THE EFFICIENCY OF THE ASYMPTOTIC EXPANSION OF THE DISTRIBUTION OF THE CANONICAL VECTOR UNDER NONNORMALITY

Transcription

1 J. Japan Statist. Soc. Vol. 38 No THE EFFICIENCY OF THE ASYMPTOTIC EXPANSION OF THE DISTRIBUTION OF THE CANONICAL VECTOR UNDER NONNORMALITY Tomoya Yamada* In canonical correlation analysis, canonical vectors are used in the interpretation of the canonical variables. We are interested in the asymptotic representation of the expectation, the variance and the distribution of the canonical vector. In this study, we derive the asymptotic distribution of the canonical vector under nonnormality. To obtain the asymptotic expansion of the canonical vector, we use a perturbation method. In addition, as an example, we show the asymptotic distribution with an elliptical population. Key words and phrases: Asymptotic distribution, canonical correlation analysis, canonical vector, elliptical population, perturbation method, small sample. 1. Introduction In multivariate statistical analysis, the distributions of latent roots and latent vectors of certain symmetric matrices constructed from the sample covariance matrix are important in some cases and have been studied by many authors. These studies can be used as the basis for the canonical correlation analysis, which is an approach that characterizes the correlation structure between two sets of variables. Considering the distribution of the canonical correlation with the assumption of the multivariate normal population, asymptotic expansions of the distributions were studied by Sugiura (1976), Fuikoshi (1977, 1978), Muirhead (1978) and others. The distributions of a function of latent roots of the sample covariance matrix in nonnormal populations were studied by Fuikoshi (1980), Muirhead and Waternaux (1980), Fang and Krishinaiah (1982), Siotani et al. (1985), Seo et al. (1994) and others. The distribution of the canonical vector was studied by Eaton and Tyler (1994), Boik (1998), Anderson (1999), Taskinen et al. (2006) and others. This paper deals with the asymptotic expansion of the canonical vector under nonnormality. Let us denote x =(x 1, x 2 ) as p + q dimensional variables with mean µ and covariance matrix ( ) ( ) x 1 Σ 11 Σ 12 (1.1) Var = Σ =, x 2 Σ 21 Σ 22 where x 1 and x 2 are the vectors of the p and q components respectively, and Σ 11 is a p p matrix. With no loss of generality, we may assume that p q. We Received October 19, Revised April 7, Accepted June 13, *Faculty of Economics, Sapporo Gakuin University, 11, Bunkyodai, Ebetsu, Hokkaido , Japan.

2 452TOMOYA YAMADA denote ρ i as the i-th canonical correlation, and α i and β i as the i-th canonical vectors of x 1 and x 2, respectively, which satisfies (1.2) (1.3) Σ 12 Σ 1 22 Σ 21α i = ρ 2 i Σ 11 α i, α iσ 11 α = δ i, Σ 21 Σ 1 11 Σ 12β i = ρ 2 i Σ 22 β i, β iσ 22 β = δ i, where ρ 2 1 ρ2 p and δ i is the Kronecker s delta. Given a sample x k = (x 1k, x 2k ) of n i.i.d. observations from x, let S be the sample covariance divided as ( ) S 11 S 12 (1.4) S =, S 21 S 22 the i-th sample canonical correlation r i and the i-th sample canonical vectors a i and b i of x 1 and x 2, respectively, satisfy (1.5) (1.6) S 12 S a i = ri 2 S 11 a i, a is 11 a = δ i, S 21 S b i = ri 2 S 22 b i, b is 11 b = δ i, where r1 2 r2 p. We also define α i and a i as the th coefficient of α i and a i respectively. Studies on the distribution of the canonical vector have been cited earlier. Boik (1998) obtained the limiting distribution of n(a i α i ) using an Edgeworth expansion. Anderson (1999) discussed the limiting distribution of n(a i α i ) with multivariate normal population. In addition, Taskinen et al. (2006) obtained the limiting distribution of n(a i α i ) with an elliptical population. However, they obtained only the first order asymptotic distribution of the canonical vector, i.e. the limiting distribution of n(a i α i ). We know from numerical study that the asymptotic bias of the canonical correlation n(ri 2 ρ2 i ) is large in a small sample, and this effects the accuracy of the asymptotic distribution. Therefore, Seo et al. (1994) gave the second order expansion of the canonical correlation. It seems that the asymptotic bias of the canonical vector would also be large in a small sample. Therefore, we need to evaluate the asymptotic bias and higher-order expansion of the distribution. Ogasawara (2007) studied the asymptotic expansion of the distribution of typical estimators in canonical correlation analysis under nonnormality, and obtained the asymptotic expansion of estimators up to O p (n 1 ) using an Edgeworth expansion. These estimators involve elements of canonical vectors. In this paper, we derive the asymptotic expansion of the canonical vector using a perturbation method (see, e.g. Sugiura (1973), Siotani et al. (1985)), obtain the asymptotic distribution of the canonical vector using Kaplan s tensor notations (Kaplan (1952)), and compare the obtained results with the simulation results. To obtain the distribution of y (i) = n(a i α i ) n(h(s) h(σ)), we consider the following assumptions:

3 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 453 (i) For some integer s 3, all the derivatives of h, of order s or less, are continuous in the neighborhood of S = Σ, and the 2s-th order moments of x exist. (ii) The (p + q)(p + q +1)/2 dimensional random vector z =(x 2 1,...,x 1x p+q, x 2 2,...,x p+qx p+q 1,x 2 p+q) satisfies Cramer s condition lim sup E{exp(it z )} < 1, t where denotes the Euclidean norm. Then it is well known (see, e.g. Bhattacharya and Ghosh (1978), Ichikawa and Konishi (2002)) that the distribution of y (i) can be expanded up to order n (s 2)/2. Therefore, in order to find an asymptotic expansion of y (i) = n(a i α i ), we need to evaluate only the first few cumulants of y (i) in expanded forms. Here, we define µ(y,...,y k ) and κ(y,...,y k ) to be the moment and the cumulant of the vector y transformed from x by the inner nonsingular linear transformation, respectively, where y denotes the -th element of y. Also,,...,y(i) p ) by µ(y (i) k 1,...,y (i) k )=µ (k 1,...,k ) (y (i) ) and κ(y (i) k 1,...,y (i) k )=κ (k 1,...,k ) (y (i) ) respectively. In general, for 2, κ (k 1,...,k ) (y (i) )=O(n ( 2)/2 ). For example, the first three cumulants and the relationships between cumulants and moments of y (i) take the form of we shall denote the -th moments and cumulants of y (i) =(y (i) 1 (1.7) (1.8) (1.9) κ () (y (i) )=n 1/2 d (i) + O(n 3/2 )=µ () (y (i) ), κ (,k) (y (i) )=σ (i) k + O(n 1 )=µ (,k) (y (i) ) µ () (y (i) )µ (k) (y (i) ), κ (,k,l) (y (i) )=n 1/2 d (i) kl + O(n 3/2 ) (3) = µ (,k,l) (y (i) ) µ () (y (i) )µ (k,l) (y (i) ) µ () (y (i) )µ (k) (y (i) )µ (l) (y (i) ), respectively. Then it is known that sup P (y (i) (1.10) B) ψ(x )dx = o(n 1/2 ), B B where the supremum is over all Borel sets B, ψ(x )=φ(x; Σ (i) ) 1+ 1 d (i) n H() (x )+ 1 (1.11) d (i) 6 kl H(,k,l) (x ), φ(x ; Σ (i) ) is the density function of N p (0, Σ (i) ), Σ (i) =(σ (i) k ), and,k,l H (i 1,...,i ) (x )φ(x ; Σ (i) )=( 1) ( / x i1 ) ( / x i )φ(x ; Σ (i) ).

