Download Hayakawa, T.; (1970)On the distribution of a trace of a multivariate quadratic form in the multivariate normal sample."

This research was supported in part by the National Science Foundation
under Grant No. GU-2059 and the Sakko-kai Foundation.
ON THE DISTRIBUTION OF A TRACE OF A MULTIVARIATE QUADRATIC FORM
IN THE MULTIVARIATE NORMAL SAMPLES
by
Takesi Hayakawa
University of North Carolina at Chapel Hill
Institute of Statistical Mathematics, Tokyo
Institute of Statistics Mimeo Series No. 682
Department of Statistios
Univerasity of North CaroZina at Chapel Hill
APRIL 1970
On the Distribution of a Trace of a Multivariate Quadratic Form
in the Multivariate Normal Samples
Takesi Hayakawa
Department of Statistics
University of North carolina
Chapel Hill, North Carolina 27514
This paper considers the derivation of the p.d.f. of the trace of a noncentral multivariate quadratic form using a polynomial P (T,A) defined by
Ie
e
the author and compares with the results of Katz et al and Ruben.
The com-
plex case is also discussed.
KEY WORm
Non-central multivariate quadratic form
generating function
Kronecker product
representation
Hermitian matrix.
CLAsSIFICATIm Nlt8ER:
LID
This :research b1as suppoFted in part by the Nationa't Science Foundation
undeF GFant No. GU-2059 and the Sa1<ko-kai Foundation.
ON THE DISTRIBUTION OF A TRACE OF A MULTIVARIATE QUADRATIC FORM
IN THE MULTIVARIATE NORMAL SAMPLES
Takesi Hayakawa
University of North Carolina at Chapel Hill
Institute of Statistical Mathematics, Tokyo
1.
INTRODUCTION.
Recently the probability density function (p.d.f.)
of latent roots of a multivariate non-central quadratic form and of a trace
of it were obtained by the use of a polynomial
Hayakawa [3].
P (T ,A)
introduced by
K
However, it only shows that if we use the polynomial
P (T,A),
K
we ean represent the p.d.f. 's in terms of a power series representation.
e
In
this paper, we discuss another type of representation, that is, r-type rep;esentation of the p.d.f. and give a more concrete form than Hayakawa [3].
By using this representation, we can compare with the results of Ruben [8]
and Katz et al [7].
We also give a p.d.f. for the case of the complex vari-
abIes.
2.
(tnSn)
NoTATIONS PHD SM USEFlL RESULTS.
real arbitrary matrices each of rank
i
positive definite symmetric (p.d.s.) matrix.
polynomial
(1)
PK(T,A)
T
and
m, and let
U be
A be an
mxn
nxn
Hayakawa [3] defined a new
as follows;
~(-TT')P (T,A) = 'lfmn
<-.1,: f ~(-2iTU')~(-UU')C
U
K
e
Let
(UAU')dU,
K
This research was supported in part by the National Science Foundation
under Grant No. GU-2059 and the Sakko-kai Foundation.
2
where
JC
Ie
is a partition of k
= (kl'k2 , ... ,km),
k
into not more than m parts, i.e.,
= k l +. u+km,
1s a zonal polynomial of
UAU'
k
~
l
k
2
~
... ~ k
m
~
0,
corresponding to a partition
and
JC
CJC(UAU')
of k,
James [5].
PJC(T,A)
has the following properties.
=
(2)
P (O,A)
(3)
PIe (T,I)
n
(4)
IpK (T,A) I
Ie
(n )
2 JC
=
~
eJC (A) Ie K(In )
t
HK (T),
et!t(TT,)(n )
2 Ie
eK(A)/e K (1n ),
where
(a)
and
K
H (T)
K
(a)
=
n
•
a{a+l) ••• (a+n-l)
=
r(a+n)
rea) ,
is a generalized Hermite polynomial of matrix argument
The generating function of
P (T ,A)
T.
is given by
Ie
(5)
L
K
P (T,A)C (W')
K
K
k t (n) C (1 )
2 K K m
and the right hand side (R.B.S.) of (5) converges absolutely with respect to
U.
d(BI)
onal groups
H (T)
K
and
and
d{H )
2
Oem}
PIe(T,A)
are the orthogonal invariant measures on the orthog-
and O(n) ,
respectively.
