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-1';-\
....,..-~,:_.,
.,}
ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS OF THE
LIKELDIOOD RATIO CRITERIA
~FOR
COVARIANCE MATRICES
by
Nariaki Sugiura
University of North Carolina
and Hiroshima University
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Institute of Statistics M1meo Series No. 574
March 1968
This research was supported by the National
Science Foundation Grant No. GU-2059 and the Sakkokai Foundation, and the Mathematic Division of the
Air Force Office of Scientific Research Contract No.
AF-AFOSR-68-1415.
DEPARTMENT OF STATISTICS
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Chapel Hill, N. C.
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....... "'"'
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1.
Intr..9d2!?}jon.
In our previous paper [11], we have proved the un-
biasedness of the modified likelihood ratio (= modified LR) test (i) for the
equality of covariance matrix to a given matrix and (ii) for the equality of
two covariance matrices.
We have also shown the unbiasedness of the LR test
(iii) for sphericity and (iv) for the equality of mean vector and covariance
matrix to a given vector and matrix.
In this paper asymptotic expansions of the distributions of the test
criteria for (i) and (iv) both under hypothesis and alternatives are derived,
by inverting the characteristic function directly.
!he
asym~totic
expansion
of the power function of the LR test for sphericity (iii), is obtained by
using the differential operator due to Welch [12], and also the limiting nonnull distribution of the LR test for the equality of k covariance matrices is
derived in a similar way.
This method has been shown to be useful in other
problems in multivariate analysis by Ito [3], Siotani [8], Okamoto
[5],
and etc.
All the limiting non-null distributions of these test criteria are shown
to be normal distributions, whereas the limiting distributions under hypothesis
are X2-distributions as in Anderson [1].
This shows the difference between
the LR test criteria for covariance matrices and the criterion for the linear
hypothesis as in Sugiura and Fujikoshi [10].
Let P X 1 vectors Xl' ••• ,
~
be a random sample from a p-variate normal
distrihution with unknown mean vector p and covariance matrix t (non-singular).
The LR criterion for testing the hypothesis Hl : t • to against the alternatives K : t ". EO' for some given positive definite matrix to' is given by
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A = (e/N)Np/?ls~~1IN/2 etr - (1/2)~~]S,
(2.1)
where etr means exp tr and S
test is not unbiased.
= ~=l (Xa-X)(Xa-X)', X = (l/N)~=lXa. This
LR
However, ir we modiry this criterion by reducing the
sample size N to the degreesor rreedom n
= B-1,
it has some desirable property,
that is, the unbiasedness is shown by Sugiura and Nagao [11] and the monotonicity or the power function with respect to p characteristic roots or
u.~l is established by Nagao [4].
So we shall consider the asymptotic ex-
pansion or the modiried La statistic
A* instead or A.
The limiting distribution or the statistic -2 log A under the hypothesis is
the X2-distribution with p (p+l)/2 degrees or rreedom, which can be seen in
Anderson [1, p. 267].
The hth moment or the statistic 1* under alternative K
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is given by
,
where rp(x) =
tion or
(2.4)
-2
log
~(p-l)/4 n~=l
r(x - (0-1)/2).
Hence the characteristic func-
A* under the hypothesis Hl is expressed as
n itpn r p (n(l-2it)/2)
CH (t) = (2e)
r (nj2)
1
p
(l_2it)-np (l-2it)/2 .
We shall use the following asymptotic formula for the gamma function as in
Anderson [1, p. 204].
log r(x+h) = 10g(2u)
i
+ (x+h-i) log x - x -
This holds for large value or x with rixed h.
~
(_l)r B H (h)
-k-l
rr
+ O(x
).
r=l r(r+l) x
L
The Bernoulli polynomial Br (h)
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of degree r is given by ~eh~/(e~-l)
(2.6)
B.3(h)
3 2
= h3 - 2'h
j",.
+"211
4
3
2
1
B4 (h) = h - 2h + h - 30 .
