ASYMPTOTIC EXPANSIONS OF THE Paa:R FUNCTIONS OFTHE LIKELIHOOD RATIO TESTS FOR MULTIVARIATE

LIHEAR HYPOTHESIS AND IRDEPDDEllCE

by

NariakiSugiuraUniversity of NOrth Carolina

and Hiroshima University

Institute of Statistics Mimeo Series No. 563

January, 1968

This research was supported by theNational Science Foundation GrantNo. 00-2059 and the Sakko-kai Foundation.

DEPARTMENT OF STATISfiCSUJlIVERSITY OF NORTH CAROLIRA

Chapel Hill, N. C.

SUMMARY. Asymptotic non-null distribution of the likelihood ratio

criterion for testing the linear hypothesis in multivariate analysis

-1is obtained up to the order N , "Where N means the sample size, by

using the characteristic function expressed by the hypergeometric

function of matrix argument. This result holds without any assumpticn

on the rank of noncentrality matrix. Asymptotic non-null distribution

of the likelihood ratio test for independence between two sets of

variates is also obtained up to the order N-t .

~~t~~. Let each column vector of p xN matrix X be distributed

independently according to p-variate normal distribution with the

comnxm covariance matrix 1: and E(X) = SA, Where A is a known s X N

matrix of rank s and e is unknown p X s matrix. Then the multivariate

linear hypothesis is defined by asking, under this model, Whether the

parameters 8 satisfies the hypothesis H: 9B=O, where B is a known s X b

matrix (b < s) and 8B is assumed to be estimable. By making an appro-

priate orthogonal transformation from X to Y by Y = X!, we can obtain

the canonical form of the linear hypothesis. Each column vector of

Y = (Yl

, ••• , YN) is distributed independently according to the normal

distribution with the COlDlOOn covariance matrix 1:. The hypothesis H

and alternatives K are specified by

HI E(Yj

) = 0 j=1,2, ••• ,b

(Ll)

K: E(Yj

) = 0 j=s+l, ••• ,N.

The likelihood ratio test for this problem is expressed by

(1.2) " = (Is I / Is +Sh I f/2e e

The matrix S is the sum ofe

square due to error and the mtrix Sh is the sum of square due to the

hypothesis. Hence under the alternative K, Se is distributed accordjng

to the Wishart distribution with N-s degrees of freedom and Sh is

distributed according to the noncentral Wishart distribution with b

degrees of freedom and noncentrality matrix of n = (!)M'1:-1 where

A =E(Yl , ••• , Yb), as in Constantine [4].

Posten and Bargmann [8] obtained the asymptotic expansion of the

power function P(-2p log" < x) up to the order N-2 under the

assumption that the noncentrality matrix n is of rank 2, where p is a

correction factor such that under H the first remainder term of lowest

degree will disappear if we approximate the statistic -2p log " by lvariate with bp degrees of freedom, that is, p is determined by pH = N-s

+ (b-p-l)/2 (Anderson [1, p. 208]).

On the other hand Constantine [4] showed that the hth moment of

the ratio of the determinants Isel / \Se+shl under K could be expressed

by the hypergeometric function of matrix argument. His result can be

expressed by our notation as follows.

(1.3) F ( h N-s+b _ n),I 1 h; + 2 ; u

where rp(x) and the hypergeometric function lFl are defined by

(1.4 ) r (x) = ~(p-l)/4 n p r(x-(a-l)/2)p a=l

2

IF1 (a;b ;Z)

The function C (Z) is called a zonal polynomial of symmetric matrix ZX

corresponding to the partition K = (kl,k2, ••• ,kp),kl+k2+••• +kp=k,

ki~ 0, i = 1, ••• , p and )' means the sum of all such partition~ It

(K)is a kth degree homogeneous symmetric polynomial of p characteristic

roots of Z.

By using this result we shall show the asymptotic expansion of

PK(-2p log A< x) up to the order N-l without restricting the rank of

n. We further require some formula for zonal polynomials in Constantine

[4].ClO

(1.5) II-zl-a= I

~k=O

•(1.6) c!o(Z)

= k~ ~

(a) C (Z) I k!K K

C (Z) I k!K

= etr z.

