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An asymptotic test for redundancy of variables in the comparison of two covariance matrices
Statistics & Probability Letters 4 (1986) 123-126 North-Holland
April 1986
AN ASYMPTOTIC TEST FOR REDUNDANCY OF VARIABLES COMPARISON OF TWO COVARIANCE MATRICES
IN THE
Bernhard N. F L U R Y
Department of Statistics, Unieersi(vof Berne. Sidlerstrasse 5, CH-3012 Berne, Switzerland Received August 1985 Revised November 1985
Abstract: Let '~1 and ~2 denote the p.d.s, covariance matrices of two p-variate normal populations, let 2~1>/?~2>/-.. >/Xp > 0 denote the characteristic roots of ~11~2, and fla. . . . . /3p the associated characteristic vectors. An asymptotic chi squared test statistic is derived for the hypothesis that some m characteristic vectors depend only on q ( < p) variables, i.e. have zero coefficients for p - q variables. An extension to elliptical distributions and robust estimators is sketched. Kevwords: eigenvectors, spectral decomposition, normal distribution, elliptical distribution.
1. Introduction Let 2 l and ~2 d e n o t e the p.d.s, covariance matrices of two i n d e p e n d e n t p-variate n o r m a l vectors X 1 a n d X 2. If ~1 ~ 2 , then it m a y be interesting to analyze those eigenvectors of ~ ( t ~ 2 which are associated with characteristic roots not equal to unity. M o r e specifically, we m a y ask whether an eigenvector/37 associated with root ~j has some zero coefficients, since in this case the linear c o m b i n a t i o n s f l / X 1 and fl/X 2 having ratio of variances var[ B}X 2] / v a r [ B } X 1] = Bi~2 f l J f l / ~ t / 3 , = ~,j d e p e n d only on a subset of the p variables. This type of p r o b l e m arises from a m e t h o d of analyzing the linear c o m b i n a t i o n s associated with extreme variance ratios; see F l u r y (1985) for a d e s c r i p t i o n of this m e t h o d a n d a p p l i c a t i o n s to quality control. M o r e formally, p a r t i t i o n the e i g e n v e c t o r s / 3 / of ~ ~'~2 into the first q a n d the last p - q c o m p o nents, i.e. write fls = (fill, fl]2)', where fljl has dimension q. Let v d e n o t e a subset of m (~< q) This work was done under contract no. 82.008.0.82 of the Swiss national Science Foundation in the Department of Statistics, Purdue University.
distinct integers b e t w e e n 1 and p such that h i ~ ?~j for all i ~ v, j ~ v. Then we define the hypothesis of simultaneous redundancy of (the last) p - q variables for m eigenvectors as
H,,(p,q):
~j2 ~--~-0
for all
j~v.
(1)
This hypothesis states that the eigenvectors a~sociated with the m roots ?~j, j ~ v, d e p e n d o n l y on the first q variables. T h e p u r p o s e of this article is to derive an a s y m p t o t i c test for H,,(p, q) b a s e d on W i s h a r t matrices. Tyler (1981) has considered the following rather general situation: Let M be a p × p - m a t r i x symmetric in the metric of the p.d.s, matrix F (i.e. F M is symmetric), with eigenvalues ~1 ~< • • • ~< ?~p. Let w denote a subset of m integers from { 1, 2 . . . . . p } (1 ~< m < p ) . Let A d e n o t e a fixed p × r - m a t r i x w i t h r a n k ( A ) = r. U n d e r the a s s u m p t i o n m i n i ~ ,,,j~ w [?~i - ?~/1 > 0, Tyler treats the following two h y p o t h e s e s on M : for r ~< m, the hypothesis H0: The c o l u m n s of A lie in the subspace generated by the set of eigenvectors of M associated with the m roots 2~, for i ~ w. F o r r >1 m, the hypothesis H*: The eigenvectors of M associated with the
0167-7152/86/$3.50 ~ 1986, Elsevier Science Publishers B.V. (North-Holland)
123
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
roots X, for i ~ w lie in the subspace generated by the columns of A. For testing H 0, Tyler (1981) has proposed a statistic T,(A) (where the subscript n refers to the sample size), and has shown its asymptotic distribution under fairly general conditions to be chi squared with r ( p - m) degrees of freedom. Moreover, T,,(A) is invariant under the transformation A ~ A B for any nonsingular r × r-matrix B. The problem of testing H3 can then be approaches as follows: Let B denote a fixed p × ( p - r)-matrix whose columns are orthogonal to those of A, i.e. A ' B = O . Then H~' can be rephrased as H0: The columns of B lie in the subspace generated by the set of eigenvectors of M ' associated with the p - m roots ),~ for i ~ w. Tyler (1981) has applied his technique to principal c o m p o n e n t and canonical correlation analysis. In the next section we are going to apply Tyler's a p p r o a c h to the hypothesis of r e d u n d a n c y H,,(p, q) in the comparison of two covariance matrices. The notation and terminology used here parallel closely that of Tyler (1981).