4 454 TOMOYA YAMADA Using the above result, we obtain the asymptotic expansion of the distribution of y (i). In Section 2, we first assume that ( ) ( ) Σ Σ 11 Σ 12 I p P (1.12) = Σ 21 Σ = 22 P, I q where P = [diag(ρ 1,...,ρ p ) : 0] (ρ 1 ρ p 0), and give the first three moments of the canonical vector using the perturbation method. Next, we extend the general covariance matrix Σ given by (1.1), obtain the cumulants, and derive the higher-order expansion of the asymptotic distribution of n(a i α i ) in Section 3. In addition, as an example of a nonnormal distribution, we give the asymptotic distribution under an elliptical population, and compare the results of Taskinen et al. (2006) and Anderson (1999) in Section 4. Lastly, the simulation results are presented in Section Asymptotic expansion of the first three moments of the canonical vector when Σ = Σ We first assume that x =(x1, x2 ) is a p+q variate with covariance matrix Σ. Let S be the sample covariance of x divided as ( ) S S11 = S 12 (2.1) S21 S 22, η i =(0,...,0, 1, 0,...,0) be the ith canonical vectors of x1, and e i be the i-th sample canonical vectors of x1, respectively, which satisfies (2.2) (2.3) Σ 12Σ 22 1 Σ 21η i = ρ 2 i Σ 11η i, S 12S 22 1 S 21e i = r 2 i S 11e i. Furthermore, (2.2) and (2.3) are equivalent to (2.4) (2.5) Σ 1/2 11 Σ 12Σ 1 22 Σ 21Σ 1/2 11 η i = ρ 2 i η i, S 11 1/2 S 12S 22 1 S 21S 11 1/2 e i = r 2 i e i, where η i = Σ 11 1/2 η i and e i = S11 1/2 ei. In this section, we derive the first three moments of y (i) = n(e i η i ). Let U =(u i )= ( ) n(s Σ V Z )= Z, W and denote the cumulant of x =(x 1,...,x p+q) by κ(x,...,x k )=κ k and κ(x s,...,x k t )=κ k s t, which were suggested by Kaplan (1952). Also, we denote (n),...,k and (n) a sum of n similar terms, determined by suitable permutations of the indices,..., k and all indices, respectively. Then, we have the following lemma.

5 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 455 Lemma 1. Let x be a vector of the p+q variate with finite sixth moments. If ρ 2 i is simple, the first three moments of y (i) are expressed as (2.6) (2.7) (2.8) where d (i) = µ () (y (i) )=n 1/2 d (i) + O(n 3/2 ), µ (,k) (y (i) )=σ (i) 2 k + O(n 1 ), µ (,k,l) (y (i) )=n 1/2 d (i) kl + O(n 3/2 ), { (i) f 1 ( = i) 2 ( i), σ(i) k 2 = 3 ( = k = i) 2 i 4 ( = k i) 1 2 i 5 ( k = i) i ki 6k ( k i) 1 8 g(i) ( = k = l = i), d (i) kl = 1 4 i 2 +2g(i) 6 + g(i) 7 ( k = l = i) 1 (2) 2 i ki 3k + g(i) 8k + 9k (, k i, l = i) (3) i ki li 4kl + 10kl (, k, l i),k,l,k, i =(ρ 2 i ρ 2 ) 1, 1 = ( ki 2 12k 2 ki 13k ), 2 k i = i 22 2 i 23 + i k i, 3 = 1 4 (κi 4 +2), 4 5 ki 26k, k i, 2 ki 24k + 3 i 25 = ρ4 i (κ i 22 +1)+2ρ3 i ρ (κ ii+p+p ρ i ρ )+ρ 2 i ρ 2 (κ i+p+p 22 +1)+ρ 2 i (κ i+p 22 +1) 2ρ 2 i ρ (κ i+p ρ ) 2ρ i ρ 2 (κ +pii+p ρ i )+ρ 2 (κ i+p 22 +1) 2ρ 3 i (κ ii+p ρ i ) 2ρ 2 i ρ (κ i+pi ρ )+2ρ i ρ (κ ii+p+p ρ i ρ ), = κi+p 31 ρ + κ ii+p 211 ρ i κ ii+p+p 211 ρ i ρ κ i 31 ρ2 i,

6 456 TOMOYA YAMADA 6k = κi+pk 211 ρ 2 i 2κ ii+pk 1111 ρ 3 i + κ ik 211 ρ4 i + κ ii+p+pk 1111 ρ i ρ (ρ 2 i +1) κ i+pk 211 ρ 2 i ρ κ i+p+pk 211 ρ 2 i ρ + κ ii+pk+p 1111 ρ i ρ k (ρ 2 i +1) κ ik+p 211 ρ 2 i ρ k κ i+pk+p 211 ρ 2 i ρ k + κ i+pk+p 211 ρ ρ k 2κ ii+p+pk+p 1111 ρ i ρ ρ k + κ i+p+pk+p 211 ρ 2 i ρ ρ k, 11 = κi k i (κ ik 22 +1), 12k =(κikk+p ρ k )ρ k +(κ kii+p ρ i )ρ i (κ ii+pkk+p ρ i ρ k )ρ i ρ k 1 2 (κki )(ρ 2 i + ρ 2 k ), 13k = ρ2 k (κik+p 22 +1)+2ρ k ρ i (κ ii+pkk+p ρ i ρ k ) 2ρ i ρ 2 k (κk+pii+p ρ i ) + ρ 2 i (κ i+pk 22 +1) ρ k (ρ 2 i + ρ 2 k )(κikk+p ρ k ) 2ρ 2 i ρ k (κ i+pkk+p ρ k ) ρ i (ρ 2 i + ρ 2 k )(κkii+p ρ i )+ρ 2 i ρ 2 k (κi+pk+p 22 +1) + ρ i ρ k (ρ 2 i + ρ 2 k )(κii+pkk+p ρ i ρ k )+ 1 4 (ρ2 i + ρ 2 k )2 (κ ik 22 +1), 21 = κi 31 + κi 31 + k i, κ ki 211, 22 = ρ iρ (κ i+p+p 31 + κ +pi+p 31 + ρ (κ +pi q k ρ i (2κ ii+p κ i+p 31 + q k i, κ k+pi+p+p 211 ) κ k+pi+p 211 ) ρ i (κ i+p 31 + k q k i κ k+pi+p 211 )+ q k=1 κ ki+p 211 )+ρ (2κ i+p κ i+p 31 + (ρ 2 i κ i+pi ρ i ρ (κ ii+p+p κ i+p+p 211 )+ρ 2 κ +pi + ρ k (2κ ikk+p 1111 ρ i κ i+pkk+p 1111 ρ κ i+pkk+p 1111 ) + k i, 211 ) κ k+pi 211 k i κ ki+p 211 ) ( 5 8 ρ2 i + 3 ) ( 8 ρ κ i ρ2 i + 5 ) ( 8 ρ κ i ρ2 i ρ + 1 ) 4 ρ2 k κ ki 211 k i, 1 2 (ρ κ i+p ρ i κ i+p 31 ρ i ρ κ i+p+p (ρ2 i + ρ 2 )κ i 31 ), 23 =2ρ iρ κ ii+p+p ρ 2 i κ i+pi 211 2ρ 2 i ρ κ i+pi+p 211 ρ i (ρ 2 i + ρ 2 )κ ii+p 211 ρ 2 i ρ (κ i+pi+p κ i+p 31 ) ρ 3 i (κ i+p 31 + κ ii+p 211 )+ρ3 i ρ (κ i+p+p 31 + κ ii+p+p 211 ) ρ2 i (ρ 2 i + ρ 2 )(κ i+pi κ i 31 ), 24k = ρ kρ κ i+pkk+p 1111 ρ 2 k ρ κ k+pi+p ρ i ρ k κ i+pkk+p ρ i ρ k ρ κ i+p+pkk+p 1111