Applying the formula (2.5) to each gamma function in
(2.7)
log C (t) = H
1
~
(~.4),
we get
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k
(_2)rB
log (l-2it) r;l «l-2it)-r -l)+O(n-k-l),
r=l r(r+l)n
Where Br +l = ~~=l Br +l «1-<1)/2). This formula implies the as:vmptotic expansion
of the characteristic function fur -2 log
"'*.
(2.8)
C (t)
Hl
= (l-2it)-P(P+l)/4[1+B2n- l
(l-2it)-1_l) + (l/6)n-2(3B~-4B3)
. (l-2it)-2_6B~(l-2it)-1+(3B~+4B;»+(l/6)n-3{(4B4-4B2B3+~)
.(l-2it)-3+B2(4B3-3~)(l-2it)-2+B2(4B;+3~)(l-2it)-1
(4B4+4B2B;+~))]
4
+ o(n- ),
where
B2 = P(2p2+3p-l)/24
B;
= - P(p-l)(p+l)(p+2)/32
3
2
B4 = p(6p4
+15p
~lOp -3Op+3)/480.
Inverting this characteristic function, using the well-known fact that
(l_2it)-f/2 is the characteristic function of the X2 -distribution with f
degrees of freedom, we have the following theorem.
~eorem
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(~r/r:)Br(h). Some of these are
listed below;
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= ~;=o
Under the hypothesis Hl : ~ • ~O' the distribution of the
modified LR criterion -2 log ).* defined by (2.2) can be expanded as:vmtotically
as
2.1.
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(2.10)
P(-2 log "'A*
+ (1/6)n
+
-2
~
= P(~ ~
z)
z) + B2n-
l
(P(~+2 ~
z,) -
P(~ ~ z»
2
2
2
2
2
2
«3 B2 - 4B )P(Xr+4:S z)-6B2 P(Xr+2 ~ z)+(3B2 +4B3 )P(Xr ~ z»
3
(1/6)n-3((4B4-4B2~+~)P(X;+6
:s z)
+
B2(4~-3B~)P(~+4 ~
z)
_2
2
3
2
-4
+B2 (4B +3B2)P(x.r+2 ~ z)-(4B4 +4B2B +B2 )P(x.r ~ z») + O(n ),
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3
where ~ means the x2-variate with f degrees of freedom and f = P(P+l)/2;
the constant B is given by (2.9).
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Now we shall consider the asymptotic expansion of the power function of
.l.
The characteristic function of _2n- 2 log A* under
this criterion -2 log "'A*.
alte:rnative K can be written from fornmla (2.3) for the moment of "'A* as
l
Applying the asymptotic formula (2.5) for the gamma function to each term of
..[nit)/rp (n/2), we have
r
(2.12)
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(2.11)
r p ({n/2)-
.1
r. P «n/2l- "nitl
log r (n/2)
r=l n
p
+
2k
.
2
2 '\ (2/tl r { 2 it
=..[n~tp log n - pt +p ~ r 2
r+l r+2)
t+r l
}
k (4)~
.l.
_ \'
r+l (1- .}it) -r-1) + 0(n-k-2) •
~ (+l) n r
n
r=l r r
Since the asymptotic formula -logII-Z/nl = ~=ln-rtr (zr)/r + o(n-k~l) holds
for any positive definite matrix Z, we have
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substituting these two expressions to (2.11), we can see 'that
log C {t)=- ~nit (logIEE~ll+ tr(I_EE~1»_t2tr{I_EE~1}2
(2.14)
2k
+\
K1
{-1)r+2
{-l)r+l
(2ipr rp(p+l) + 2p{i t)2
+ 2 it 2{ tr EEO
_tr r.EO
}]
~ r 2 L~ (r+l) (r+2)
()
r+2
r+l
r.l n
k { 2)rB
_\
~
r=l
r+l;
r{r+l)nr
{(l- 2it)-r_ l ) + o{n-k - i ),
~
which implies that the statistic A** = -2n-; log A*
~n(tr(EE;l-I)
- log
1~11J converges in law to the normal distribution with mean zero and
variance"2 = 2tr {I-I:to-1)2 and further it enables us to expand the characteristic function of
A**/"
up to any order asymptotically.