Put m = PN = N-s+(b-p-l)/2 and let m tend to infinity instead of N as

in Posten and Bargmann [8], we can express the characteristic function

of -2p log A as

C(t) = E[e-2it p log A ]

= E[ ISe ,-itm liSe+Sh I-itm]

3

r (~ (1-2it)- ~) r (!!l +~)- ;p 2 P 2 IFl(-itm; 2~1-2it)+~;-0)- r (!!l _ b-p-L)r (~(1-2it)+ b+;p+l) ~

P 2 4 P 2 4

Under the hypothesis H, the noncentrality matrix 0 = 0 and the

hypergeometric function IF1 is equal to unity. So the first four r

functions give us the characteristic function under H, which we shall

denote by Cl(t) and IFI by C2 (t). Cl(t) can be expanded by the usual

manner due to Box [3] (Anderson [1, p. 204]). Applying the formula

m (_l)rB (h)(1. 8) log r(x+h) = log ,J2; + (x+h-!) log x-x- I r;~1 +o( Ix I-m-~

r=lr(r+l)x

which holds for large Ix I and fixed h with the Bernoulli polynomial

2Br(h) of defree r, B2(h) = h -h+(1/6), to Cl(t), we have

log Cl (t)~-1l0g (l-2it)-iI {1l2(b:Pt'2a)_~(-b-pi",2a)l~~t -to{m-2).

Cl=l

The second term of the above expression vanishes to get

This is the reason Why we use the correction factor p. Now we shall

consider the second term C2(t) of (1.7).

(-imt) C (-0)K K

TTm{1-2it)J/2 + {b+P+l)/4)K. k!

This follows from the definition (1.4). The coefficient of each term

can be arranged by the order of m.

4

(loll) P ( a-1X a-l) ( a-I )(-itm) = n_ -itm - --- -itm - ---2 +1 •• ~-itm - --- + k -1K <X=L 2 2 a

(1.12) ([m(1-2it)/2] + (b+p+l)/4)K

P= fm(1-2it) lk

[l + 2. f~ k _ ,ka(a-ka )1 2 J m{1-21t) L~ ~ 2a=l

+ O(~2)} ] .

differentiating

Hence we can write C2(t) as

Now we shall evaluate the each term of the above infinite series. By00

(1. 6) we can see exp(x tr Z) =I }'xkcK(Z) /k!,

~o ~1this equality with respect to x to get

(1.14)

00

(tr z) etr Z =I }' CK(Z)/(k-l)! •

k=l 6<5This formula is used by Fujikoshi [5], in deriving the asymptotic

expansion of generalized variance under the noncentral case. By (1. 5)

we have

5

(1.15)

co

=I )'k::O (;<)

which holds for any number n and a, Z is a p X P positive definite

matrix such that all characteristic root of Z/n lies in the interval

(0,1). This is satisfied, if we consider n morderately large. The

left handside can be expanded asymptotically in another way by the

formula -log II - n-1zl = tr(Z/n) + tr(z/n)2/2 + 0(n-3).

(1.16) II - n-lzjna =exp (-na log II - n-lzl)

= (etr a Z) (1 + (a/2n)tr Z2 + 0(n-2».Comparing the coefficients of each term of the order n-1 in (1.15) and

(1.16), we can get

p

I ka(ka-<X)}= a2tr z2(etr a z).

a=1.

Applying the formula (1.14) and (1.17) to the expression of C2(t) in

(1.13), we have

(1.18) C2 (t) = exp [(2it / (1-2it» tr oJ

X [1 - 1. {(b+P+l)it tr 0 + 2it tr 02 } + 0(-m12) J.. m (l_2it)2 (1_2it)3

Combining this with the expression of C1(t) in (1.9), we can obtain

asymptotic expansion of the characteristic fUnction C(t) in terms of

noncentral X2 distribution.

6

(1.19) C(t) = (1_2it)-bP/2 exp [{2it/ (1-2it)} tr 0]

[1+ 1 {(b(i+l)tr 0 _ 1 (l>+P+l tr 0 _ tr 0.2)

X m 2 1-2it) (1-2it)2 2

tr 0.2

} ~ ]- :3 + ~m2) •(1-2it)

By inverting this characteristic function with the well known result

that (1_2it)-f/2 exp [2it ~2/(1-2it)] is the characteristic fUnction

of the noncentral X2 distribution with f degrees of freedom and

noncentrality parameter ~2, we can finally obtain the following

theorem.