April 1986
Muirhead (1982, p. 82), n , S , - Wp(n,, Z , )
(i = 1 . 2 ) .
(2)
Then, for n, -~ oo, (n-~,( S, - ,Y,,) converges in distribution to a r a n d o m matrix with mean 0. The asymptotic covariance matrix of vec(S,- ~i) is
(U_+1,,,,,)(z,®z,)
(3)
(i=1,2),
where l(p,p) denotes the commutation matrix or permuted identity matrix of order p2 × p 2 see Izenman (1975, p. 258) and Muirhead (1982, p. 113). Put n = n I + n 2 and suppose that n 1 and 172 go to infinity such that the limits lira n / n ,
k,=
(i=1,2)
(4)
nj ~
are b o u n d e d away from 0 and oo. By expanding S2S11 in a Taylor series about M = G2,~{ 1 we get the approximation
~ ( S2S{ 1 - M ) -¢n[(S2-~2)Z{'-M(Sl-Z,),Y,['
].
(5)
As n goes to infinity, this converges in distribution to a normal matrix
(6)
N = CGN2 - v"k N, 2. Derivation of the test of redundancy Our hypothesis H,,(p, q) of simultaneous red u n d a n c y of p - q variables for the m eigenvectors/3,. (i E v) of ,~a ~ ~:2 can now be formulated in the form H~ of Tyler's approach by putting A = (~), where 0 is a ( p - q) x q-matrix of zeros. Putting B = (t,0 ',) , this is equivalent to Ho: The columns of B lie in the subspace generated by the eigenvectors a, (i ~ v) of M = ~72~?{-1. For convenience we will from now on write w for the complement of v, that is, w contains the p - m indices not in v. We are now going to derive the test for H 0 in terms of the eigenvectors a, of S 2 S~ ~ and then relate the resulting test statistic to the eigenvectors b~ of $1-1 S 2. Assume now that S~ and S 2 are independent sample covariance matrices from normal samples of size n~ + 1 and n, + 1, i.e., in the notation of 124
with mean zero, where N l and N= are the asymptotic distributions of {n-~M(Sx-Y~I)Y-,~ 1 and n 2 ( ~ - "~2 )'~{ 1 respectively. The covariance matrix of v e c ( N ) is qJ = k, cov(vec( N, )) + k 2 cov(vec( N 2 )) = k~2{ ~ ® M Z = + k = , ~ ~M ® Z :
+ (k, + k 2 ) l , p , p ) ( M ® M ' ) .
(7)
Let the eigenvectors a, of M be normalized such that aiN21a, = 1 ( i = 1 . . . . . p ). Noting that M is symmetric in the metric of ~_~ l, the eigenprojections are , P, = aiaiN 21
(i = 1 . . . . .
p).
(8)
Following Tyler (1981, p. 729), we define (7,,=
E Z
(X,-Xj)
'~®Pj
(9)
tGw jet,
(notice that C , does not depend on the particular
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
convention used for eigenvectors associated with multiple roots, as long as ming~ , . ~ , , iX, - ?hi > 0 is satisfied). Tyler's statistic is based on the asymptotic distribution of P, = E2,~,P,, where P, is the sample analog of (8). The r a n d o m vector ~/n v e c [ ( l p - P , ) B ] converges in distribution to normal (0, q'0(B)) (Tyler, 1981. T h e o r e m 4.1), where
(lO)
q,0(e) = (n'® t~)C,t,<, ( e ® t~).
