ASYMPTOTIC EXPANSIONS OF THE Paa:R FUNCTIONS OF THE LIKELIHOOD RATIO TESTS FOR MULTIVARIATE LIHEAR HYPOTHESIS AND IRDEPDDEllCE by NariakiSugiura University of NOrth Carolina and Hiroshima University Institute of Statistics Mimeo Series No. 563 January, 1968 This research was supported by the National Science Foundation Grant No. 00-2059 and the Sakko-kai Foundation. DEPARTMENT OF STATISfiCS UJlIVERSITY OF NORTH CAROLIRA Chapel Hill, N. C.
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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