We shall write it
up to order n -1 •
(2.15)
where the coefficients
Al and
~
of each term are given by
Al = it,.-l P{p+l)/2 + (2/3),.-3{it)3(p+2tr - 3tr )
2
3
A = (1/B),.-2(it)2p(P+l) (p2+P+4)+(1/3),.-4(it)4(p(p2 +P+2)_3P(P+1)tr
2
2
+ 2{p2+p-4)tr3+6tr4) + {2/9),.-6(it)6{p+2tr -3tr )2,
2
3
with the abbreviated notation trj = tr(~l)j.
By inverting this character-
istic function, we can get the following theorem.
Theorem 2.2.
Under the alternative (1:1: "to' the distribution of the
modified LR criterion -2 log A* defined by (2.2) can be expanded asymptotically as
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p«lff~~) [-2 log h*-n(tr(tt~l-I) - log Itt~ll)] ~ z)
(2.16)
-:. t(z) - Wn [3~-1 t'(z)P(p+l) + 4~-3 t"'(z)(p+2tr -3tr )]
2
3
+ . 7~ri [9~-2t"( z)P(P+l)(p2+p+4 )+24~-4t(4) (z) (p(p2+P+2)-3P(p+l)tr
2
+ 2(p2+p-4)tr3+6tr4) +
where tr.
J
16~-6.(6)(z)(p+2tr3-3tr2)2] + o(n-3 / 2 ),
= tr(Et~l)j and ~2 = 2tr(I_tt~1)2; t(r)(z) means the rth derivative
of the standard normal distribution function t(z).
It may be interesting to note that the asymptotic mean and variance of
the statistic -2n-i log h* vanish, if we put E
= EO
in the above theorem.
This shows some singularity of the limiting distribution at the hypothesis.
The statistic -2n-! log h* converges to zero under the hypothesis, in fact,
-2 log h* converges in law to the X2-distribution.
3.
~ansion
ofI
•••
•
••
the distribution of the criteri0J,l for 1: = 1:0 IQd
J! = '"0 .
~.~
..-..
mean vector '" and covariance matrix t be denoted by Xl'
H2: 1: = 1:0
~,
•.• ,~.
The LR
and IJ = "'0 against alternatives
f 1:0 or '" f fila' where Ib and FLo are a given positive definite matrix
2
and a given vector, is expressed as
K : 1:
where S =~=l (Xa-X)(Xa-X) I and X = (l/N)Z:=lXa .
For this problem, it has
been shown by Sugiura and Nagao [11] that the LR test without modification
is unbiased.
So we shall consider the asymptotic expansion of the distribution
of the LR statistic h.
as
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Let a p-variate random sample of size P. from the normal distribution with
statistic for testing the hypothesis
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The hth moment of h under alternative K is expressed
2
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Nh 1:-l( ~-j;10 )( j;1-j;10 ) I
etr[ 2'"'"
0
X
where n
~
~
N-l.
(I-hl:c-l( 1:-1+h.t-1)-1 )),
O
The characteristic function of -2 log Aunder the hypothesis
is given from (3.2) as
r p «n/2)-Nit)
Np(1-2it)/2
r (n!2)
(1-2it)
•
p
Applying the asymptotic formula (2.5) for the gamma function to each term of
r p «n/2)-Nit)/rp (n/2), we have
k
C (t)
H
2
= (1_2it)-(P/2)-P(P+l)/4
exp[-
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r::::l
+ 0 (N-k-l) ,
where Br +l t = ~::::1 Br+l(~/2). Noting that this expression is of the same
form as (2.7), we can immediately obtain the following theorem.
Theorem 3.1.
Under the hypothesis
~:
1: • 1: and P -_ PO' the distribut0
ion of the LR criterion -2 log A given in (3.1) can be expanded asymptotically
as
(3.5)
P(-2 log A ~ z) :::: P(~ ~ z) + B2N-l(P(~+2 ~ z)
-P(x~ ~ z)} + (1/6>N-2 ((3B t2 - 413) P(~+4 ~ z) - 6B22 P(X~+2 ~ z)
2
+(3B2 +4B3)
P(~ ~ z)) +(1/6).-3 ((4B4~4B2B3+B23) P(~~ ~ z)
2
3 - 3B2 )
+B2 (4B
P(~+4 ~ z) + :&2(413 + 3B22 ) P(~+2 ~ z )
- (4B4 + 4:&213 + B23 )
P(~ ~z»)
4
+ 0(n- ),
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where f
= P + p(p+l)/2 and the constant B~ is given by B2 = p(2p2 + 9P + 11)/24,
B = -p(p+l)(p+2)(p+3)/32 and B4 = p(6p4+45p3+llOp2 +9Op+3)/480.