7 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR ρ i ρ 2 k ρ κ k+pi+p+p ρ 2 k κk+pi ρ i ρ κ ki+p+p 211 ρ i ρ 2 k κk+pi+p (ρ2 k + ρ2 )(ρ k κ ikk+p ρ i κ ki+p 211 ρ i ρ k κ i+pkk+p 1111 ) 1 2 (ρ2 i + ρ 2 k )(ρ kκ ikk+p ρ κ ki+p 211 ρ k ρ κ i+pkk+p 1111 ) (ρ2 i + ρ 2 k )(ρ2 k + ρ2 )κ ki 211, 25 = ρ iρ κ i+p+p 211 ρ i ρ 2 κ +pi+p 211 ρ i ρ 2 κ i+p 31 + ρ 2 κ +pi 211 ρ 3 κ +pi 31 ρ 3 κ i+p 211 ρ i ρ 2 κ +pi+p ρ i ρ 3 κ +pi+p 31 + ρ i ρ 3 κ i+p+p (ρ (ρ 2 i + ρ 2 )κ i+p 211 ρ i ρ (ρ 2 i + ρ 2 )κ +pi 211 (ρ 2 i + ρ 2 )ρ 2 κ i 31 ), 26k = ρ kκ ikk+p ρ i κ ki+p 211 ρ i ρ k κ i+pkk+p (ρ2 i + ρ 2 k )κki 211. Since the expression of g is complicated, it is provided in Appendix A. Proof. We first give the asymptotic expansion of ei, which is the latent vector of R = S11 1/2 MS11 1/2, in the same manner as Seo et al. (1994), where M = S12 S 22 1 S21. Let r2 1 r2 p be the latent roots of the matrix R. Then M can be expanded as where Furthermore, M = Ω + n 1/2 M (1) + n 1 M (2) + O p (n 3/2 ), Ω = diag(ρ 2 1,,ρ 2 p), M (1) = ZP + PZ PWP, M (2) = PW 2 P PWZ ZWP + ZZ. (2.9) R = S 11 1/2 MS 11 1/2 = Ω + n 1/2 R (1) + n 1 R (2) + O p (n 3/2 ), where R (1) =(r (1) i )=M (1) 1 2 ΩV 1 2 V Ω, R (2) =(r (2) i )=M (2) 1 2 VM (1) 1 2 M (1) V V 2 Ω ΩV V ΩV. The perturbation method is a useful method to obtain the asymptotic expansion of the latent roots and the latent vectors. Since ri 2 and ei are the latent root and the latent vector of the characteristic equation Rei = ri 2e i, respectively, it is known (see, e.g. Siotani et al. (1985) pp. 176, Chapter 4.6) that (2.10) ri 2 = ρ 2 c 2 ki i + c ii ρ 2 k ρ2 i k i + O p (n 3/2 ),

8 458 TOMOYA YAMADA (2.11) (2.12) c i e i( i) = ρ 2 ρ2 i e ii =1 1 2 k i c iic i (ρ 2 ρ2 i )2 + 1 ρ 2 ρ2 i c 2 ki (ρ 2 k ρ2 i )2 + O p(n 3/2 ), k i c ki c k ρ 2 k ρ2 i + O p (n 3/2 ), where R =(ri )=S 11 1/2 S12 S 22 1 S21 S 11 1/2, e i is the -th coefficient of e i and c i = ri δ iρ 2 i. Using (2.11), (2.12) and (2.9), the asymptotic expansion of can be written as e i (2.13) e i = η i + n 1/2 s 1i + n 1 s 2i + O p (n 3/2 ), where s 1i =(s 1i1,...,s 1ip ), s 2i =(s 2i1,...,s 2ip ), and r(1) i s 1i = ρ 2 ( i) ρ2, i 0 ( = i) r(2) i ρ 2 ρ2 i s 2i = 1 2 k i r(1) ii r (1) i (ρ ρ2 i )2 ρ 2 ρ2 i k i r (1) ki r(1) k ρ 2 k ρ2 i ( i) r (1) 2 ki (ρ 2 k ( = i) ρ2 i )2 Finally, the asymptotic expansion of e i, which is the latent vector of (2.3), is obtained as follows: e i = S 1/2 11 ei = η i + n (s 1/2 1i 1 ) 2 V η i + n (s 1 2i 12 Vs 1i + 38 ) V 2 η i + O p (n 3/2 ), e i ( i) =n (s 1/2 1i 1 ) 2 v i + n 1 s 2i O p (n 3/2 ), e ii =1 1 2 n 1/2 v ii + n 1 s 2ii vik k=1 v k v ik 1 2 k=1. v k s 1ik k i v ik s 1ik + O p (n 3/2 ), where e i is the -th coefficient of e i. Then the following formulas are useful for calculating the moments (see, e.g. Kaplan (1952), Kendall and Stuart (1969)): k i E(u ab )=0, E(u ab u cd )=m(ab, cd) n 1 κ abcd + O(n 2 ), E(u ab u cd u ef )=n 1/2 m(ab, cd, ef)+o(n 3/2 ),

