U U = n q I q V 3 From C.4, C.5, Φ + Φ F Ũ + U U F Φ can be exanded as q V 4 + O n. C.5 Φ + Φ F Ũ + U U F Φ = I q + Λ Φ V + ΛΦ V ΛΦ + ΛΦ V ΛΦ + ΛΦ U Φ + Φ U ΛΦ n q ΛΦ V 3 ΛΦ q ΛΦ V 4 ΛΦ + O n. By using the above equation, we can exand log Φ + Φ F Ũ + U U F Φ as log Φ + Φ F Ũ + U U F Φ = trλ Φ V n q q trλ Φ V 4 + n q trλ Φ V 3 n q trλφ U + O n. C.6 9

20 Also, by using V 3, Ũ can be exanded as Ũ = Therefore, we can exand I q + Ũ U U as I q + Ũ U U = n q I q + I q V 3 + O n. C.7 n q n q + O n. n q By using the above equation, log I q + Ũ U U can be exanded as log I q + Ũ U U = q log n q From C., C.6, and C.8, T LR can be exanded as T LR = q log + log Φ + n q n q trλ Φ V 3 q + trv V 3 + q V 4 n q n q trv 3 q trv 4 + O n. C.8 n q trλ Φ V q trv 3 trλ Φ V 4 n q n q trλφ U + O n. C.9 Then, we can derive the following equation from C.9: n q T LR q log log Φ n q = trλ Φ n q V tr Λ Φ + I q V 3 n q n q + tr I q Λ Φ V 4 + trλφ U + O n /. C.0 It is easy to see that trλ Φ V trλ4 Φ 4 d N0,, tr n q n q Λ Φ + tr[ q n Λ4 + n n q n q I q V 3 q n Λ + ] d N n Iq Φ 4 tr I q Λ Φ V 4 trφ 4 trλφ U trλ Φ 4 d 0, U0 c = 0 c N 0, c 0 c N0,,, c c, 0

21 and log Φ = log I + ΓΓ, trφ = tr I + ΓΓ q, where U is the degenerate distribution, and c and c are the limit of /n and q /n, resectively. Hence, we can derive the non-null asymtotic distribution of the LR criterion by normalizing the asymtotic variance. C. Case of the LH criterion Next, we rove the case of the LH criterion. From the equation 9 in Lemma 3.4, T LH can be exressed as follows: T LH = trũ DD. By using C., we can derive the following equation: T LH = trf Ũ F + trũ U U. C. It is easy to see that + λ Λ = O because of + λ Λ = From C.3, + λ F can be exanded as q λi i= + λ q <. + λ F = + λ Λ + + λ ΛV + V + λ Λ + + λ U + O n /. C. From C.7 and C., we can exand + λ F Ũ F + λ as + λ F Ũ F + λ = + λ Λ + + λ Λ V n q + + λ ΛV + λ Λ + + λ ΛV + λ Λ + λ ΛV 3 + λ Λ n q + + λ Λ + λ U + + λ U + λ Λ + O n. C.3 From C.3, + λ trf Ũ F can be exanded as follows: + λ trf Ũ F = n q n tr + λ Λ q + tr + λ Λ V n q n q n q tr + λ Λ V 3/ 3 + n q tr + λ ΛU + O n. C.4

22 By using V 3 and V 4, Ũ U U and + λ trũ U U can be exanded as and Ũ U U = q V 4 + q I q n q + O n, + λ trũ U U = q + λ q n q q + tr + λ V 4 n q q n q V 3 V 4 q V 3 n q q n q 3/ tr + λ V 3 q n q 3/ tr + λ V 3 V 4 + O n. C.5 From C., C.4 and C.5, we can derive the following equation: + λ n q T LH n q trλ q n q n q n q = tr + λ Λ V tr + λ Λ + + λ I q V 3 n q n q n q + + λ trv 4 + tr + λ ΛU + O n /. C.6 It is easy to see that trλ V trλ4 d N0,, tr n q n q Λ + tr q n Λ4 + n n q n q I q V 3 trv 4 d q n Λ + d N n Iq trλu trλ 0, U0 c 0 N 0, c q c 0, c N0,,, c c and trλ = trγγ, trλ 4 = tr ΓΓ. From the above equations, we can derive the result of the case of the LH criterion by normalizing the asymtotic variance.