3
The limiting distribution of -2 log A is stated in Anderson [1, p.268].
Now we shall consider the asymptotic expansion of the distribution of -2 log A
under the alternative
K2
K2'
The characteristic
fUnction of
can be obtained from the moment of A in (3.2) as C (t)
Ka
-2N-!
log A under
= C~) (t) C~) (t),
where
C~)(t) = etr~it E~l(~_~o)(~-~)'(I
The first factor
C~)(t)
+ 2itN-l
E~1(E-l_2itN-tE~1)-l).
has a similar expression to the characteristic
fUnction C (t) in (2.11).
The same computation gives us the following
K
1
expression
(3.7)
leg
Applying the formula
factor
(3.8)
c~)(t),
K2
(I
-t Z)-1
- N
2k+l N-r/2 Zr + 0 (-k-l)
= E;=o
N
to the second
we have
log C(2)(t)
l
= ~h
~it (II~-~O
u )'E0
• Eo-l( EEa-l)r+l( ~-IJO )
(It~-~O
It )
+ 0 (-k-,)
N
•
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Hence we have an asymptotic formula for the characteristic function log
C (t) by adding the expression (3.7) and (3.8).
K2
~*
=-2N-! log
~ - ~N (tr(Et~l
I~ll
- I) - log
This shows that the statistic
+
(~-~o)IE~l(~_~o»
is dis-
tributed asymptotically according to the normal distribution with mean zero
E~lEt~l(~_~O)} and further
and variance -r2 = 2(tr(Et~1 - I)2 + 2(~-~o)'
the characteristic function of
where the coefficients
~
and
~*/-r
~
can be expanded asymptotically as
are given by
~ = (1/6) (3-r- l it p(p+3) + 4-r-3 (it)3 (P+6d2+2tr3 - 3tr2»
A
2
= (1/72)(9-r-2(it)2p(P+3)(p2+3p+4)+24-r-4(it)4(p(P+l)(P+2)
+
6p(p+3)d2+24~+6tr4+2(p+4)(p-l)tr3-3p(p+3)tr2}
+l6-r-6(it)6 (P+6d +2tr3-3tr2
2
with the abbreviated notations dr
Inverting this characteristic
lJ ,
= (~-~o)'E~l(t&~l)r(~_~o)
and trj=tr(t&~l)j.
function we immediately obtain the following
theorem.
Theorem 3.2.
Under the alternative
~:
E
!' Eo
..
or
~
!'
-
~O'
the distri-
bution of the LR criterion -2 log A given in (3.1) can be expanded asymptotically as
P({l/~N-r)(-2 log ~ - N(tr(Et~l-I) - log IEt~ll+do}] :s z )
= .(z) - ~ (4-r-3t " , (z )(p+6~+2tr3-3tr2) + 3-r- l t' (z) p(p+3)]
+ ~ (16'-r-6t(6) (z )(P+6d2+2tr3-3tr2)2+24-r-4t(4) (.z) (p(p+l)(p+2)
+ 6p(p+3)dZ+24d3+6tn4+2(p+4)(p-l)tr3-3p(p+3)tr2}
+9-r-2t"(z)p(P+3)(p2.3p+4)] + Q(N-3/ 2 ),
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where d ~ (~-~o)'E~l(tE~l)r(~_~o)' tr j ~ tr(tE~l)j and T2=2(tr(EE~1-I)2+2dl};
r
~(r)(z) means the rth derivative of the standard normal distribution function
~
(z).
4.