A detailed discussion of
may be found in Hayakawa [3].
We give here some useful lemmas which will be applied to the representation of the p.d.f. of a trace of a non-central quadratic form.
3
lfr.t1A 1.
(6)
• • .AJ _ _J . AJ
],
-'1c-2 -"'k-1 -"'k-1
where
.f.
D
1,2, ••• ,k
and
AO -
PRooF:
0f
for convenience.
From the definition of
P (T ,A),
I<:
we construct the generating func-
tion of
k
co
(7)
l
(-x)
k-O
k!
l
I<:
• Wtg;')
'IT
=
P (T,A)
I<:
I
e.tIt(-2iTU')et'1t.(-UU')et'1t.(xUAU') dU
U
det(I_xA)-(m/2) e.tIt(TT').
4
= det(I_xA)-(m/2)
~(T(I-(I-XA)-l)T'),
IlxAll
where
HAll
means the maximum value of the absolute value of the charac-
teristic roots of
A.
We expand the R.B.S. of (7) with respect to
II xAll
< 1
<1,
by noting that
x
using
x2
log de.t(I-xA)
III
2
xtJt.A + T Vt.A +... +
xk
k
k
J:JtA + •••
and
-1
(I-xA)
•
I
+
xA
k k
+ x 22
A + ••• + x A + •••
Hence
R.H.S.
III
e.xp[- ~ log det(I-xA)] e.tJt(-xTA(I-xA)-l.r,)
Here we set
and
A
O
=
0J
for convenience.
5
We can obtain the value of
dents of
k
x
(_l)kt P (T ,A)
Ie Ie
on the two sides of (7).
by comparing the coeffi-
By differentiating the left hand
x and by setting x. 0, we have
side with respect to
(_l)k t P (T,A).
K Ie
Let us divide the R.B.S. into two parts such that
R.B.S.
•
~xp(g(x»
•
{l + g(x) +
exp(f(x»
1 g2 (x)
2T
• {1 + f (x) +
1 gk (x) + ••• }
+ ••• + kT
2~ f2 (x) + ... } ,
where
OIl
g(x)
The degrees of
k+l,
R.B.S.
x
f(x)·
•
L
j-k+l
A
j
x.1
in the second factor are all greater than or equal to
and so there is no contribution to the coefficient of
Bence we need only consider the first factor.
.~,
g'(O)
g"(O)
•
k
x
in the
We have
2~, ••• , g(k)(O) • k!~,
and
g(O)
•
0
Now we differentiate
g(k) (0)
•
(=!a'
k
by definition).
times the power series of
k!~,
I
taO
(~)g(l)(X)g(k-l)(X)1
xeO
g(x)
and set
x · O.
6
In the same way we have
and
(gP(x»(k)\
•
0,
for
P
x-O
~ k+l.
Henee combining the results, we have
which completes the proof.
fxAr.9LES:
k .. 1:
k-2:
P (T,A) ... (-l)A
l
l
!
(2)
CI
-
;
tJtA + .tJtTAT',
P(2)(T,A).2I[~2+i~)
2
... 21[.tJt(: A2 - TA T') +
i<.tIt(~
A - TAT') }2],
7
etc.
REMARK.
If we set
A = I.
n
then we have inunediately from Hayakawa
[4. (18)]
2 PK(T.!)
n
K
3. THE PID.F. OF
•
l
H (T)
K
bt
K
•
I:-1ux'.
(_I)k L(mn/2)-1 (~T') •
k
Let
X be an mxn
(m
S
n)
matrix
whose density function is given by
(8)
Where
I:
is an
mxn (m S n)
is an
mxm p.d.s. matrix.
matrix such that
B
is an
E(X) • K
nxn
p.d.s. matrix and M
and rank M. m.
Let
A
be
an nxn p.d.s. matrix.
I.m1A 2.
(PotUe:r series ztep:resentation.)
p.d.f. (8). then the p.d.f. of the latent roots
E-~XAX'E~ is given by
Let
X be distributed with
A· diag(Al •••• 'A~
of
8
2
n(m 12)~(_ ~-~'I-l)
(detA)~(n-m-l)
______
2_-,-"'r.=_
(9)
r (n)r (~(det2AB)(m/2)
m2
r (a) _ n%m(m-l) nm
PRooF:
See Hayakawa [3].
m
r (a _ a-l)
and
2
a-l
(Ai-Aj)
i<j
m2
where
n
Next we give another type (say, r-type) expression for this p.d.f.