9 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 459 E(u ab u cd u ef u gh )= (3) m(ab, cd)m(ef, gh)+o(n 1 ), where (2.14) (2.15) m(ab, cd) =κ abcd + κ ac κ bd + κ ad κ bc, (12) m(i, kl, st) =κ iklst + κ iks κ lt + κ iks κ kt + κ ik κ s κ lt. (4) (8) Using the above formulas and κ a 2 = κa+p 2 =1,κ aa+p = ρ a and κ ab =0(b a, a+p) in our setting, the first three moments can be calculated. 3. Main result We extend to the general covariance matrix Σ, and give the asymptotic distribution of the canonical vector by considering the relation between a i and e i, and between α i and η i, respectively. Let A =(α 1,...,α p ), B =(β 1,...,β q ) and α i =(α i1,...,α ip ). Then we state the theorem that relates to the asymptotic distribution of the canonical vector. Theorem 1. Let x be a vector of the p+q variate with finite sixth moments, and a i and α i be the i-th latent vectors that are calculated using (1.2) and (1.5), respectively. If ρ 2 i is simple under the assumption (i) and (ii), then the cumulative distribution function of y (i) = n(a i α i ) can be expanded for large n as (1.10) where (3.1) (3.2) (3.3) d (i) =(d (i) 1,...,d(i) p ) = A d (i), Σ (i) =(σ (i) k )=A Σ (i) A, d (i) kl = (3) α s α tk α ul d (i) stu (i) d s,t,u σ(i) kl, d (i) = (d (i) 1,...,d (i) p ), Σ (i), and d (i) kl are given in (2.6), (2.7) and (2.8), respectively. Remark 1. The expectation of y (i) is written as E(y (i) ) = n 1/2 d (i) + O(n 3/2 ). In other words, the expectation of a i is given by E(a i )=α i + n 1 d (i) + O(n 2 ). Remark 2. Boik (1998) described the limiting distribution of n(a i α i ) using an Edgeworth expansion. Σ (i) in this theorem presents using Kaplan s tensor notations. In addition, this theorem includes the asymptotic expansion of the distribution of y (i) for the small sample.

10 460 TOMOYA YAMADA Proof. From (1.10), we may obtain the first three cumulants of y (i) = n(ai α i ) to prove this theorem. For the proof, we first consider the relation between α i and η i. There exist orthogonal matrices H and Q such that (3.4) Σ 1/2 11 Σ 12 Σ 1/2 22 = H PQ, where P is defined in (1.12). Substituting A = H Σ 1/2 11 and B = QΣ 1/2 22,we obtain AΣ 11 A = I p, BΣ 22 B = I q, AΣ 12 B = P, where I p is the p dimensional identity matrix. By the transformation ( ) x A 0 = Lx where L =, 0 B we may assume that x is p + q variables with covariance matrix (1.12). In this case, (1.2) and (1.5) become (2.2) and (2.3) respectively. Hence, we obtain the first three moments of y (i) as µ () (y (i) )=n 1/2 µ (,k) (y (i) )= s,t µ (,k,l) (y (i) )=n 1/2 s α s d (i) s + O(n 3/2 ), α s α tk σ (i) 2 st + O(n 1 ), s,t,u α s α tk α ul d (i) stu + O(n 3/2 ), by the transformation a i = A e i, and also obtain the first three cumulants of y (i) by (1.7) (1.9). Furthermore, the asymptotic distribution of a i marginal distribution of n(a i α i ) as follows. is given by considering a Corollary 1. Let a i =(a 1i,...,a pi ) and α i =(α i1,...,α ip ). Then, under the same conditions of theorem 1, the cumulative distribution function of n(ai α i ) can be expanded for large n as (3.5) n(ai α i ) Pr y σ (i) 1/2 =Φ(y) n 1/2 d (i) σ (i) 1/2 Φ(1) (y)+ d(i) 6σ (i) 3/2 Φ(3) (y) + o(n 1/2 ), where Φ(y) is the cumulative distribution function of N(0, 1), and Φ (k) (y) is the k-th derivative of Φ(y).

11 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 461 Remark 3. We can consider the Cornish-Fisher expansion by corollary 1. An asymptotic expansion for the ξ percentile of the coefficient of y (i) n(a i α i )/σ (i) 1/2 can be obtained by (3.6) y ξ = u ξ + 1 (d (i) n d(i) (u2 ξ 1)) + o(n 1/2 ), where u ξ is the upper ξ percentile of N(0, 1). Remark 4. Ogasawara (2007) described the asymptotic expansion of the distribution of the coefficient of the canonical vector using the Edgeworth expansion. This corollary presents using Kaplan s tensor notations. 4. An example of the asymptotic distribution of the canonical vector with an elliptical population As an example of the asymptotic distribution of the canonical vector, we consider the case where x is distributed as an elliptical distribution EL p+q (µ, Λ) (see, e.g. Muirhead (1982)). Then its density function and characteristic function are of the form f(x ; µ, Λ) =c p+q Λ 1/2 g((x µ) Λ(x µ)), φ(t) = exp{it µ}ϑ(t Λt), for some function g and ϑ, respectively, where c p+q is a positive constant. If there exist, E[x ]=µ, Σ Cov[x ]= 2ϑ (0)Λ. Let the kurtosis parameter be κ {ϑ (2) (0)/(ϑ (0)) 2 } 1 and ϕ {ϑ (3) (0)/(ϑ (0)) 3 } 1. Then we obtain the asymptotic distribution of n(a i α i ) under an elliptical population as follows. Lemma 2. Let x be a vector of the p + q variate that is distributed as an elliptical distribution EL p+q (µ, Λ), and the original covariance matrix be of the form (1.1). Then the cumulative distribution function of y (i) = n(a i α i ) can be expanded for large n as (1.10), where d (i) =(d (i) 1,...,d(i) p ), d (i) 3 = 8 (3κ +2)+1 8 (κ +1) c 1im α i, where Σ (i) = 1 4 (3κ +2)α iα i +(κ +1) d (i) kl = d(i) iii α iα ik α il + (3) s i i,s c 2im α m α m, d (i) (3),k,l iss α iα sk α sl d (i) σ(i) kl, d (i) iii = 1 (15ϕ 9κ +8) (3κ +2) (3κ + 2)(κ +1) c 1im,