Theorem 1.1. The non-null distribution of the likelihood

ratio criterion (1.2) for multivariate linear hypothesis defined by

(1.1) can be approximated asymptotically up to the order m-l by

( 2(2) ) !Jb+p+l (2 (2)p( -2p log A < x) = P Xr ~ < x + iill 2 tr 0 P Xr+2 t> < x)

_(b+~+l trn _ tro2)p(~+4(52) < x) _ trn2p(~+6(t>2) < x)} + o(m-2),

where

m = PN = N - s + (b-p-l)/2, f = bp/2, e2 = trO = trAAt Z-l /2

and ~(52) means the noncentral l variate with f degrees of freedom

and noncentrality parameter t:P.If we specialize the rank of 0. to two, we can easily recognize

the agreement of the result by Posten and Bargmann [8] and ours,

after minor change of notation.

close connection between multivariate linear hypothesis and the test

7

for indepence between two sets of variates. Because we can reduce

the test for independence to that of the linear hypothesis by taking

conditional distribution of one set of variate for given another set.

MOnotonicity property of the power function of the test criteria

was proved by this fact in Anderson and Gupta [2]. Thus we can ex-

pect that the same is true in asymptotic expansion.

Let pX 1 vectors Xl"",XN be a random sample from multivariate

normal distribution with mean vector J.I. and covariance DRtrix E. Put

N N

S = I (Xex-X) (Xex-X) t, X = N-l I Xex and let us partition 1; and S into

ex=l cx=l

Pl and P2 rows and columns (Pl + P2 = p) as

E=(I:u ~),~l ~2

Without loss of generality we can assume Pl .5. P2 • The likelihood

ratio test for the hypothesis of independence H ~ = O(Pl XP2)

against all alternatives K:~ f 0 is given by

A = (Is IIlsllIIS22I)N/2 = II - Sll-l 812 822-1

S21 1N/2,

Pl 2 N/2 .This can be also expressed as n

j=l (l-r j ) by us~ng the sample

canonical correlations defined by the Pl characteristic roots of

-1 -1Sll S12 S22 S2l' First we shall show the mment of the statistic

A under K in the convenient form for the expansion,

Theorem 2.1. Under the alternative K, the mment of likelihood

ratio statistic can be expressed as

8

(2.2)

is a P:t X P.t diagonal. matrix,

PI2 h N-l 2n (l-Pj ) 2FI(h,h;h+ -2;P ),

j=l

2 2 2 )where P = diag (~ "'" PPI

diagonal. element P~ means the polulation canonical correlation and

the hypergeometric function 2FI(~,a2;b;Z) is defined by

Proof. Considering the conditional distribution of the first PI

components of the sample for given P2 second components in the canonical

set up of Z, Constantine [4] showed that the joint distribution of

I - r1

2, ••• , I - r 2 is the same as that of PI characteristic rootsPI

of WI (W 1+ WI) -1, where PI X PI matrices W' and W' have the Wishart

distribution for given Y(P2

X (N-I)) with cozmoon covariance matrix

r= diag (1-P12, ••• ,1-P 2) such that VV' is central with N-P2-1PI

degrees of freedom, WI is noncentral with P2 degrees of freedom and

noncentrality matrix n = r-l JftrI' A' /2 and that they are distributed

independently for given fi' (P2 X P2), which has the Wishart distribution

with N-l degrees of freedom, covariance matrix I, and the PI X P2 matrix

It. is given by

A.

Hence we can express the conditional moment of Isl/(Islll IS22 I) for

given yy' from (1.3) by simple change of notation.

9

Applying Kummer transformation formula IFl(a;b;Z)=(etr Z) IF1(b-a;b;-Z)

by Hertz [6] to the second expression and taking expectation by YY'

with the Wishart distribution W (N-l,I), we can writeP2

(2.4)

CKi"Nr-1AYI')d(YI') •

By the fOrmula r {etr(_RS»)lslt -(P+l)/2C (ST)dS=r (t)(t) IRI-tc (TR-l ),"..13>0 K P K K

which holds for p X P positive definite matrix R, S and P X P any

symmetric matrix T, established by Constantine [4], we have

Applying again the Kummer transformation formula 't!1 (al ,a2 ;b;Z) =b-~-a

II - ZI 2 't!l(b-~,b-~;b;Z) due to Hertz [6] and the identity

II + A'r-1AI = II - p2 1-1 to the above expression, we have

10

Combining this result with (2.3), we can get the theorem.

From this theorem, we shall expand the non-null distribution of

the likelihood ratio statistic -2 log ~ asymptotically. It is known

by Olkin and Siotani [7] that the limiting distribution of

.[N {( Is II Isll lls22 I) -11: II 1~111~2I) is normal with mea.n 0 and variance

4( 11: 1/ 1~111~21)2tr 1:u-1~2~ -l~l' so -2W-1(10g ~ - log n~:l

(1_Pj2)N/2) has the same limiting distribution with mean 0 and

variance 4 I;:l P~. This situation is somewhat different from the

previous section. Under the hypothesis the test statistic -2p log ~

is recommended instead of -2 log A, where the correction factor p

is determined so tha.t the first remainder term disappears in the

asymptotic expansion. We have pN=N-(3/2)-(Pl+P2)/2 (Anderson [1, p.