THEOREM I.
rr-type representation.)
Let
d.f. (8), then the p.d.f. of the latent roots
I""'XAX'I"'" is given by, for
IIABII
X
be distributed with p.
A .. diag(Al, ••• ,Am)
of
< p,
2
n(m 12)~(_ lE-~-~')
______
2_ _......,,-_
(10)
r (n)r (m)(de.t2AB)m/2
m 2
m 2
e.tJc.(-..!..
2p
The power series converges absolutely for
PRooF:
where
•A
We decompose
H
l
y ..
I""'XA~
A~ Cl.
A > O.
as
is an orthogonal matrix of order
column are positive and
A)(detA)~(n-m-l)
,Ii,,;, ~
W\o'"1:f (A '
l
~
.... ,Am)
m whose elements of the first
and
A , ••• ,Am
l
are latent
9
roots of
yyt
= I:-%XAXtt-~,
and
L
is an
mxn
Stiefel matrix such that
LL' = I .
By inserting this decomposition into (8), we have the joint p.d.
f. of A,
HI
m
and
L:
(11)
If we set
L -+ LH ,
2
H €O(n),
2
variant with respect to
O(n)
J!iBJ!2.
Hence
d(L)e.tJL(_lA).1..
I f
where
C=
2p
LL'·I
2m
Ul (LH )t = 1
and
m
2
2
del)
remains in-
Then the integral with respect to
is the same form as (5) with
- pI)~ ,
f
H •
2
then
U·
12 A~L
and T ...
Oem)
and
iT-~MB-IA-~(C-l
.
eVL[_l
2
A~LH
2
(C- l _
I)H'L'A~
p
2
Oem) O(n)
m
The proof of the absolutely convergence is easily achieved by using (4),
which completes the proof.
NoTE.
Since
I
Al>···>Am>O
e.tJL(- Jl
2p
A)(detA)~(n-m-l) n
(Ai-Aj)C (21 A)dA
i<j
~
10
...
we have the following relations.
(12)
III
e.tIt(- ~t-lm-~')(detAB/p)mn/2.
I
This formula can be obtained from (7) directly if we replace
T with
-kI-~MB-lA-~(C-l - ~)-~,
and
Hayakawa [3] obt'ained the p.d. f. of
~,
Vtt-~XAX'I:-';; as
-p,
respectively.
a power series
We give this as Lemma 3 for comparison wi th Theorem 2.
representation.
l.Et+1A 3.
C- l -
A with
x with
(POtJ'er
Series representation.)
p.d.f. (9), then the p.d.f. of
T ...
~
Let
A
be distributed with
is given by
(13)
e.tIt(-
tz:-~-~')
T(mn/2)-1
r (mu) (de.t2AB) m/ 2
2
PRooF:
r
1
<i)k
k-O k I (mn)
2k
L P1/i2I:-~MB-lA4-C~ ,e-1),
Ie
See [3].
THEOREM
2.
(r-type representation.) Let
f. (10), then the p.d.f. of
T'"
~A
A be distributed with p.d.
is given by
11
(14)
-1
C
where
C
I)
-P ,
A~BJ!z.
III
The series converges absolutely for
PRooF:
T >
By applying a Fourier transform to
T
o.
III
Vc.A
and inverting it, we
Q.E.D.
obtain (14) easily.
-
4. THE
p.d. f. of
RELATION WITH A llUVARIATE QUADRATIC FORM. We can derive the
.tItt-!xAx'
by anotheT way.
We denote
-Xl
x
X
III
2
•
•
m
x
III
Xa
III
(xa l' •• "x00 ). and
(Xl ,x2 ' ... ,xm)
mean
uct of
and
II
,
M
ll2
III
t
and
B.
.
,
•
llm
•
(lla l, ••• ,ll~ ),
ll· (lll'll2'''.' llm)'
and a covariance matrix
II
a
M as
],11
x
where
and
X
t8B,
On the otheT hand,
where
then
Qt
a
x
III
l,2, ••• ,m.