12 462TOMOYA YAMADA d (i) iss = si(1 ρ 2 i ){(ρ 2 i 4ρ 2 i ρ 2 s +3ρ 2 s)ϕ c 3is κ +2ρ 2 s(1 ρ 2 i )} c 2is(29κ + 14)(κ +1)+(κ +1) 2 3 si(1 ρ 2 i ) 2 ρ 2 s(ρ 4 s +3ρ 2 i ρ 2 s 4ρ 2 i ) c 2 2is c 2is c 1i =2 i c 3i 2 ic 4i +3=4 2 i(1 ρ 2 i )(2ρ 4 ρ 2 i ρ 2 ), c 2i = 1 4 (2 ic 3i + 2 ic 4i +1)= 2 i(1 ρ 2 i )c 3i, c 1im c 3i = ρ 2 i + ρ 2 2ρ 2 i ρ 2, c 4i =4(ρ 2 i ρ 2 ρ 2 i ρ 2 )(ρ 2 i + ρ 2 1)+(ρ 2 i + ρ 2 ) 2., Remark 5. Taskinen et al. (2006) described the asymptotic distribution of y (i) (e.g. Corollary 1). In their description, ASV (Ĉ11; F 0 ) = 3κ + 2 and ASV (Ĉ12; F 0 )=κ +1ifĈ is a sample covariance matrix; therefore, Σ(i) given in lemma 2 coincides completely with Taskinen et al. (2006). In addition, in the normal population, that is κ =0,Σ (i) also coincides with Anderson (1999). Proof. In an elliptical population, (3) (15) κ abcd = κ σab σ cd, κ abcdef =(ϕ 3κ) σab σ cd σ ef. Therefore, the first three moments are obtained from lemma 1 by setting σ ii =1, σ ii+p = ρ i, σ i = κ ik = 0. In particular, we note that for i, E(r (1) i v i) = 1 2 c 3i + O(n 1 ), E(r (1) 2 1 i )= 4 c 4i + O(n 1 ), where r (1) is defined in (2.9). Then we have the first three moments of y (i) as follows. i 3 d (i) = 8 (3κ +2)+1 8 (κ +1) c 1im ( = i), 0 ( i) 1 (3κ +2) ( = k = i) σ (i) 2 4 k = (κ +1)c 2i ( = k i), 0 (Others) 1 (15ϕ 9κ +8)+3g(i) 5 ( = k = l = i) 8 d (i) kl = i 3 + g(i) 8 +2g(i) 9 ( = k i, l = i) 0 (Others),

13 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 463 where 3 =(1 ρ2 i ){(ρ 2 i 4ρ 2 i ρ 2 +3ρ 2 )ϕ c 3i κ +2ρ 2 (1 ρ 2 i )}, 5 = 9 32 (3κ +2) (3κ + 2)(κ +1) c 1im, 8 = 3 8 c 2i(3κ + 2)(κ +1) {c 2 2i 2 i(1 ρ 2 i ) 2 ρ 4 }(κ +1) (κ +1)2 c 2i c 1im, 9 = 1 4 c 2i(5κ + 2)(κ +1) 2 3 iρ 2 i ρ 2 (1 ρ 2 i ) 2 (1 ρ 2 )(κ +1) 2. By (3.1) (3.3), we have the first three cumulants of y (i) ; therefore the proof is complete. Lastly we give the asymptotic distribution of a i under an elliptical population. Corollary 2. Let x be a vector of the p + q variate that is distributed as an elliptical distribution EL p+q (µ, Λ), and the original covariance matrix be of the form (1.1). Then the cumulative distribution function of n(a i α i ) can be expanded for large n as (3.5) where d (i) = 3 8 {(p +2)κ + p +1} (κ +1) c 1im α i, σ (i) = 1 4 (3κ +2)α2 i +(κ +1) d (i) =3 s i α i α 2 sd (i) iss c 2im αs, 2 + α3 id (i) iii 3d (i) σ(i). 5. Simulation study To study the efficiency of the asymptotic expression of the first three cumulants, and the accuracy of approximation of sample canonical vectors under an elliptical population, we compared these values with the previous results. A Monte Carlo simulation was performed for n = 50, 100 and 200, and each of the following three elliptical populations: the multivariate normal (M.N.), the multivariate t with degrees of freedom ν =12(M.T 12 ), and the multivariate t with degrees of freedom ν =9(M.T 9 ). Note that κ and ϕ for normal, and multivariate t are given by κ M.N. κ =0, ϕ M.N. ϕ =0,

14 464 TOMOYA YAMADA κ M.T κ f = 2 ν 4, ϕ M.T ϕ f = (ν 2) 2 (ν 4)(ν 6) 1. In our setting, κ 12 =1/4, κ 9 =2/5, ϕ 12 =13/12, and ϕ 9 =34/15. The simulation results were based on 10,000 replications for each sample size, distribution and situation. Using corollary 2 we study the accuracy of the coefficient of the canonical vectors in the following two cases: (a) ρ 2 1 =.8, ρ2 2 =.2, α 1 =(1, 0) and α 2 = (0, 1), and (b) ρ 2 1 =.8, ρ2 2 =.2, ρ2 3 =.19, α 1 =(1, 0, 0), α 2 =(0, 1, 0) and α 3 =(0, 0, 1). We note that d (i) an asymptotic bias of order n 1 only when = i; therefore we only evaluate a 11 and a 22. Also, under our setting, d (i) d (i) = d(i) iii α3 i 3d (i) σ(i) and d (i) depend only on α i, so that there is = 1 8 is simply written as (15ϕ 9κ +8) (3κ +2)2. This tells us that d (i) does not depend on p, q and ρ i, and that d (i) tends to be large as κ tends to be large. However, this property does not hold in the general canonical vector since d (i) includes d(i) iss, which depends on p and ρ i. The previous result is the limiting distribution of y (i) ; therefore we compare the accuracy between u ξ and y ξ, where u ξ is the upper ξ percentile of N(0, 1), and y ξ is defined by (3.6). The accuracy of the asymptotic distribution in case (a) is listed in Table 1. Since y (i) is adusted by dividing by σ (i) 1/2, the accuracy gives the same performance regardless of the distribution. In addition, we calculate the mean(mean), the variance(var) and the skewness(skew), and compare with E = α ii + d(i) i n, V = σ(i) ii n, γ = d(i) iii n 2 V 3/2. These simulation results are listed in Table 2. In each sample and distribution, it can be seen from Table 1 that y ξ performs better than u ξ. In particular, if κ is large, which means that the tail of the density function is heavy, u ξ would perform worse. This causes the asymptotic bias and skewness to be large if κ tends to be large, from Table 2. In addition, these values would be large if p or q tend to be large. From Table 2, the simulation values of the mean and the skewness are close to our expected values; therefore, y ξ performs well because our result includes the asymptotic bias and skewness. In addition, we note that u ξ in a 22 is worse; however, y ξ still performs well. The case of (b) gives the accuracy and simulation result when the second and third canonical correlation are nearly the same. The problem with canonical correlation analysis is that it cannot be estimated if multiplicity exists. Therefore, we assume that the canonical correlation has no multiplicity. Thus, only the results of the first canonical vector are shown in Tables 3 and 4. Our results show better performance even if the second and third canonical correlation are nearly the same.

15 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 465 Table 1. The accuracy of the asymptotic distribution in the case (a). α a 11 M.N. n =50 y α u α y α u α y α u α M.T 12 n =50 y α u α y α u α y α u α M.T 9 n =50 y α u α y α u α y α u α a 22 M.N. n =50 y α u α y α u α y α u α M.T 12 n =50 y α u α y α u α y α u α M.T 9 n =50 y α u α y α u α y α u α This simulation is designed as ρ 2 1 =0.8, ρ2 2 =0.2, α 1 =(1, 0) and α 2 =(0, 1). M.N. denotes the multivariate normal, M.T f denotes the multivariate t with the degree of freedom f.