239]).

Put m = pN and let m tends to infinity instead of N. Then the

characteristic function of the statistic -2pm-1(10g ~ - log

n;:1(1"P~)N/2) is obtained by putting h = .. iWm in (2.2).

P -P +1 P +p +1r (! _ .[mit + 1 2 )r (! + 1 2 )Pl 2 4 PI 2 4

(2.7) C(t) = P _p +1 P +p +1r (! + 1 2 )r (! _ .[mit + 1 2 ).Pl 2 4 Pl 2 4

(..,[mi t k (..,[mit) K •

P +p +1(! r: 0t + 1 2 )2 .....,nu. 4 I(

k!

Applying the asymptotic formula for log r(x+h) in (1.8) to each of

the four r fUnctions, we get

11

(2.8) First Factor = 1 + !fro P1P2 +0(12).m

By the same way as (1.11) and (1.12),

Pl

(-Jmit) K = (-Jmit)k[l+(~t) I ka(a-kJ + o(m-l )]

a.l

(2.9)P +p +1 .

(~_ J"mit + 1 2 ) = (~)k[l _~ k + 0(1)].2 . 4 K 2 "m m

Hence we have (-J"mit)K(-J"mit)j<<m/2)-J"mit + (Pl+P2+l)/4)K =Pl

(_2t2)k[1+m-!(2itk + (it)-l I ka(a-ka )} + O(m-l )], which gives,

CX=1

with the formula (1.14) and (~17), the second factor of (2.7).

(2.10) 'tll(-J"mit, -J"mit; (m/2)-J"mit + (Pl+P2+l)/4;p2)

= (eKP(_2t2tr p2)} [1 + 4m-!(it)3 (tr p2 _ tr p4) + o(m-l )].

Combining this expansion with (2.b), we have the following expansion

01' tne cnaracteristic tunctlon.

22 1 .3 2 4~c.ll) ~(t) = (exp(-2t tr P »[1 +Ji (it P1P2 + 4(lt) (tr P -tr P )}

which implies the following theorem.

Theorem 2.2. The power fUnction of the likelihood r~tio test

for testing the independence between two sets of variates can be

expanded asymptotically up to the order m-l in the following way.

12

Pl

m = N - (3/2 ) - (Pl +P2)/2 and T =~ = 2<L p~r~.. Then we have

j=l

Pl1 {PIP2 1 ~ \' 4 } 1(2.12) p(~ < x) = ~(x) -rm --:r- ~t(x)+ ~ ~t"(x)(T-4/,p) +O(m-),

j=l

where ~(x) means the distribution fUnction of the standard normal

distribution and ~Xx), ~ttXx) are its derivatives.

In case p = 2 the problem reduces to test that the correlation

coefficient vanishes. It would be of interest to compare numerically

with another formula in Ruben [9].

13

REFERENCES

[1]. Anderson, T. W. (1958). An Introduction to MUltivariate

Statistical Analysis. Wiley, New York.

[2]. Anderson, T. W. and Das Gupta, S. (1964). MOnotonicity of the

power fUnctions of some tests of independence between two

sets of variates. Ann. Math. Statist. 35, 206-208.

[3]. Box, G. E. P. (1949). A general distribution theory for a

class of likelihood criteria. Biometrika, 36, 317-346.

[J~]. Constal1tine, A. G. (1963). Some non-central distribution

problems in multivariate analysis. Ann. Math. Statist.

34, 1270-1285.

[5]. Fujikoshi, Y. (1967). Asymptotic expansion of the generalized

variance. (Reported in the Meeting of Math. Society of

Japan. OCtober, 1967).

[6]. Hertz, C. S. (1955). Bessel functions of matrix argument.

Ann. Math. 61, 474-523.

[7]. OD~in, I. and Siotani, M. (1964). Asymptotic distribution of

functions of a correlation matrix. Stanford University,

Technical Report No.6.

[8]. Posten, H. o. and Bargmann, R. E. (1964). Power of the likelihood­

ratio test of the general linear hypothesis in multivariate

analysis. Biometrika. 51, 467-480.

[9]. Ruben, H. (1966). Some new results on the distribution of the

sample correlation coefficient. J.R. Statist. Soc. B 28, 513­

525.