M o o r e - P e n r o s e inverse
[~,,(8)] +=
=
(11)
C~v(,~r~llM ® ,~2 )Cw
s2 I,,,=l, '-'b,
x~
~2 ll~i® ~2P/,
(12)
=
kl~.i~.., + k2~. 2
(x - x , ) ~
8'z;'eB®z~
'
,
k l ) t ' ~ ' / + k2~'2'
.
Gj has rank p - q. vanishes thanks to l{p,p)). Using the generalized inverse we therefore get [q~o(B)] + = E
.]CP
(x,-x,)
(18)
n E b;B B'
/]
ktl~--+ k,l,l,__ b,b; B , ,
(l,
l,)-
B*
b,. (19)
(13)
where G, = B ' Z 2 1 ~ .
=
,~,,
E ~J ® "~2~"' jcu
( j = l . . . . . p).
where b/ is the j-th eigenvector of S t ~S,. normalized such that b;Slb ~ = 1. Thus we get r,,(B)
and therefore
Y'. Y', ,~,,,,E,,
(17)
N o t i n g that $2 i~ = $2 lal(S 2 lal ), ( j = 1 . . . . . p), we see from the spectra decomposition theorem that
and
~o(8)=
n Y', t r a c e ( 6 ; IB'P,,!S 2 lPP,,B)
= n Y'. t r a c e ( d / 1 B ' S . ' ~ B ) . let'
h,Xj
,~w jE,, ( X , - X,) ~
(~6)
T,,(B) = ,, [vec( h n)]'[~.( n)] +vec(hn)
c',.(z; ~ ® M&)c,,.
=ZE
E 67' ® ~,'s~_ '
Using L e m m a 2.2.3(iii) in Muirhead (1982) and the fact that p,, = y ~ , a , a l S f l = 1, ~2 ~,,aja¢S2 1 = I , - ~,, the test statistic proposed by Tyler (1981, p. 730) is
Using the orthogonality of the P,, we get
= E E "~21pi ® "~'2Pjt ,~. ~,, ( X , - Xj):
April 1986
~
1
a,a I Z 2 lB.
J
(14)
(Note that the last term of (7) the c o m m u t a t i o n property of fact that the M o o r e - P e n r o s e of P, is P / = ~ (i = 1 . . . . p),
G, ' ® ~ ' Z 2'
We are now going to use the special structure of B to simplify (19), and. to mark the fact that T,,(B) is a statistic for testing the Redundancy of p - q variables for the eigenvectors ,B, (i ~ c), we will call it R,,(p, q) from now on. Partition b, as
h,
= (b,, 't it,,: J
in q and p - q components, then
n,,(p,q)
(15)
as can be verified easily. Replacing Z,, aj and hj by the corresponding sample quantities S,, a] and l/, (that is, the aj and lj are eigenvectors and eigenvalues of $2S~1), we get a c o n s i s t e n t e s t i m a t e ~ 0 ( B ) and its
(2o)
JEt'
--½
t,,~b5
b,~. (21)
By Theorem 5.3 of Tyler (1981), R A p , q) is asymptotically distributed as chi squared with m( p - q) degrees of freedom under H 0. Note that 125
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
(21) depends on the correct normalization of the eigenvectors b i. They must be normalized such that b~Slb ~ = c (i = 1 . . . . . p) for some c > 0. The most convenient way of doing this is of course to use the standard convention b~Slb ~= 1.
3. Generalization to elliptical distributions and robust estimates The result of the preceding section can easily be generalized to elliptical populations with finite forth moments. Assume that the sample covariance matrices S 1 and S 2 stem from elliptical populations with kurtosis parameters x 1 and ~2, where x, is defined such that 3x~ is the kurtosis of any marginal distribution (Muirhead, 1982, p. 41). Then the asymptotic covariance matrix of ~"n,~ v e c ( S , - Z,) is (1
+
Ki)( Ip2-]- t(p,p,)(~_11 ~ ~-~i)
+xi(vec ~ , ) ( v e c ~ , ) '
( i = 1, 2)
(22)
(Tyler, 1981, p. 733) instead of (3). However, the terms K, (vec ~ , ) (vec ~ , ) ' cancel out in the computation of ~ 0 ( B ) (equation (13)), and so the only change in R,.(p, q) is that the constants k, = n / n , have to be replaced by k* = k,(1 + K,). If xl = x2 = x (say), then the correct statistic can be written as R*(p,
q)=(l+K)
1R,,(p, q).