Let
is distributed with
denotes a Kronecker prod-
.tItXAX' .. In
x Ax'
a·1 a
a
III
x[I aDA] x , •
m
Hence the problem is reduced to the one of the univm:iate non-centTal
e
quadratic form.
To compare with the Tesults of KDtz et a1 [7] and Ruben
(8), we can assume that
A is a diagonal matTix whose diagonal elements
12
The p.d.f. of tJtxAx'
is derived as follows.
J exp[- i<x-~)(~-leB-l)(x-~)']
1
(15)
mn
(211')T
•
n
m
(dett)2'(detB)2'
mn
(211')T
dX
Tax(I QiDA)x'
m
n
m
(dett)2'(de.tAB)2'
J exp[-
i(E-1QiDC-l)x'
T-xx'
From the above integral, we derive two types of the representation.
Tt£OREM 3.
(POIJJer series representation.)
p.d.f. (8), then the p.d.f. of
T·~'
Let
X be distributed with
is given by (16)
exp[- ~(E-lQiDB-l)~,]
T(mn/2)-l
mn
n
m
22 r (1111) (dUE) 2'(detAB)2'
2
where
and
P (.,.)
k
c.
PRooF:
m· 1 in the definition of PK' (T ,A)
ifni;.
As for Lemma 2.
CoROLLARY,
(17)
is a polynomial for
The p.d.f. of
T = m-lXAX'
exp[- ~(t-~n-l)~,]
T(mn/2)-1
mn
m
22 r{mn) (de.tAB)2'
2
is given by
13
We can easily show that (17) is the same form as (13).
Here we compare with the results of Kotz et a1 [7].
Katz et a1 showed
the following Lemma.
~.
(Kota et aZ.)
with mean . 0
T
x· (xl' ••• ,x )
n
and covariance matrix
diag(~, ••• ,an)'
p.d.f. of
Let
a
1
~ a
2
I
~ ••• ~ an >
= (x+b)A(x+b)'
and
n
0,
be normally distributed
A be a diagonal matrix, i.e.
and
b · (bl, ••• ,b ),
n
then the
is given by
00
I
(18)
k=O
a P (_l)k (T)(n/2)+k-l
1
k
2
2r(n+ k)
2
are determined by
The
where the recurrence relation
(20)
k-l
r
r-O
l~
with
p
bk
(ag
and
b
aP/k,
k -r r
2
=- '2 Li=tl(1-kb t )(at )
b:
-k
can be obtained.
of Katz et a1 should be changed to above form.)
To compare with Theorem 3 and Lemma (Kotz et a1), we set
B = I
n
in (16) and we have
1:
=Im
and
14
1
o yn [ - ";\1JlJ
-...,...
2 I ].
(16) I
QC)
T (mn/2)-1
~
f
L
k-O kl(T)k
etIt(=
Now replacing
(T)L
2' -11k (1
7V'"
leA-1)
~')
A by
1 fiJA
m
and b
by
lJ
in Lemma (Katz et 81), we have
the following form.
QC)
!
(18) ,
(19) I
aP (_l)k (T)(mn/2)+k-l
k=O
k
I
aPe k
k:aO
-
2
aP
0
=
(21) ,
bP
k
=
r
(detA) -(m/2) e.tJt(-
n
!
,
ri
]~
(detA) -(m/2) e:tp[ -!
1
(1-e/a ) -(m/2)
2j =1 l-e/aj i=l ij jel
j
=
it
(20)'
1
2r(mn+ k)
2
1 k
(a)
j=l
j
m
!
i=l
1
2
MM')
2
(l-klJ ij )·
Therefore, by comparing with (16) I and (18)', we have the following
relation.
(22)
where
k
1-1
(-1) tKPK (12M,A )
form for
ai
is given by Lemma 1.
Hence (22) gives an explicit
not involving a recurrence relation.
We can also easily check
that if we insert (22) into the left hand side of (19)', then we have (7).
Next we compare with the f-type representation.
15
THEOREM
4.
(r-type representation.)
the p.d.f. (8), then the p.d.f. of
Let
T • ..tIr.XAX'
X be distributed with
is given by
exp(- ~(E-l~B-l)~,)
e.xp( - ..!.. T) T(m/2)-l
•
2p
n
m.
r(~)(de.tE)2(de;t2AB)2
(23)
PROOF:
From (15) and the proof of Theorem 3, we can easily show (23).
