EXPECTATIONS AND ASYMPTOTIC DISTRIBUTIONS
FOR THE j-th esf's OF TWO MATRICES
by
D. J. de Waal*
Department of Statistias
of No~th CaroZina at Chapet Hitt
Unive~sity
~d
Unive~sity
of the OPange
F~ee
State
Institute of Statistics Mimeo Series #982
February, 1975
* Research partially supported by the C.S.I.R.
EXPECTATIONS AND ASYMPTOTIC DISTRIBUTIONS FOR THE j-th esf's OF TWO MATRICES
D. J. de Waa.l*
Univepsity of Nopth CaroZina
Univepsity of the Opange Fpee State
1. INTRODUCTION
be distributed N(M; E 0 l q) , then for q ~ p B = XXI
uuI
is distributed W(E,q,n) , n = E-l vu~
•
Let A(p x p) be distributed indeLet
X(p
pendently as
x
q)
W(E,n) , then the first matrix we are interested in is
VI ;:; A-1 B.
1.1
Since
plim nV l = n
n~
we \'lil1 write the j-th elementary symmetric function of nV l ' Le. tr j nV 1 ,
under the assumption of linearity (see Madansky and Ok1in (1969)) as
1.2
The second matrix we are interested in is the matrix
1.3
where A is partitioned as
A = (All (q x q)
A
2l
Al2 )
A22
* This research was partially supported by the C.S.l.R.
-2-
and A1l . 2
A1l • 2
~
= All
-1
- A12A22A2l'
W(L ll . 2 ,n)
-1
G = A12A22A2l'
and conditional on AZ2
G ~ W(L l1 . 2 , P - q, 6)
~
in a similar way than A.
It is well known that
independent of All • 2 that
-~
-~
= Lll.ZSA22BiLll.2'
S
-1
= L1Z L22
.
L
is partitioned
If we assume (Sugiura (1969)) that
it follows that
-~
-~
plim nAIl ZGA II 2 =
n-+oo
•
•
e•
Hence, under the assumption of linearity we will write
3tr.e
J
1.4
ae
It may be pointed out that
VI
is the matrix associated with test of
means and V2 with the test of independence.
Asymptotic distributions for the
j-th esf's in these two cases were considered in de Waal (1975) but due to an
overlooked error in equation 2.7 of de Waal (1974) these results have to be
reconsidered.
We shall consider here the asymptotic distributions of
is however interesting to know more about
E(~i) ,
~l
and
~Z.
It
i = 1,2 , and therefore we
shall first, in section 3, consider these expectations before we derive the
asymptotic distributions in section 4.
will be given.
In section 2 a few preliminary results
-3-
2. PRELIMINARY RESULTS
Lemma 2.1. de Waal (1973)
atr.E
. 1
a~ = (_I)J-
2.1
~
1. (_I)j-i Ei - 1 tr . . E •
J-1
i=1
Lerrma 2.2. Let F = X'A -1 X for X defined in section 1 and q
R(q x q)
2.2
any positive definite symmetric matrix, then
E etr(it nRF)
Proo:
f
= II - 2itRI-~
etr{itRn(l - 2itR)-I}
·
F = XI,A-1 X:;: .*,
*-1 X:;,
* . where
SInce,
!!to .'A-
A* ,... W(Iq,n)
and A.
< p ,
we can assume
E=I
+
O(n- 1 )
X*,... _
N(Ek2M, I
0 I ) and
p
q
E-~M in the densities of X
and M as
From Crowther (1974) the density of
F
= X'A-1X
conditional on A
is given by
IAI~q etr(-~1'M) IFI~(p-q-l)
00
L L .
k=O
( -~F (Y .-
where the expectation is taken w.r.t. the density
Hence
1
EyC
K
K (41') Kkl
. i M) I A (Y -
fi
-.!. M))
.fi
,
F > 0
-4-
etr(-~I~)Ey
=
r (J§(n+q))
p
r (Jm)(-int)J;pq
.w
IRI-2
etr{2i~t R-lCY - i
1:2
M) 'A(Y -
-! M)}
12
etr C-J;M 1M)
p
E II - ~cy - -! M)R-ICY _ -! M)II-J§Cn+ q)
y
1nt
1:2
1:2
= C-2it) -.wq IRI-~
etr( -#1IM)Ey
etr{Z~t (Y
-
~ M) I
(Y _ -! M)R- I }
+
O(n- l )
1:2
=
(-2it)-tpq IRI-~ etr(-#1IM)etrC- 4~t M1MR- l )
II - 2~tR-II-tpetr{=
Substitute
II -
l
Sit(R-lM1MR-ICI - ZitR-l)-l}+DCn- )
2itRI-tp etr{it M'MeI - 2itR)-IR} + D(n- l )
E-l§M for M and the lemma is proved.