(23)
Similarly, if affine-invariant M-estimates of location and scatter are used instead of the usual sample covariance matrices, Tyler (1982) has shown that the resulting asymptotic normal distribution is again of the form (22), with (1 + x,) and xs replaced by some constants ch~ and o2,, respectively, where the 0/, depend on the population and
126
April 1986
the type of estimator used. Again, the o2, will disappear in the further computations. If oll = or2 = 01, say, then the statistic R**(p, q)=o~-lR,,(p,
q)
(24)
(where the M-estimates are now being used in the definition of R,.(p, q)) is again asymptotically distributed as chi squared on m ( p - q) degrees of freedom if H,,(p, q) holds. The details of the proof of the above statements are omitted, since the technique is completely analogous to Section 4 of Tyler (1983). Moreover, x in (23) and o t in (24) can be replaced by consistent estimates ~ and 61 without affecting the validity of the asymptotic results.
Acknowledgement I wish to thank Alan Izenman, David Tyler and an anonymous referee for helpful comments on an earlier draft of this article.
References Flury, B.N. (1985), Analysis of linear combinations with extreme ratios of variances, Journal of the American Statistical Association 80, to appear. lzenman, A.J. (1975), Reduced-rank regression for the multivariate linear model, Journal of Multivariate Analysis 5, 248-264. Muirhead, R.J. (1982), Aspects of Multivariate Statistical Theoo' (Wiley, New York). Tyler, D.E. (1981), Asymptotic inference for eigenvectors, Annals of Statistics 9, 725 736. Tyler, D.E. (1982), Radial estimates and the test for sphericity, Biometrika 69, 429-436. Tyler, D.E. (1983), A class of asymptotic tests for principal component vectors, Annals of Statistics 11, 1243 1250.
April 1986
AN ASYMPTOTIC TEST FOR REDUNDANCY OF VARIABLES COMPARISON OF TWO COVARIANCE MATRICES
IN THE
Bernhard N. F L U R Y
Department of Statistics, Unieersi(vof Berne. Sidlerstrasse 5, CH-3012 Berne, Switzerland Received August 1985 Revised November 1985
Abstract: Let '~1 and ~2 denote the p.d.s, covariance matrices of two p-variate normal populations, let 2~1>/?~2>/-.. >/Xp > 0 denote the characteristic roots of ~11~2, and fla. . . . . /3p the associated characteristic vectors. An asymptotic chi squared test statistic is derived for the hypothesis that some m characteristic vectors depend only on q ( < p) variables, i.e. have zero coefficients for p - q variables. An extension to elliptical distributions and robust estimators is sketched. Kevwords: eigenvectors, spectral decomposition, normal distribution, elliptical distribution.
1. Introduction Let 2 l and ~2 d e n o t e the p.d.s, covariance matrices of two i n d e p e n d e n t p-variate n o r m a l vectors X 1 a n d X 2. If ~1 ~ 2 , then it m a y be interesting to analyze those eigenvectors of ~ ( t ~ 2 which are associated with characteristic roots not equal to unity. M o r e specifically, we m a y ask whether an eigenvector/37 associated with root ~j has some zero coefficients, since in this case the linear c o m b i n a t i o n s f l / X 1 and fl/X 2 having ratio of variances var[ B}X 2] / v a r [ B } X 1] = Bi~2 f l J f l / ~ t / 3 , = ~,j d e p e n d only on a subset of the p variables. This type of p r o b l e m arises from a m e t h o d of analyzing the linear c o m b i n a t i o n s associated with extreme variance ratios; see F l u r y (1985) for a d e s c r i p t i o n of this m e t h o d a n d a p p l i c a t i o n s to quality control. M o r e formally, p a r t i t i o n the e i g e n v e c t o r s / 3 / of ~ ~'~2 into the first q a n d the last p - q c o m p o nents, i.e. write fls = (fill, fl]2)', where fljl has dimension q. Let v d e n o t e a subset of m (~< q) This work was done under contract no. 82.008.0.82 of the Swiss national Science Foundation in the Department of Statistics, Purdue University.
distinct integers b e t w e e n 1 and p such that h i ~ ?~j for all i ~ v, j ~ v. Then we define the hypothesis of simultaneous redundancy of (the last) p - q variables for m eigenvectors as
H,,(p,q):
~j2 ~--~-0
for all
j~v.