23 C.3 Case of the BNP criterion Finally, we rove the case of the BNP criterion. From the equation 9 in Lemma 3.4, T BNP can be exressed as follows: T BNP = trdd DD + Ũ. By alying the formula of inverse of the sum of matrices to DD + Ũ, the above equation can be exressed as follows: T BNP = trdd U3 tr DD U3 F Φ Φ + Φ F U3 F Φ Φ F U3 = trf F U3 + tru U U 3 tr F U3 F Φ Φ + Φ F U3 F Φ Φ F U3 F tr U U U 3 F Φ Φ + Φ F U3 F Φ Φ F U3, C.7 where U 3 = Ũ + U U. By using V 3 and V 4, U 3 can be exanded as U 3 = n q = I q + n q V 3 + n q I q n q V 3 From C.8, we can exand the following equations as q n q V 4 V 4 + O n n q tr + λ Λ V tr + λ F F U3 = tr + λ Λ + n q tr + λ Λ V 3 + tr + λ ΛU + O n, n q tr + λ U U U 3 = + λ q n q tr + λ V λ tr q. C.8 tr + λ Λ V 4 tr + λ F U3 F Φ Φ + Φ F U3 F Φ Φ F U3 F = tr + λ Λ 4 Φ + C.9 V 4 + O n, tr + λ Λ 4 Φ Λ 6 Φ 4 V n q C.0 n q tr + λ Λ 4 Φ Λ 6 Φ 4 V 3 tr + λ Λ 4 Φ Λ 6 Φ 4 V 4 + tr + λ Λ 3 Φ Λ 5 Φ 4 U + O n, n q C. 3

24 and tr U U U 3 F Φ Φ + Φ F U3 F Φ Φ F U3 = trλ Φ + tr Λ Φ Λ 4 Φ 4 V 3/ n q tr Λ Φ Λ 4 Φ 4 V 3 + tr [ n q Λ Φ + Λ 4 Φ 4 ] V 4 + tr ΛΦ Λ 3 Φ 4 U 3/ + O n. C. From C.7, C.9, C.0, C. and C., we can derive the following equation: + λ 3/ T BNP q trλ Φ n q = + λ tr Λ Φ I q Λ Φ V n + λ q n q tr Λ Φ n q n q + λ n q n q tr n q Λ Φ n q + + λ tr ΛΦ I q Λ Φ U + O n /. Λ 4 Φ 4 + I q V 3 Λ 4 Φ 4 n q I q V 4 It is easy to see that tr Λ Φ I q Λ Φ V trλ4 Φ 8 d N0,, n q n q q n trλ4 Φ 8 + n d N 0,, d c c n q tr n q tr n q n q Λ Φ n q n q Λ 4 Φ 4 + n q n q U0 c 0 c N 0, c 0 c n q I q V 3 q n trλ Φ 8 + n trφ 8 Λ Φ n q n q Λ 4 Φ 4 n q n q I q V 4 trφ 8 tr ΛΦ I q Λ Φ U trλ Φ 8, N0,, and trλ 4 Φ 8 = tr ΓΓ I + ΓΓ 4, 4

25 trλ Φ 8 = tr ΓΓ I + ΓΓ 4, trφ 8 = tr I + ΓΓ 4 q. From the above equations, we can derive the result of the case of the BNP criterion by normalizing the asymtotic variance. D Proof of Theorem 3. D. Case of the LR criterion From the equation 9 in Lemma 3.4, T LR can be exressed as follows: T LR = log I q + Ũ U U. D. By using V 4, U U can be exressed as follows: U U = U U + U U = U U + q I q + q V 4, where U = U, U and U U = q I q + q / V 4. Then, by using C.7, Ũ U U can be exanded as Ũ U U = n q q I q + U U V 3 + q V 4 + O / n /. n q Therefore, from D. and D., we can derive the following equation: T LR q log = q trv 3 n q n q q + trv 4 + tru U + O n /. It is easy to see that q trv 4 and q q χ q q q q d N0, q, tru U χ q. D. Hence, we can derive the result of the case of the LR criterion by normalizing the asymtotic variance. 5

26 D. Case of the LH criterion From the equation 9 in Lemma 3.4, T LH can be exressed as follows: T LH = tru U Ũ. From D., we can derive the following equation: n q q T LH = q n q n q trv 3 q + trv 4 + tru U + O n /. Hence, we can derive the result of the case of the LH by normalizing the asymtotic variance. D.3 Case of the BNP criterion From the equation 9 in Lemma 3.4, T BNP can be exressed as follows: T BNP = tru U U U + Ũ. By using V 3 and V 4, U U + Ũ can be exressed as U U + Ũ = n q I q + n q V 3 + q V 4 + U U, Then, we can exand U U + Ũ as U U + Ũ = n q I q V 3 Therefore, the following equation can be derived. q V 4 U U + O n. U U U U + Ũ = q n q I q V 3 q V 4 + V 4 + U U + O / n /. q q Using the above equation, we can drive the following equation. T BNP q = q n q n q trv 3 n q q + trv 4 + n q tru U + O n /. Hence, we can derive the result of the case of the BNP criterion by normalizing the asymtotic variance. References [] Chung, P. K., Zhao, Y., Liu, J. D. & Quach, B. 07. A canonical correlation analysis on the relationshi between functional fitness and health-related quality of life in older adults. Arch. Gerontol. Geriat., 68,

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28 Fuzi and its three rocessed roducts on mitochondrial growth couled with canonical correlation analysis. J. Ethnoharmacol., 53,