CoRou.ARv.
The p.d.f. of the
T =
tIt}';-~'
is given by
(24)
We can show easily that (24) is the same form as (14).
Ruben [8] gave a r-type representation of a quadratic form which is
obtained in the following lemma.
l.Ef.t.tA.
p.d.f. of
(Ruben)
T
= (x+b)A(x+b)'
IlO
kIo
(25)
e
The
{
Under the same condition of Lemma (Kotz et al), the
is given by
a C e-(T/2p)T(n/2)+k-l (1)(n/2)+k
k
2 (n/2)+k
are determined by
r (~+ k)
2
P
16
(26)
Hence the recurrence relation
a~
•
k-l
r b~_r a~/2k,
~.
r-O
k ~ 1,
with
can be obtained.
To compare with Theorem 4 and Lemma (RubE!;ll), we set
B
=In
in (23) and we replace
A with
I eA
m
and
b
t
with
I
D
1.1
m
in (26).
Then we have
(23) ,
e.xp(-
1
":;'1.12
1.1')
!!!l
!!
22 f(mn) (de.tA) 2
e.xP(_"!") T(mn/2)-1
2p
2
(leA-1-lIp)
•
e
and
e.xP(_.1..) T(mn/2)-1
2p
and
17
r
(25) ,
k-O
c e-(T/2p)T(mn/2)+k-1 (1)(mn/2)+k
~
2 (mn/2)+kr (mn + k)
2
P
(26) ,
1 n
exp[- 2
1:
j=l
m
1-6
l-(l-P/a )6
2
1: ~ij]·
i=l
j
Therefore, by comparing with (23)' and (25)', we have the following
relation.
a:
(27)
=
:r
(detA/p)-(m/2)e;tIt(_ ~')
1 -% -1
%-1
IKP K (~ (A -lIp) ,A -lIp)
and
c
~,
an explicit form for
1: PK(~4(A-1_I/P)%,A-l_l/p)
K
is given by Lemma 1.
Hence (27) gives
not involving a recurrence relation.
We can
easily check that if we insert (27) into the left hand side of (26)', then
we obtain (7) by changing T
and
A into the appropriate variables.
From (27), we have the relation
E(Qka2k (L/Qlf»
2k {2k-l) II
(28)
\'
1
= ~ PK (72
-1
%-1
MA '2 (A -lIp) ,A -tIp),
_Jr:
where
L
X
ij
and
=
n
1
m
1: 7i1: 1: ~ij
j=l
j i-l
x ij '
Q •
m
1:
n
1
1
2
1: ('8- p)xij ,
i eo 1 j e 1
j
are independent normal variables with zero mean and unit variances,
H ·(Y)
2k
1·3••• (2k-l).
is an Hermite polynomial of order
2k,
and
(2k-1)!!.
18
NoTE.
M .. 0
If we set
and
B· I
n
in (16) t then we have the p.d.f.
of a central case. given in Bayakawa [2J by using the zonal polynomials.
5. THE CDPLEX M,LTIVARIATE QUADRATIC
FORM. In this section, we
shall state the above results for the complex Gaussian distribution studied
by Goodman [1]. James [5] and Khatri [6].
Let
are
m,
trix.
and
T
U be
mxn
respectively. and
We define
PK (T .A)
(m
:S
n)
A be an
'll'mn
K
CK (UAU')
where
nxn
positive definite Hermitian ma-
as follows:
m(-TT ')P (T ,A) .. (_l)k
(29)
complex arbitrary matrices whose rank
f
etIt(-i (TU '+U'f , ) )eVt(-W')e (UAl'J')dU,
U·
is a zonal polynomial of a Hermitian matrix
K
UAU'.
Then we
can show that
(30)
PK
(O.A) .. [n] C (A)le (I ).
KK
I<:
n
(31)
PI<: (T,I)
.. HI<: (T).
n
:s eVt(TT')[n] I<:
(32)
C (A)/eK (In ),
I<:
where
m
[n]
and
H(T)
K
I<:
=
n
(a-a+l)k'
a-I
a
is a generalized complex Hermite polynomial of a matrix argument
T.