16 466 TOMOYA YAMADA Table 2. Simulation results and approximations to mean, variance and skewness in the case (a). M.N. M.T 12 M.T a 11 mean E var V skew γ a 22 mean E var V skew γ Table 3. The accuracy of the asymptotic distribution in the case (b). α M.N. n =50 y α u α y α u α y α u α M.T 12 n =50 y α u α y α u α y α u α M.T 9 n =50 y α u α y α u α y α u α This simulation is designed as ρ 2 1 = 0.8, ρ2 2 = 0.2,ρ2 3 = 0.19, α 1 = (1, 0, 0), α 2 = (0, 1, 0), and α 3 =(0, 0, 1). M.N. denotes the multivariate normal, M.T f denotes the multivariate t with the degree of freedom f.

17 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 467 Table 4. Simulation results and approximations to mean, variance and skewness in the case (b). M.N. M.T 12 M.T mean E var V skew γ In some of the simulation results (data not shown), y ξ performs better than u ξ in all cases; nevertheless the accuracy of a i does not perform well in the case where ρ 1 is small, ρ i is close to another canonical correlation, or p is large. We will work on improving these cases in the future. 6. Conclusion In this paper, we derived the asymptotic distribution of the canonical vector, and compared it with a Monte Carlo simulation. The following results are obtained. Our main result is the asymptotic distribution of the canonical vector under nonnormality. With no conditions on the distribution, our result was a complicated expression because it required many cumulants; however, for the elliptical distribution, it can be easily solved. To check the accuracy of the approximation, we performed a Monte Carlo simulation and compared it with our results. The approximation results were in agreement with the simulation results. We assumed that the population canonical correlation is simple, because we had to solve the characteristic equation of the nonsingular matrix. It is necessary to use theoretical formulas for multiple roots, which will be addressed in a future study. Appendix A: Expression of the third moment An expression of the third moment is more complicated than the first and second moments. This is caused by the complicated asymptotic expansion of y (i) = n(e i η i ). To obtain the third moment, we denote some formulas as follows. Let Z 1 = Z and Z 2 = Z and E(s (1) ab s(2) cd )=m(s(1) ab,s(2) cd )+O(n 1 ), E(s (1) ab s(2) cd s(3) ef )=m(s(1) ab,s(2) cd s(3) ef )+O(n 1 ), where s (1), s (2) and s (3) denote Z 1, Z 2, W or V and s (i) =(s (2) k ). We note that m(s (1) ab,s(2) cd ) and m(s(1) ab,s(2) cd,s(3) ef ) are also expressed by κ k s t if we use the formula of (2.14) and (2.15). In addition, we define θ(s (1), s (2) )=m(s (1) ab,s(2) cd ),

18 468 TOMOYA YAMADA γ 1 (s (1), s (2), s (3) )=m(s (1) ii s(2) ii s(3) ki ), γ 2 (s (1), s (2), s (3) )=m(s (1) ii s(2) i s(3) ki ), γ 3 (s (1), s (2), s (3) )=m(s (1) i s(2) ki s(3) li ). Then, the third moment of y (i) is given as 1 8 g(i) ( = k = l = i) where d (i) kl = 1 4 i 2 +2g(i) 6 + g(i) 7 ( k = l = i) 1 (2) 2 i ki 3k + g(i) 8k + 9k (, k i, l = i) (3) i ki li 4kl + 10kl (, k, l i),k,l 1 = κ i 6 +12κ i 4 +4(κ i 3) 2 +8, 2 = γ 1(V, V, V )ρ 2 i + γ 1 (V, V, W )ρ i ρ γ 1 (V, V, Z 2 )ρ i γ 1 (V, V, Z 1 )ρ, 3k = ρ2 i 31k ρ i 32k + g(i) 33k, 4kl = ρ3 i 41kl ρ2 i 42kl +3ρ i 43kl g(i) 44kl {, 5 = 1 3 t 5 (ii, ii, mi, mi) ( mi t 4 (ii, ii, mi, mi) 2 2 mit 3 (ii, ii, mi, mi)), { 6 = 1 3 h 6 (ii, i, mi, mi) ( mi h 5 (ii, i, mi, mi) 2 2 mih 4 (ii, i, mi, mi)), { 7 = 1 4 i q h 9 (ii, ii, im, m) ( 1 i 8 ρ2 i 3 8 ρ2 + 1 ) 4 ρ2 m t 5 (ii, ii, m, mi) i (t 4 (ii, ii, m, mi)+t 4 (ii, ii, im, m)),k,

19 8k = 3 8 9k = 1 2 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR i mi t 3 (ii, ii, m, mi) 2 it 3 (ii, ii, i, ii) t 5 (ii, ii, m, mi) 1 2 h 3 (i, ki, mi, mi) { mi t 4 (ii, ii, m, mi), ( mi h 2 (i, ki, mi, mi) 2 mih 1 (i, ki, mi, mi)), i q h 8 (ii, ki, im, m) ( 1 i 8 ρ2 i 3 8 ρ2 + 1 ) 4 ρ2 m h 6 (ii, ki, m, mi) i (h 5 (ii, ki, m, mi)+h 5 (ii, ki, im, m)) + i mi h 4 (ii, ki, m, mi) 2 ih 4 (ii, ki, i, ii) kl = i h 6 (ii, ki, m, mi) 1 2 q h 7 (ki, li, im, m) mi h 5 (ii, ki, m, mi), ( 1 i 8 ρ2 i 3 8 ρ2 + 1 ) 4 ρ2 m h 3 (ki,li,m,mi) i (h 2 (ki,li,m,mi)+h 2 (ki, li, im, m)) + i mi h 1 (ki,li,m,mi) 2 ih 1 (ki,li,i,ii) h 3 (ki, li, m, mi) 1 2 mi h 2 (ki, li, m, mi), 31k = γ 2(V, V, V )ρ 2 i + γ 2 (V, W, V )ρ i ρ + γ 2 (V, V, W )ρ i ρ k + γ 2 (V, W, W )ρ ρ k, 32k = {γ 2(V, Z 2, V )+γ 2 (V, V, Z 2 )}ρ 2 i