(1)
This hypothesis states that the eigenvectors a~sociated with the m roots ?~j, j ~ v, d e p e n d o n l y on the first q variables. T h e p u r p o s e of this article is to derive an a s y m p t o t i c test for H,,(p, q) b a s e d on W i s h a r t matrices. Tyler (1981) has considered the following rather general situation: Let M be a p × p - m a t r i x symmetric in the metric of the p.d.s, matrix F (i.e. F M is symmetric), with eigenvalues ~1 ~< • • • ~< ?~p. Let w denote a subset of m integers from { 1, 2 . . . . . p } (1 ~< m < p ) . Let A d e n o t e a fixed p × r - m a t r i x w i t h r a n k ( A ) = r. U n d e r the a s s u m p t i o n m i n i ~ ,,,j~ w [?~i - ?~/1 > 0, Tyler treats the following two h y p o t h e s e s on M : for r ~< m, the hypothesis H0: The c o l u m n s of A lie in the subspace generated by the set of eigenvectors of M associated with the m roots 2~, for i ~ w. F o r r >1 m, the hypothesis H*: The eigenvectors of M associated with the
0167-7152/86/$3.50 ~ 1986, Elsevier Science Publishers B.V. (North-Holland)
123
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
roots X, for i ~ w lie in the subspace generated by the columns of A. For testing H 0, Tyler (1981) has proposed a statistic T,(A) (where the subscript n refers to the sample size), and has shown its asymptotic distribution under fairly general conditions to be chi squared with r ( p - m) degrees of freedom. Moreover, T,,(A) is invariant under the transformation A ~ A B for any nonsingular r × r-matrix B. The problem of testing H3 can then be approaches as follows: Let B denote a fixed p × ( p - r)-matrix whose columns are orthogonal to those of A, i.e. A ' B = O . Then H~' can be rephrased as H0: The columns of B lie in the subspace generated by the set of eigenvectors of M ' associated with the p - m roots ),~ for i ~ w. Tyler (1981) has applied his technique to principal c o m p o n e n t and canonical correlation analysis. In the next section we are going to apply Tyler's a p p r o a c h to the hypothesis of r e d u n d a n c y H,,(p, q) in the comparison of two covariance matrices. The notation and terminology used here parallel closely that of Tyler (1981).
April 1986
Muirhead (1982, p. 82), n , S , - Wp(n,, Z , )
(i = 1 . 2 ) .
(2)
Then, for n, -~ oo, (n-~,( S, - ,Y,,) converges in distribution to a r a n d o m matrix with mean 0. The asymptotic covariance matrix of vec(S,- ~i) is
(U_+1,,,,,)(z,®z,)
(3)
(i=1,2),
where l(p,p) denotes the commutation matrix or permuted identity matrix of order p2 × p 2 see Izenman (1975, p. 258) and Muirhead (1982, p. 113). Put n = n I + n 2 and suppose that n 1 and 172 go to infinity such that the limits lira n / n ,
k,=
(i=1,2)
(4)
nj ~
are b o u n d e d away from 0 and oo. By expanding S2S11 in a Taylor series about M = G2,~{ 1 we get the approximation
~ ( S2S{ 1 - M ) -¢n[(S2-~2)Z{'-M(Sl-Z,),Y,['
].