The generating function of
PI<: (T ,A)
is given by
19
I I
(33)
U(m)
U(n)
~
• I I
k-O
Ie
ebr.(-SU Atf'5'+U SU A%f'+TA-tu'S'ij')d(U )d(U )
2 2
P (T.A)C
Ie
1 2
2
1
(m
~
n)
of order m and
(SS')
kl[n] KCK (Im)
complex arbitrary matrix,
n,
2
_Ie
The R.B.S. of (33) converges absolutely with respect to
an mxn
1
respectively, and
IiK (T)
l
d(U )
l
invariant measures over the unitary groups
A detailed discussion of
U
U(m)
and
and
S,
U
2
S
U{n),
are the unitary
respectively.
may be found in Hayakawa [4].
lEf.tiA 2.
I
(34)
PK(T,A) •
f(
where
and
,..
A •
O
PRooF:
Let
given by
0,
for convenience.
Similar to Lemma 1.
X be an
mxn (mSn)
is
are unitary matrix
d(U )
2
and
where
complex matrix whose denisty function is
20
(35)
where
I
is an
whose rank 1s
an
nxn
M is an
mxm p. d. Hermite matrix,
m,
and
B is an
p.d. Hermitian matrix.
nxn
We denote
.. .2 ,
xa
setting
= (xa l'xa 2' ••• 'xan)
x = (xl'~' ••• ,xm)
= xCI 8A)x I
tit
and
m
with mean
x
has an
P and covariance
mn
A be
l
1J
2
•
•
,
1J
m
and
and
1J
M =
xm
Let
and M as
X
x
where
complex matrix
p.d. Hermitian matrix.
Xl
X
mxn
1J
a
1J.
= (pa l' ••• 'Pan ),
(PI P2 • • Pm) ,
a · 1,2, ••• ,m.
By
we rewrite .t!lXAf1
dimensional complex Gaussian distribution
Then by applying the same method as in
IQ)B.
the real variate case, we have the following theorem.
THEOREM
5.
(P01JJer series representation.)
the p.d.f. (35), then the p.d.f. of
Up(-p(I: -1QDB-1)}.I')
(36)
where
and
Pk ( • , •)
C co A"iBA1t•
is a polynomial for
THEOREM 6. (r-type
T
=.t'JLXAI 1
Let
X be distributed with
is given by
mn-l
T
m· 1 in the definition of ""
P (T,A)
~pFesentation.)
K
Let
p.s.f. (35), then the p.d.f. of T"~'
X be distributed with the
is given by
21
(37)
where
D. t-
1
ec- l
m n Ip
- I 0)1
and
IIABII
< p.
Theorem 5 and 6 can be proved in the same way as the real variate case.
However, since the procedure is exactly the same, we will omit it.
AcKNOWl..EIlGt-£N.
The author wishes to express his sincere thanks to
Professor Norman L. Johnson who read carefully the original version and
gave some advice.
REFERENCEs.
[1].
Goodman, N.R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction).
Ann. Math. Statist.~ 34, 152-177.
[2).
Hayakawa, T. (1966). On the distribution of a quadratic form in a
multivariate normal sample.
Ann. Inst. Statist. Math.~ Tokyo~ 18, 191-201.
(3] •
Hayakawa, T. (1969). On the distribution of the latent roots of a
positive definite random symmetric matrix l.
Ann. Math. Statist. Math.~ Tokyo~ 21, 1-21.
[4].
Hayakawa, T. (1970).
On the distribution of the latent roots of a
comp Z~ Wishart matri3: (Non-centraZ case).
Institute of Statist. Mimeo Series No. 667. University of North
Carolina at Chapel Hill.
[5).
James, A.T. (1964). Distribution of matrix variates and latent roots
derived from normal samples.
Ann. Math. Statist., 35, 475-501.
[6].
Khatri, C.G. (1966). On certain distribution problems based on
positive definite quadratic ft.mctions in normal vectors.
Ann. Math. Stati8t.~ 37, 468-479.
22
[7].
Kotz, S., Johnson, N.L. and Boyd, D.W. (1967). Series representations of distr1butions of quadratic forms in normal variable
I I. Non-central case.
Ann. Math. Statist., 38, 838-848.
[8].
Ruben, H. (1962). Probability content of regions under spherical
normal distribution IV: the distribution of homogeneous and
non-homogeneous quadratic functions of normal variables.
Ann. Math. Statist., 33, 542-510.