E~Jj.on.J>!.. th~.. ~j~rj bu.s-j~~ ~-th~ .f.I.i.iH..~
:f.£.r..~~fJJt~·
The LR statistic for testing the sphericity hypothesis H :E -:.. rJI against
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the alternatives ~: E ,. r,.2I, where rJ is unspecified, from a p-variate
random sample of size N with normal distribution having common covariance
matrix E, is
~. ISIN/ 2 (~trs)-NP/2,
(4.1)
where S
= ~=l{Xa-X)(Xa-X)'.
Unbiasedness of this test criterion was
established by Gleser [2] and Sugiura and Nagao [11], whereas we already
know that in order to get unbiasedness in the k-sample situation, the statistic
(4.2)
is preferable instead of ). by Sugiura and Nagao [11].
H3'
the hth moment of
(4.3)
h
E[).* IH3 ]
~*
Under the hypothesis
is expressed as
h r{pn/2)r ({n/2)+h)
= p p rT(i)n!2)~)r (n!2)
p
,
which implies the asymptotic expansion of the null distribution of this
criterion as in Anderson [1, p.263].
(4.4)
P{-2p10g).*
~ Z} = P{~ ~z) + (1/288p2m2)(p+2)(P-1)(P-2)
x{2p3+6p2+3 p+2) (p{ ~+4 ~ z) - p{ ~
where f
= (p{p+1)/2}-1,
:s z )}
+ O{m-3),
m = pn and p = 1 - {2p2+p+2)/6pn.
The correction
factor p is so chosen that the term of order m- 1 in the expansion vanishes.
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It may be remarked that in the previous two sections we cannot use the
correction factor p as in this case.
The reason is that the hth moment of
the test statistic has two gamma functions containing h in (4,3), but only
one gamma function containing h in (2.3) and (3.2).
We shall now consider the asymptotic expansion of the non-null distribution of the LR criterion -2plog
ternatives cannot be written
A*. Since the hth moment of A* under al-
explicitl~
some different technique is necessary,
It is provided by using differential operators.
distribution wp(n,t) under
K3'
Noting that S has the Wishart
we can write the characteristic function of
-2pm-llog A* as
J
10 I (n-p-l)/2
..&..1
-.[mit
etr
p
ItJn/2 (tr s/prJ'mit
1.::-1
-~
S dS
•
By the transformation S-. HSH , for some orthogonal matrix H of order p, we
may assume t~. r • diag (~, ••• ,~) where Ajare characteristic roots of t.
Put U = (l/m)S, then the statistic U converges in probability to r as m
tends to infinity.
(4.6)
CK-(t)
-~
=
We can express the characteristic function as
np/2 .,,[mitp
m
p
r (n/2)2np/2Irl n!2
J
p
Transform the variable U to D and R by
!LJ(n-p -l)/2.,,[mit
-J
;/iDi t
etr{ - ~ U)dU •
(tr U)
p
u=Dtm,i such
diagonal and diagonal element is gi ven by that of U.
leu/o(D,R) I • ID I(P-l)/2
and trr-1u=
that the
Then
trr-~, so we have
II!l. trix
D is
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(4.7)
etr( - ~-ln)dD.
.[mitp
= ( trA)
Put f{A)
A
= diag
(~, •.• ,~).
and ~ = diag (~/~~, .•• ,~/~>p), where
Noting that the diagonal matrix D converges in pro-
bability to r as m tends to infinity, we shall expand the function f(D) in
the expression (4.7) about D =
r, that is, f(D) = etr[{D-r)~]f{A1A= r. This
gi ves by taking the integration regarding the operator
~
as constant as in
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Okamoto [5] etc.,
J
(4.8)
IDI{n/2)-J"mit-1{etr -
~ r-~)
f{D)dD
rmit)Petr(-r~- (~
p
= (2/m)n /2 -J"mitp r(n/2) ..
..[mit)loglr- l -
~lJf{A>lAI-I'.
Hence we have the following expression of the characteristic function for
-2plog
"A*.
elL (t)
-~
.[mit
= (:p)
p
r
({n/2) -J"mit)
2
p
(mit etr(-r~-{~ ..[mit)logII- mr~I.)
r (n/2)lr
.
p
. f{A) 1,A,-r.