20 470 TOMOYA YAMADA + {γ 2 (V, Z 1, V )+γ 2 (V, W, Z 2 )}ρ i ρ + {γ 2 (V, Z 2, W )+γ 2 (V, V, Z 1 )}ρ i ρ k + {γ 2 (V, Z 1, W )+γ 2 (V, W, Z 1 )}ρ ρ k, 33k = γ 2(V, Z 2, Z 2 )ρ 2 i + γ 2 (V, Z 2, Z 1 )ρ i ρ k + γ 2 (V, Z 1, Z 2 )ρ i ρ + γ 2 (V, Z 1, Z 1 )ρ ρ k, 41kl = γ 3(V, V, V )ρ 3 i + {γ 3 (W, V, V )ρ + γ 3 (V, W, V )ρ k + γ 3 (V, V, W )ρ l }ρ 2 i + {γ 3 (W, W, V )ρ k + γ 3 (W, V, W )ρ l }ρ i ρ + {γ 3 (V, W, W )ρ i + γ 3 (W, W, W )ρ }ρ k ρ l, 42kl = {γ 3(Z 2, V, V )+γ 3 (V, Z 2, V )+γ 3 (V, V, Z 2 )}ρ 3 i + {γ 3 (Z 1, V, V )+γ 3 (W, Z 2, V )+γ 3 (W, V, Z 2 )}ρ 2 i ρ + {γ 3 (Z 2, W, V )+γ 3 (V, Z 1, V )+γ 3 (V, W, Z 2 )}ρ 2 i ρ k + {γ 3 (Z 2, V, W )+γ 3 (V, Z 2, W )+γ 3 (V, V, Z 1 )}ρ 2 i ρ l + {γ 3 (Z 1, W, V )+γ 3 (W, Z 1, V )+γ 3 (W, W, Z 2 )}ρ i ρ ρ k + {γ 3 (Z 2, W, W )+γ 3 (V, Z 1, W )+γ 3 (V, W, Z 1 )}ρ i ρ k ρ l + {γ 3 (Z 1, V, W )+γ 3 (W, Z 2, W )+γ 3 (W, V, Z 1 )}ρ i ρ ρ l + {γ 3 (Z 1, W, W )+γ 3 (W, Z 1, W )+γ 3 (W, W, Z 1 )}ρ ρ k ρ l, 43kl = {γ 3(Z 2, Z 2, V )+γ 3 (Z 2, V, Z 2 )+γ 3 (V, Z 2, Z 2 )}ρ 3 i + {γ 3 (Z 1, Z 2, V )+γ 3 (Z 1, V, Z 2 )+γ 3 (W, Z 2, Z 2 )}ρ 2 i ρ + {γ 3 (Z 2, Z 1, V )+γ 3 (Z 2, W, Z 2 )+γ 3 (V, Z 1, Z 2 )}ρ 2 i ρ k + {γ 3 (Z 2, Z 2, W )+γ 3 (Z 2, V, Z 1 )+γ 3 (V, Z 2, Z 1 )}ρ 2 i ρ l + {γ 3 (Z 1, Z 1, V )+γ 3 (Z 1, W, Z 2 )+γ 3 (W, Z 1, Z 2 )}ρ i ρ ρ k + {γ 3 (Z 1, Z 2, W )+γ 3 (Z 1, V, Z 1 )+γ 3 (W, Z 2, Z 1 )}ρ i ρ ρ l + {γ 3 (Z 2, Z 1, W )+γ 3 (Z 2, W, Z 1 )+γ 3 (V, Z 1, Z 1 )}ρ i ρ k ρ l + {γ 3 (Z 1, Z 1, W )+γ 3 (Z 1, W, Z 1 )+γ 3 (W, Z 1, Z 1 )}ρ ρ k ρ l, 44kl = γ 3(Z 2, Z 2, Z 2 )ρ 3 i + {γ 3 (Z 1, Z 2, Z 2 )ρ + γ 3 (Z 2, Z 1, Z 2 )ρ k + γ 3 (Z 2, Z 2, Z 1 )ρ l }ρ 2 i + {γ 3 (Z 1, Z 1, Z 2 )ρ k + γ 3 (Z 1, Z 2, Z 1 )ρ l }ρ i ρ + {γ 3 (Z 2, Z 1, Z 1 )ρ i + γ 3 (Z 1, Z 1, Z 1 )ρ }ρ k ρ l, and h, t and m are given by h 1 (ab, cd, ef, gh) = ab cd t 1 (ab, cd, ef, gh)+ 1 2 { cdt 2 (ab, cd, ef, gh) + ab t 2 (cd, ab, ef, gh)} t 3(ab, cd, ef, gh), h 2 (ab, cd, ef, gh) = ab cd t 2 (ef, ab, cd, gh)+ 1 2 { cdt 3 (ef, ab, cd, gh)

21 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR ab t 3 (ef, cd, ab, gh)} t 4(ef, ab, cd, gh), h 3 (ab, cd, ef, gh) = ab cd t 3 (ef, gh, ab, cd)+ 1 2 { cdt 4 (ef, gh, ab, cd) + ab t 4 (ef, gh, cd, ab)} t 5(ab, cd, ef, gh), h 4 (ab, cd, ef, gh) = cd t 2 (ab, cd, ef, gh)+ 1 2 t 3(ab, cd, ef, gh), h 5 (ab, cd, ef, gh) = cd t 3 (ab, ef, cd, gh)+ 1 2 t 4(ab, cd, ef, gh), h 6 (ab, cd, ef, gh) = cd t 4 (ab, ef, gh, cd)+ 1 2 t 5(ab, cd, ef, gh), h 7 (ab, cd, ef, gh) =ρ e ρ g h 71 (ab, cd, ef, gh)+h 72 (ab, cd, ef, gh), ρ g h 73 (ab, cd, ef, gh) ρ e h 73 (ab, cd, gh, ef), h 71 (ab, cd, ef, gh) = ab cd t 61 (ab, cd, ef, gh)+ 1 2 { cdt 62 (ab, cd, ef, gh) + ab t 62 (cd, ab, ef, gh)} t 63(ab, cd, ef, gh), h 72 (ab, cd, ef, gh) = ab cd t 71 (ab, cd, ef, gh)+ 1 2 { cdt 72 (ab, cd, ef, gh) + ab t 72 (cd, ab, ef, gh)} t 73(ab, cd, ef, gh), h 73 (ab, cd, ef, gh) = ab cd t 81 (ab, cd, ef, gh)+ 1 2 { cdt 82 (ab, cd, ef, gh) + ab t 82 (cd, ab, ef, gh)} t 83(ab, cd, ef, gh), h 8 (ab, cd, ef, gh) =ρ e ρ g h 81 (ab, cd, ef, gh)+h 82 (ab, cd, ef, gh) ρ g h 83 (ab, cd, ef, gh) ρ e h 83 (ab, cd, gh, ef), h 81 (ab, cd, ef, gh) = cd t 62 (ab, cd, ef, gh)+ 1 2 t 63(ab, cd, ef, gh), h 82 (ab, cd, ef, gh) = cd t 72 (ab, cd, ef, gh)+ 1 2 t 73(ab, cd, ef, gh), h 83 (ab, cd, ef, gh) = cd t 82 (ab, cd, ef, gh)+ 1 2 t 83(ab, cd, ef, gh), h 9 (ab, cd, ef, gh) =ρ e ρ g t 63 (ab, cd, ef, gh)+t 73 (ab, cd, ef, gh) ρ g t 83 (ab, cd, ef, gh) ρ e t 83 (ab, cd, gh, ef), t 1 (ab, cd, ef, gh) = (3) m1 (ab, cd)m 1 (ef, gh), t 2 (ab, cd, ef, gh) =m 2 (ab, cd)m 1 (ef, gh)+m 2 (ab, ef)m 1 (cd, gh) + m 2 (ab, gh)m 1 (cd, ef), t 3 (ab, cd, ef, gh) =m 3 (ab, cd)m 1 (ef, gh)+m 2 (ab, ef)m 2 (cd, gh) + m 2 (ab, gh)m 2 (cd, ef), t 4 (ab, cd, ef, gh) =m 3 (ab, cd)m 2 (ef, gh)+m 3 (ab, ef)m 2 (cd, gh)