(5)
As n goes to infinity, this converges in distribution to a normal matrix
(6)
N = CGN2 - v"k N, 2. Derivation of the test of redundancy Our hypothesis H,,(p, q) of simultaneous red u n d a n c y of p - q variables for the m eigenvectors/3,. (i E v) of ,~a ~ ~:2 can now be formulated in the form H~ of Tyler's approach by putting A = (~), where 0 is a ( p - q) x q-matrix of zeros. Putting B = (t,0 ',) , this is equivalent to Ho: The columns of B lie in the subspace generated by the eigenvectors a, (i ~ v) of M = ~72~?{-1. For convenience we will from now on write w for the complement of v, that is, w contains the p - m indices not in v. We are now going to derive the test for H 0 in terms of the eigenvectors a, of S 2 S~ ~ and then relate the resulting test statistic to the eigenvectors b~ of $1-1 S 2. Assume now that S~ and S 2 are independent sample covariance matrices from normal samples of size n~ + 1 and n, + 1, i.e., in the notation of 124
with mean zero, where N l and N= are the asymptotic distributions of {n-~M(Sx-Y~I)Y-,~ 1 and n 2 ( ~ - "~2 )'~{ 1 respectively. The covariance matrix of v e c ( N ) is qJ = k, cov(vec( N, )) + k 2 cov(vec( N 2 )) = k~2{ ~ ® M Z = + k = , ~ ~M ® Z :
+ (k, + k 2 ) l , p , p ) ( M ® M ' ) .
(7)
Let the eigenvectors a, of M be normalized such that aiN21a, = 1 ( i = 1 . . . . . p ). Noting that M is symmetric in the metric of ~_~ l, the eigenprojections are , P, = aiaiN 21
(i = 1 . . . . .
p).
(8)
Following Tyler (1981, p. 729), we define (7,,=
E Z
(X,-Xj)
'~®Pj
(9)
tGw jet,
(notice that C , does not depend on the particular
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
convention used for eigenvectors associated with multiple roots, as long as ming~ , . ~ , , iX, - ?hi > 0 is satisfied). Tyler's statistic is based on the asymptotic distribution of P, = E2,~,P,, where P, is the sample analog of (8). The r a n d o m vector ~/n v e c [ ( l p - P , ) B ] converges in distribution to normal (0, q'0(B)) (Tyler, 1981. T h e o r e m 4.1), where
(lO)
q,0(e) = (n'® t~)C,t,<, ( e ® t~).
M o o r e - P e n r o s e inverse
[~,,(8)] +=
=
(11)
C~v(,~r~llM ® ,~2 )Cw
s2 I,,,=l, '-'b,
x~
~2 ll~i® ~2P/,
(12)
=
kl~.i~.., + k2~. 2
(x - x , ) ~
8'z;'eB®z~
'
,
k l ) t ' ~ ' / + k2~'2'
.
Gj has rank p - q. vanishes thanks to l{p,p)). Using the generalized inverse we therefore get [q~o(B)] + = E
.]CP
(x,-x,)
(18)
n E b;B B'
/]
ktl~--+ k,l,l,__ b,b; B , ,
(l,
l,)-
B*
b,. (19)
(13)
where G, = B ' Z 2 1 ~ .
=
,~,,
E ~J ® "~2~"' jcu
( j = l . . . . . p).
where b/ is the j-th eigenvector of S t ~S,. normalized such that b;Slb ~ = 1. Thus we get r,,(B)
and therefore
Y'. Y', ,~,,,,E,,
(17)
N o t i n g that $2 i~ = $2 lal(S 2 lal ), ( j = 1 . . . . . p), we see from the spectra decomposition theorem that
and
~o(8)=
n Y', t r a c e ( 6 ; IB'P,,!S 2 lPP,,B)
= n Y'. t r a c e ( d / 1 B ' S . ' ~ B ) . let'
h,Xj
,~w jE,, ( X , - X,) ~
(~6)
T,,(B) = ,, [vec( h n)]'[~.( n)] +vec(hn)
c',.(z; ~ ® M&)c,,.
=ZE
E 67' ® ~,'s~_ '
Using L e m m a 2.2.3(iii) in Muirhead (1982) and the fact that p,, = y ~ , a , a l S f l = 1, ~2 ~,,aja¢S2 1 = I , - ~,, the test statistic proposed by Tyler (1981, p. 730) is
Using the orthogonality of the P,, we get
= E E "~21pi ® "~'2Pjt ,~. ~,, ( X , - Xj):
April 1986
~
1
a,a I Z 2 lB.