. 1nteresting
.
I t 1S
to note from 2.2 that ignoring terms
D(n- I )
nRF has
a noncentral Wishart distribution with p degrees of freedom, noncentrality
parameter M'E-IM and covariance matrix R. The distribution of tr nRF
ignoring terms DCn- l ) will be denoted by x2 (R,l§Q) where Q = M'E-lM. If
pq
2
R = I then it follows that xpq (I,~) = ..pq C~rQ) , the noncentral chi-square
x:
distribution with pq degrees of freedom and noncentrality parameter ~rQ.
The density of Y = tr nRF ignoring terms D(n- l ) can be written in the
fo 1I0011ing form:
-5-
r :W)y*Pq-l
2.5
';R/Wr(Wq)
2-tq(q-l)
(2~i)~q(q+l) JR(Z»O
..
etr (z) Iz I
-W
(-W)K
00
L LK (-W )
kI
2q k .
k=O
CK(-~(I - Z-ln)R-l)dZ,
where the integration is taken over
yk
y > 0 ,
Z = V + iW symmetric with V > Vo > 0
fixed and W ranges over all symmetric matrices.
The density is not hard to
derive if the integral (Khatri and Pillai (1968))
2.6
J
lul-t(p-q-l) a (U)C (U)du ... dU
K
2
q
q
D
rq(w)rq(~q)(W)KCK(Iq)
= ~---~2~---------------~
~~q r(~q)(wq)k
D = {O < uq < ••• < u l }
and a q (U) = .TI. (u.1. - u.)
, is used after expressing the OF function in the
J
I
l.<J
noncentral Wishart density as
where U = diag(u l ,
,U )
q
~
~
2.7
q
-1
OFl(W; ~R S)
=
,
ul
2~q(q-l)
k ( 1)
(2~i)2q q+
=1
-
U
z ... -
f
R(Z»O
uq '
etr(Z)
Izl-~
Corollary 2.1. Let V = B~A-IB~ if q ~ p, R(p x p)
matrix and
2.8
Q
= E-~MM'L-~ ,
E tr(nitVR)
= II
any p.d. symmetric
then
- 2itRI-~q etr(itn(I - 2itR)-lR) + O(n-l) •
Proof: This follows from 2.2 using the fact that F(q
as a noncentral multivariate beta distribution with
of freedom while V(p x p)
distribution with
I
1
etr(%Z- nR- S)dZ .
(q,n)
x
q)
is distributed
(p, n + q - p)
degrees
is distributed as a noncentral multivariate beta
degrees of freedom.
-6-
3. EXPECTATIONS
Theorem 3.1.
If F(q x q)
distribution with (p, n
meter n = WE- 1M, then
3.1
=
E(tr.F)
J
is distributed as a noncentral multivariate beta
+
q - p)
degrees of freedom and noncentrality para-
j
1
L
(n+q-p-2)(j) i=O
~-~)
( J-1
( P - 1') (j-i)
,..
tr.~G
1
where
(a) (j) = a(a - 1) ... Ca - j + 1) .
Proof: We consider F
such that
XJ
= X'A-IX
as was defined in lemma 2.2.
Then according to the definition of
XJCp x j)
are independently normally distributed with covariance
Let
X = (XJXJ )
X the columns of
E and
E(XJ )
= MJ
F now becomes
It is obvious that
FJJ
tribution with p and
is distributed as a noncentral multivariate beta disn
+ j -
p degrees of freedom and noncentrality para-
meter
But (de Waal (1972))
3.2
j
1
~
-_-:::"'--CT:·""-) l..
(j+n-p-2) J i=O
ElF JJ I
Substitute in
3.3
=
Cp _
1') (j-i)
tr i CMI~-l~1
Jld 1'IJ ) •
-1where
IJJ
IAJJI
matrix A and
is the summation over all principal minors of order
j
of a
(Saw (1973))
3.4
the theorem follows.
Corollary 3.1.