The first factor can be expanded by using the asmptot-ic formula (2.5) for the
ga~a
function with respect to m instead of n as
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+ (1/6[m)(p2+2p-2)it + 4p(it)3) + (1/6m)(p2+2p_2)(it)2+4p(it)4)
+ o(m-3/ 2 ),
which implies
··1
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+ 4p(it)3} + (l/72m) (p2+2P_2)(p2+2p+10 )(it)2 + 8p(p2+2I>.+4 )(it)4
+ l6p2(it)6) + O(m-3 / 2 )].
The problem is how to evaluate the exponential part of CIS(t) in (4.9).
Since orCA) = Jiii1p (trA)f(A), we must regard the order of 0 as .fm. Applying
k
the formula logII-(2/m)fol = -E ~l (2/m)r tr(fo)r/r + o(m- - i ) to (4.9) we
can write
(4.12)
etr ( - fo - «n/2)- .fmit) log II - (2/m)fof) f(A)
where
Ao(o)
1
2
=m
tr(fo)
2it
- ~ tr (fo)
Al (0.) = 3rJ-m tr(fo)3 - 2;t tr(fo)2+
~:rm+2
~(~) ~ :2 tr(r~)4. ~$m tr(r~)3+
p;-ti,pil! tr (r~)2.
tr fa
Note that the result of applying each term Ao(o), Al (0) and A2 (O) to f(A) is
O(l).f(A). We can expand the expression (4.12) as
We now evaluate the first part - [exp A (o)]f(A).
o
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14
1
2 2it
exp (- tr (ro) - -r= trro}r(A)
(4.14 )
m
=
""m
L• ~! (~tr
(ro)2-
3i :
trro)r rCA)
reO
GO.
=
,
{\'.l. \'
~r~ ~
r=O
(r)
k+'=r
~t~ ~
k~)
k~
kl: •.• k:
(k ,(,)
P
. rCA),
where l:(k),(J) means the sum or all possible combinations of non-negative
integers k l , .•• , kp and '1' ... , Jp such that kl+···+kp=k and J 1+··.+,p=,·
p
2ka +J
~mit -2k-J
Since the equality na=1 0a
a rCA) • (.[mitP)2k+,(trAr p
holds
wher3 (a)k = a(a-l)
(4.15)
~
~
reO
1;.
r.
L
k+J=r
(a-k+l), we can simplify the above expression as
(V
J
r . P •
(r) (-2it;Jifmitp) r+k { trr2 Jk trr
(trA)""mt
k
mk +(J!2)
(trA)2
tr
We can easily see that (.[mitP)r+k can be expanded asymptotically with
respect to m as
(4.16)
(.[mitP)r+k = (.[mitp)r+k_(.[mitp)r+k-1(1/2) {(k)2+2rk +(r)2)
+ (.[mitp)r+k-2(1/24) (3(k)4+4(3r+2) (k)3+6r(3r+1) (k)2+12r2 (r-1)k
+ r(r-1)(3r-1)(r-2)} + o(m(r+k-3)/2),
which makes it possible to simplify the summation in (4.15) giving
(4.17)
(exp Ao(O) )f(A) = (exp Bo(A) H1 +
2
B (A.) = - pt2 (p E!:o
(trA)2
_2
B (A)
1
Jm
+
B (A.)
2
trr )
trA
2
B (AJ = _ 2(it)3p (p trr
_ trr )2 _ itp trr2
2
1
(tr.A.)
trA
(trA)2
m } rCA),
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2
x (13p trr -4 trr)+3(it)2{11p2 (tr~,2- 8p tr~trr }] •
(trA)2
trA:
(trA)
(trA)3
Applying the operator [1 + m-i Al(a) + m- l {A {a) + A (a)2/2 )] to this ex2
l
pression we can see that the exponential part or (4.9) can be written
asymptotically as
(4.18)
(1 +
Jm (Bl (A)+Al (a»
+ ~(B2{A)+Al (a)B l {A)+~{a)+Al (a)2/2)
2
+ o{m-3/ )}.(exp BO{A)} rCA)
I~.