22 472TOMOYA YAMADA and t 5 (ab, cd, ef, gh) = + m 3 (cd, ef)m 2 (ab, gh), (3) m3 (ab, cd)m 3 (ef, gh), t 61 (ab, cd, ef, gh) =m 1 (ab, cd)m 61 (ef, gh)+m 41 (ab, ef)m 41 (cd, gh) + m 41 (ab, gh)m 41 (cd, ef), t 62 (ab, cd, ef, gh) =m 2 (ab, cd)m 61 (ef, gh)+m 51 (ab, ef)m 41 (cd, gh) + m 51 (ab, gh)m 41 (cd, ef), t 63 (ab, cd, ef, gh) =m 3 (ab, cd)m 61 (ef, gh)+m 51 (ab, ef)m 51 (cd, gh) + m 51 (ab, gh)m 51 (cd, ef), t 71 (ab, cd, ef, gh) =m 1 (ab, cd)m 62 (ef, gh)+m 42 (ab, ef)m 42 (cd, gh) + m 42 (ab, gh)m 42 (cd, ef), t 72 (ab, cd, ef, gh) =m 2 (ab, cd)m 62 (ef, gh)+m 52 (ab, ef)m 42 (cd, gh) + m 52 (ab, gh)m 42 (cd, ef), t 73 (ab, cd, ef, gh) =m 3 (ab, cd)m 62 (ef, gh)+m 52 (ab, ef)m 52 (cd, gh) + m 52 (ab, gh)m 52 (cd, ef), t 81 (ab, cd, ef, gh) =m 1 (ab, cd)m 63 (ef, gh)+m 42 (ab, ef)m 41 (cd, gh) + m 41 (ab, gh)m 42 (cd, ef), t 82 (ab, cd, ef, gh) =m 2 (ab, cd)m 63 (ef, gh)+m 52 (ab, ef)m 41 (cd, gh) + m 51 (ab, gh)m 42 (cd, ef), t 83 (ab, cd, ef, gh) =m 3 (ab, cd)m 63 (ef, gh)+m 52 (ab, ef)m 51 (cd, gh) + m 51 (ab, gh)m 52 (cd, ef), m 1 (ab, cd) =ρ a ρ c θ(z 2, Z 2 )+ρ b ρ c θ(z 1, Z 2 )+ρ a ρ d θ(z 2, Z 1 )+ρ b ρ d θ(z 1, Z 1 ) ρ a ρ b ρ c θ(w, Z 2 ) ρ a ρ b ρ d θ(w, Z 1 ) ρ a ρ c ρ d θ(z 2, W ) ρ b ρ c ρ d θ(z 1, W )+ρ a ρ b ρ c ρ d θ(w, W ) (ρ2 a + ρ 2 b )(ρ cρ d θ(v, W ) ρ c θ(v, Z 2 ) ρ d )θ(v, Z 1 ) (ρ2 c + ρ 2 d )(ρ aρ b θ(w, V )ρ a θ(z 2, V )+ρ b θ(z 1, V ) (ρ2 a + ρ 2 b )(ρ2 c + ρ 2 d )θ(v, V ), m 2 (ab, cd) =ρ d θ(v, Z 1 )θ(v, Z 2 )+ρ c θ(v, Z 2 ) ρ c ρ d θ(v, W ) 1 ( ρ 2 2 c + ρ 2 ) d θ(v, V ), m 3 (ab, cd) =θ(v, V ), m 41 (ab, cd) =ρ a θ(z 2, W )+ρ b θ(z 1, W ) ρ c ρ d θ(w, W ) 1 ( ρ 2 2 a + ρ 2 ) b θ(v, W ),

23 ASYMPTOTIC DISTRIBUTION OF THE CANONICAL VECTOR 473 m 42 (ab, cd) =ρ a θ(z 2, Z 1 )+ρ b θ(z 1, Z 1 ) ρ c ρ d θ(w, Z 1 ) 1 ( ρ 2 2 a + ρ 2 ) b θ(v, Z1 ), m 51 (ab, cd) =θ(v, W ), m 52 (ab, cd) =θ(v, Z 1 ), m 61 (ab, cd) =θ(w, W ), m 62 (ab, cd) =θ(z 1, Z 1 ), m 63 (ab, cd) =θ(z 1, W ). Acknowledgements The author would like to thank the referee for a very careful reading of the manuscript and for many helpful comments that led to an improved version of the paper. Also, This research is supported by a sabbatical leave-of-absence from Sapporo Gakuin University. References Anderson, T. W. (1999). Asymptotic theory for canonical correlation analysis, J. Multivariate Anal., 70, Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion, Ann. Statist., 6, Boik, R. J. (1998). A local parameterization of orthogonal and semi-orthogonal matrices with applications, J. Multivariate Anal., 67, Eaton, M. L. and Tyler, D. (1994). The asymptotic distribution of singular-values with applications to canonical correlations and correspondence analysis, J. Multivariate Anal., 50, Fang, C. and Krishinaiah, P. R. (1982). Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations, J. Multivariate Anal., 12, Fuikoshi, Y. (1977). Asymptotic Expansions for the Distributions of Some Multivariate Tests, in Multivariate Analysis-4 (ed. P. R. Krishnaiah), North-Holland, Amsterdam, Fuikoshi, Y. (1978). Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations, J. Multivariate Anal., 8, Fuikoshi, Y. (1980). Asymptotic expansions for the distributions of the sample roots under nonnormality, Biometrika, 67, Ichikawa, M. and Konishi, S. (2002). Asymptotic expansions and bootstrap approximations in factor analysis, J. Multivariate Anal., 81, Kaplan, E. L. (1952). Tensor notation and the sampling cumulants of k-statistics, Biometrika, 39, Kendall, M. G. and Stuart, A. (1969). The Advanced Theory of Statistics Vol. 1, 3rd ed., Charles Griffin, London. Muirhead, R. J. (1978). Latent roots and matrix variates: A review of some asymptotic results, Ann. Statist., 6, Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, Wiley, New York. Muirhead, R. J. and Waternaux, C. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations, Biometrika, 67, 3l 43. Ogasawara, H. (2007). Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality, J. Multivariate Anal., 98, Seo, T., Kanda, T. and Fuikoshi, Y. (1994). The effects on the distributions of sample canonical correlations under nonnormality, Commun. Statist.-Theory Met., 23, Siotani, M., Hayakawa, T. and Fuikoshi, Y. (1985). Modern Multivariate Analysis: A Graduate Course and Handbook, American Sciences Press, Ohio.

24 474 TOMOYA YAMADA Sugiura, N. (1973). Derivatives of the characteristic root of a symmetric or a Hermitian matrix with two applications in multivariate analysis, Commun. Statist., 1, Sugiura, N. (1976). Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices, J. Multivariate Anal., 6(4), Taskinen, S., Croux, C., Kankainen, A., Ollila, E. and Oa, H. (2006). Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices, J. Multivariate Anal., 97,