If
V(p x p)
distribution with
(q,n)
3.5
=-~""'r:"'''''''
E(tr.V)
J
3.6 E(trJoV z) =
is distributed as a noncentral multivariate beta
degrees of freedom, then
f
1
(n-Z) (j)
1
f
C)
i=O
(n) (i)
i=O
(n-Z) )
(J?-~). (q
)-1
(~-~)
- i) (j-i) tr .n
1
(p _ q - i) (j-i) tr (P(I _ P) -1)
)-1
o
l
(Note that if R is defined as
= trj(R(I
_ R)-l) .)
Proof: Since Aii.2G given AZZ is distributed as a noncentral multivariate
beta distribution with
~ =
parameter
3.7
But
"
degrees of freedom and noncentrality
-~
-~
Ll1.2eAZZ8'Lllo2
' it follows from 3.5 that
E(tr~(v~IA22)) =
J
(p - q, n)
1 (0)
(n-2) )
A (p - q x p - q)
Z2
~
W(L 22 ,n)
and hence (de Waa1 (1972))
-8-
The unconditional expectation 3.6 follows easily.
4. ASYMPTOTIC DISTRIBUTIONS
Theorem 4.1.
Ignoring terms
(~l
4.1
- tr.n
J
O(n-l)
trnr.)
+
J
2
X- (r.,~)
.pq J
~
where
j
r. = (_l)j-l
J
( _l)j-i ~~ni-l t r . . n~6
L
J-1
i=l
Proof: The characteristic function of
4.2
~~
(t)
1
Using 2.8 with
Theorem 4.2.
4.3
~l
= E exp(it~l)
= exp(it trjn)etr(-itnrj)E
is given by
etr(itnVlr j ) + O(n-l)
R replaced by r j , the theorem follows.
.
Ignor1ng
terms
(/;2 -
0 ( n -1)
tr.e + treA.)
J
J
2
~ X (
q p-q
)
(A.
J
,~e)
where
j
A.
J
Proof:
= (_l)j-l L
i=l
( -1) j -i
ei - l tr . . e
J-1
From 1.4 and 2.8 the conditional c.f. of
/;2
given A22
is given by
-9-
II - ZitA·I-~(p-q)
J
etr(it6(I - ZitA.)-lA.)
J
where
J
Hence the unconditional c.f. of
+
O(n-l)
becomes
~2
But
,-~ (1
2'
)-1 A L-~ . 8 }
.
4 • 6 EAZ2etr {ltA
22 8 Lll . 2 - ltA j
j 1l z
= II
=
-~.
-~
- ZitLzz8'Lll.Z(I
- 21tA j ) -1 Aj Lll
• 28 I-~
etr{it6(I - ZitA.)-lA.}
J
J
+
O(n-l)
and under the assumption
-~
-1 -~.
Lll.2LlZL2ZLll.Z
1
=~
+
O(n
-1
) .
Substitute 4.6 in 4.5 and the theorem follows.
REFERENCES
Crowther, N. A. S. (1974): The exact noncentral distribution of a quadratic
form in normal vectors. Technical report 87, Department of Statistics,
Stanford University.
de Waal, D. J. (1972): On the expected values of the elementary symmetric
functions of a noncentral Wishart matrix. Ann. Math. Statist. 43,
344-347.
-de Waal, D. J. (1974): An asymptotic distribution for the j-th esf of the
generalised Hotelling's beta matrix. Inst. of Statist. Mimeo Sere 966,
Univ. of North Carolina.
de Waal, D. J. (1975): On the asymptotic distributions of the esf's considered
as test statistics in two multivariate tests. Inst. of Statist. Mimeo
Sere 975, Univ. of North Carolina.
-10Khatri, C. G. and Pi11ai, K. C. S. (1968): On the noncentra1 distribution of
two test criteria in mUltivariate analysis of variance. Ann. Math.
Statist. 39, 215-226.
Madansky, A. and 01kin, I. (1969): Approximate confidence regions for constraint parameters. frmZtivariate AnaZysis II edited by Krishnaish, P. R.,
216-286.
Saw, J. G. (1973): Expectation of elementary symmetric functions of a Wishart
matrix. Ann. Statist. !, 580-582.
Sugiura, N. (1969): Asymptotic non-null distributions of the likelihood ratio
criteria for covariance matrix under local alternatives. Inst. of Statist
Mimeo Sere 609, Univ. of North Carolina.