J
(14)
(Note that the last term of (7) the c o m m u t a t i o n property of fact that the M o o r e - P e n r o s e of P, is P / = ~ (i = 1 . . . . p),
G, ' ® ~ ' Z 2'
We are now going to use the special structure of B to simplify (19), and. to mark the fact that T,,(B) is a statistic for testing the Redundancy of p - q variables for the eigenvectors ,B, (i ~ c), we will call it R,,(p, q) from now on. Partition b, as
h,
= (b,, 't it,,: J
in q and p - q components, then
n,,(p,q)
(15)
as can be verified easily. Replacing Z,, aj and hj by the corresponding sample quantities S,, a] and l/, (that is, the aj and lj are eigenvectors and eigenvalues of $2S~1), we get a c o n s i s t e n t e s t i m a t e ~ 0 ( B ) and its
(2o)
JEt'
--½
t,,~b5
b,~. (21)
By Theorem 5.3 of Tyler (1981), R A p , q) is asymptotically distributed as chi squared with m( p - q) degrees of freedom under H 0. Note that 125
Volume 4, Number 3
STATISTICS & PROBABILITY LETTERS
(21) depends on the correct normalization of the eigenvectors b i. They must be normalized such that b~Slb ~ = c (i = 1 . . . . . p) for some c > 0. The most convenient way of doing this is of course to use the standard convention b~Slb ~= 1.
3. Generalization to elliptical distributions and robust estimates The result of the preceding section can easily be generalized to elliptical populations with finite forth moments. Assume that the sample covariance matrices S 1 and S 2 stem from elliptical populations with kurtosis parameters x 1 and ~2, where x, is defined such that 3x~ is the kurtosis of any marginal distribution (Muirhead, 1982, p. 41). Then the asymptotic covariance matrix of ~"n,~ v e c ( S , - Z,) is (1
+
Ki)( Ip2-]- t(p,p,)(~_11 ~ ~-~i)
+xi(vec ~ , ) ( v e c ~ , ) '
( i = 1, 2)
(22)
(Tyler, 1981, p. 733) instead of (3). However, the terms K, (vec ~ , ) (vec ~ , ) ' cancel out in the computation of ~ 0 ( B ) (equation (13)), and so the only change in R,.(p, q) is that the constants k, = n / n , have to be replaced by k* = k,(1 + K,). If xl = x2 = x (say), then the correct statistic can be written as R*(p,
q)=(l+K)
1R,,(p, q).
(23)
Similarly, if affine-invariant M-estimates of location and scatter are used instead of the usual sample covariance matrices, Tyler (1982) has shown that the resulting asymptotic normal distribution is again of the form (22), with (1 + x,) and xs replaced by some constants ch~ and o2,, respectively, where the 0/, depend on the population and
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the type of estimator used. Again, the o2, will disappear in the further computations. If oll = or2 = 01, say, then the statistic R**(p, q)=o~-lR,,(p,
q)
(24)
(where the M-estimates are now being used in the definition of R,.(p, q)) is again asymptotically distributed as chi squared on m ( p - q) degrees of freedom if H,,(p, q) holds. The details of the proof of the above statements are omitted, since the technique is completely analogous to Section 4 of Tyler (1983). Moreover, x in (23) and o t in (24) can be replaced by consistent estimates ~ and 61 without affecting the validity of the asymptotic results.
Acknowledgement I wish to thank Alan Izenman, David Tyler and an anonymous referee for helpful comments on an earlier draft of this article.
References Flury, B.N. (1985), Analysis of linear combinations with extreme ratios of variances, Journal of the American Statistical Association 80, to appear. lzenman, A.J. (1975), Reduced-rank regression for the multivariate linear model, Journal of Multivariate Analysis 5, 248-264. Muirhead, R.J. (1982), Aspects of Multivariate Statistical Theoo' (Wiley, New York). Tyler, D.E. (1981), Asymptotic inference for eigenvectors, Annals of Statistics 9, 725 736. Tyler, D.E. (1982), Radial estimates and the test for sphericity, Biometrika 69, 429-436. Tyler, D.E. (1983), A class of asymptotic tests for principal component vectors, Annals of Statistics 11, 1243 1250.