Noting that the formulas
.r
B (A)
m- J/ 2t:'(fa)" e OrCA;)_ {itp)"(trA)-liliJiitP-
J
B· (A)
e 0
trr"
hold for any non-negative integers k and J, we can compute each term in (4.8),
obtaining
(4.19)
exp(- fa - (~ - .[mi~ log II
Bo(f)
=e
f(f) (1 +
Jm cl(r)
-~
rail f(A)
I&:I'
+ ~ c2 (r) + 0 (mJm
Cl(f)=(2/3)(it)3p(-3(t2-1)2+2t3-3t2) + (1/6)it(2p2+P+2-6t2 )
»)
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16
+6(t2-1)(15t2-l4t3) + l8t 4 + 2(2p2+p-16)t3-3(2p2+p-4)t2}
+ (1/72)(it)2 (396t~ - 288t
3
2
- 24(2p2+p+8)t2+ (2p2+p+2)(2p +P+26)).
= pj-l(trrj)/(trr)j. Combining this result
j
with the equation (4.11), we can finally obtain the asymptotic formula for
with the abbreviated notation t
the characteristic function C (t) of the statistic -2pm-tlog A given by
K3
(4.2) •
2
CK.,(t) = exp rJ"mit log (! trr)P/lrlJ - pt2 (p trr 2 - :I)]
-~
p
(trr)
(4.20)
(1 +
Jm D1(r)
+
~ D2 (r) + 0 (;J-m)}
Dl(r)
= (2P/3)(it)3(-3(t2-1)2+2t3-3t2+1)+(1/2)it(p2+p-2t2)
D2 (r)
= (2/9)p2(it)6((3(t2-1)2+3t2-2t3)2
+ 1 - 6(t2 -1)2+4 t3 -6t2 ]
2
2
+ 6t4 +2(P+3)(P-2)t -(3p +3p-4)t2 +p +p+2}
3
+(1/24)(it)2(132t~-96t3-4(5p2+4p+14)t2+3p4+6p3+23p2+16p+8),
th~
A** = m-i r-2plog A* variance ~2
limiting distribution of the statistic
m log (tr(r)/p)p /Irl)] is normal with mean 0 and
= 2p(p(trr2 )/(trr)2
- I}
as m tends to infinity.
has already been obtained by Olkin and siotani(6].
characteristic function of
A**/~,
This result
GIeser [2] also gives a
similar result, but his result seems to be incorrect.
I
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el
+(p/3 )(it)4(t2-l)2(26t2-3p2_3P-8) + 2(t -1) (15t2 -14t )
2
3
Which implies that
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By inverting the
we have the following theorem, from (4.20).
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Under the alternative K : ~ ~ aFI, the asymptotic ex3
pansion of the power function of the LR criterion -2plog A* given in (4.2)
Theorem 4.1.
for sphericity is expressed as
p«1/~m~)(-2plogA*-mlog(trt/p)P/I~I)] ~ z)
(4.21)
= t(z)+
Jm (2p/3~ )t'''(z )(-3(t2-l)2+2t3-3t2+1)+(1/2~)t(z )(p2+P-2t2)]
I
~(2p2/9~6)t(6)(Z)((3(t2-1)2+3t2-2t3]2+1-6(t2-1)2+4t3-6t 2 )
+(p/3~4)t(4)(Z)(t2-1)2(26t2-3p2_3P-8)+2(t2-1)(15t2-14t 3 )
+6t4 +2 (p+3 ) (P-2)t3-(3p2+3P-4 )t +p2+p+2 )
2
+(1/24~2)tn(Z)(132t~-96t3-4(5p2+4p+14)t2+3p4+6p3+23p2+16p+8)],
Where t
j
=
pj-l(tr~j)/(trt)j, ~2=2p(t2-1) and t(r)(z) means the rth de-
rivative of the standard normal distribution function t(z).
Let
~l'
••• ,
~
wi'\ih mean vector
be a random sample from p-variate normal distribution
~
and covariance matrix
1b for a
statistic for testing the hypothesis H4 :1:1 •
ternatives K4 1 ~i ~ l:j for some i, j (i
(5.1)
A•
N /2
(~=l ISafNal a
N
where Sa =
)
I
f
~
• ••• •
~
The LR
against the al-
j), is given by
/
Is/N IN 2,
N
ex
I
a = l:j=l Xaj No
a (~j-Xa
- )( Xaj-~
- )' , -X
l:j=l
= 1, 2, ••• , k.
~
and S = ~=lSa'
N = ~=lNa. If we m:>dify this criterion by reducing the sample size Na to
the degrees of freedom no = Na-l,
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18
with n = ~=l na , we have an unbiased test in the univariate case (Pitman
[7]h and in the two sample case for arbitrary p (Sugiura dnd Nagao [11]).
The asymptotic expansion of the
distributi~n
of this criterion under the
hypothesis is stated in Anderson [1, p. 255]. The limiting non-null dis_J.
tribution of the statistic -2n 2 log A* can be obtained from the characteristic function by the same argument as in the previous section.
However,
so far as the limiting distribution is concerned, it is simpler to use the
following lemma, which is a direct extension of Siotani and Hayakawa [9].
Lemma.
for fixed P
a
f(Ul, ••• ,Uk )
Let na Ua have the Wishart distribution wp(na,lb) and Ib = P~
such that ~=l P = 1. Suppose a real-valued function
a
is continuously differentiable with respect to each variable.
Then the statistic
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el
is distributed asymptotically for large n according to the normal distribution
with mean zero and variance
2~=1 p~l
the symmetric matrix having
(l+~ab)/2) of/Ou~~)
Ua =
(U~~»
tr(o(a)f)lb}2,
where O(a)f means
as its (a,b) element for
and Kronecker delta ~ ab'
Putting f(U l , ... , Uk) = log I~=l PaUa I - ~=l Pa log IUa I in the above
lemma and noting that the equality «((l+~ij)/2}OlogIAI/Oaij) = A-I holds
for any positive definite matrix A, we have the following theorem.
For testing the hypothesis H4 : El =.•• =I1t against all alternatives, the non-null distribution of the modified LR statistic.
Theorem 5.1.
2
-Tn
.... ~
[logA* - n 10g(jEI/.Itr=1
~a
lral }]
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is asymptotically normal with mean zero and variance ~=lPatr(~~-1-I)2,
where
~ = ~=lPdEa.
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REFERENCES
[1] Anderson, T. W. (1958).
Analysis.
[2]
An Introduction to MUltivariate statistical
Wiley, New York.
Gleser, Leon J. (1966).
A note on the sphericity test.
Ann. Math.
Statist. 37, 464-467.
[3]
Ito, K. (1956).
Asymptotic formulae for the distribution of Hotelling's
generalized T~ statistic, I.
[4]
Nagao, H. (1967).
Monotonicity of the modified likelihood ratio test
for a covariance matrix.
[5]
Okamoto, M. (1963).
Ann. Math. Statist. 27, 1091-1105.
J. Sci. Hiroshima Univ. to appear.
An asymptotic expansion for the distribution of
the linear discriminant function.
[6]
Olkin, I. and Siotani, M. (1964).
of a correlation matrix.
[7]
Pitman, E. J. G. (1939).
scale parameters.
[8]
Siotani, M. (1957).
On
Ann. Math. Statist. 34, 1286-130l.
Asymptotic distribution of functions
Technical Report No.6, Stanford Univ.
Tests of hypothesis concerning location and
Biometrika 31, 200-215.
2
the distribution of the Hotelling's T -statistic.
Ann. Inst. Statist. Math. Tokyo 8, 1-14.
[9]
Siotani, M. and Hayakawa, T. (1964), Asymptotic distribution of functions
of Wishart matrix.
The Froc. Inst. Statist. Math. 12 (in Japanese),
191-198.
[10] Sugiura, N. and Fujikoshi, Y. (1968).
Asymptotic expansions of the
power functions of the likelihood ratio tests for multivariate
linear hypothesis and independence.
Submitted to Ann. Math. Statist.
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[11] Sugiura, N. and Nagao, H. (1967).
Unbiasedness of some test criteria
for the equality of one or two covariance matrices.
Submitted to
Ann. Math. Statist.
[12]
Welch, B. L. (1947).
The generalization of "Student's" problem when
several different population variances are involved.
34, 28-35.
Biometrika.
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