*This research was partially supported by the C.S.l.R.
AN ASYf1PTOTIC DISTRIBUTION FOR THE
j-th esf OF THE CHARACTERISTIC ROOTS
OF THE NON-CENTRAL WISHART r~TRIX
D.3. de Waal *
University of North Carolina
University of the Orange Free State
Institute of Statistios Mimeo Series No. 960
November, 1974
An Asymptotic Distribution for the j-th esf of the
characteristic roots of the non-central Wishart matrix
D.J. de Haa1*
University of North Carolina
University of the Orange Free State
1.
n
Introduction.
= ~ E-ll~'
and
Let
nS
M = E(X).
= nXX' for X(pXn) be distributed W(E,n,n) where
The asymptotic distribution for
considered by Fujikoshi (1968) if n
Q=
for
O(n)
-1
tr R S,
and n
of
w.r.t.n.
lsi
= 0(1) and by Sugiura and Nagao
has been
(1970) if
Fujikoshi (1970) has derived the asymptotic distribution
R any real fixed
= O(n). Fuyikoshi
for
lsi
n = 0(1n).
pxp
symmetric matrix, for both cases
n
= 0(1)
(1970) has also considered the asymptotic distribution
An attempt is made here to derive an asymptotic dis-
tribution for the j-th esf of the characteristic roots of S, denoted by trjS.
This will generalize the above mentioned papers in the sense that
for j
=1
for j
=p
The two theorems that will be proved, are the asymptotic distributions of
r
trjR -1 S to order terms llyn
for n = 0(1) and n = O(n) = ne say. Since the
characteristic roots of S are invariant under an orthogonal transformation
S
-+
HSH'
H
E
O(p) ,
* This research was partially supported by the C.S.l.R.
2
Q
can be considered as diagonal under the assumption
E
= 1.
p
The theorems
E but since R is arbitrary, E is taken as the
hold, however, for general
identity in the proofs.
2. Some useful results.
Lemma 2.1
de Waal (1973)
For any Q Cpxp)
atr.QF
J
2.1
dF
dE
*
Ers
=-*- and
dO
rs
being the Kronecker's delta, then
Lemma 2.2
Let
d
= (!(l
2
+ <5
2.2
where
2.3
Proof:
Since
* )
= tr(A.E
J rs
Corollary 2.1
dtr.R -1 E
2.4
and
J
IE
= 1\)1
) __d
rs dO
'
*
= \) l-Jtrf.E
J rs
rs
) =
C+)
dO
rs
<5
rs
dtr.R
2.5
dO;
rs
-1
l:
3
Il: = -(1+20)
v
1
* *
= v 1·
-Jtrr.E
J rs
where
2.6
and
1+20
Proof:
These results follow directly from Lemma 2.2.
Lemma 2.3
Let
and
then
2.8
2.9
2.10
Proof:
From 2.2 it is clear that
2
2
= 2trf.J
2
n
= 2trf.J
T
*2
4
trQa exp(v
j-1
trjR
-1
E)IE
= ~I
v P
= vj-1exp(vj-1tr.R-1E)E
w trA.E* I
J
r,s rs
J sr E = ~1
v P
Hence
=
Er.Sao.a:.
{vj-lexp(vj-ltrjR-lE)trnAjIIE
=
~l
rs sr
. 1
. 1
1 atrQA.
exp(v J - tr.R- E)
* J }I~
J
ao
t..
sr
+ VJ -
= exp(v-1tr.R-1)trr~trnr.
J
J
tr2a2trQaexp(yj-1trjR-1E)IE
J
+
= ~I
v
0(V- 1 )
= II
\) P
4
a
= Em,nEr,sao * ao * a* d*
mn
nm rs sr
{ j-l
\)
exp(v
j-1
tr.R
J
-1
E)trQA.
J
}
IE = -I1
v P
trA.E * trA.E * trQA.!
1 + O(V~ 1 )
J rs
J sr
J E = -1
v
5
Continuing in the same manner it is quite clear that
trka2trnaexp(vj-ltrjR-lE)IE
=!r
v p
+ O(V- l )
= exp(v-ltr.R-l)trkf~trnf.
j
j
j
and hence 2.8 follows.
Consider
tra
3
=
exp(v
j -1
E
r,s,e
-1
trjR E)IE
dO
*
a3
*
=!r
v p
* exp(v
ao dO
rs se er
j-l-l
tr.R
j
E)I E = -I1
v
It is therefore quite obvious that 2.8 is true.
Let
B = (I + 60) , then
3
tra B exp(v
=
j-l-l
trjR E)IE
E
r,s,e,
kb
rs dO
*
se
= ~(1+2e)
a3
*
*
dO kdOk
e
r
exp (v
j· - 1 - 1
tr. R E)
j
IE = 1
~(1+20)
v
1
*
*
= Er,s,e, kb rs v 3 (.J- 1) exp(v J. - 1tr.RE)trA.E trA.E k
j
J se
j e
trA.E
j
= trf.*3
(1+60)exp(tr.R -1 (1+20))
j
j
and hence 2.10 follows easily.
+ O(v- 1 )
I
*
-1
1
+O(v)
rs E = -(1+20)
v
6
3. The asymptotic distributions of trjR-1s
From the result by Anderson (1946) we have that if nS
W(I,n,Q), then for
B p.d.s.
is distributed
and scalar A
E etr(1ii" ABS)
3.1
=
1
2~
1
2"etr(-n)etr(n (1
-
2?n -1J .
-
Using the expansions
1
I - 2AB
I
Iii"
1- ~ = etr(! ~ n 1- i / 2 (2AB)i/i
0(n- 7/ 2))
+
2i =1
and
3.1 can be written as
3.2
E etr(/n ABS)
= etr(1J1 AB)etr(tA2B2)(1
-!-
+
A3trB 3
+
0(n- 1))
3/n
etr(2A Bn
In
Theorem 3.1
Let
nS
+
2
(2A) B2Q
n
be distributed as
+
W(E,n,Q)
the asymptotic distribution of
In
-1
-1
Y = --(tr.R
S - tr.R E)
J
T
J
is given by
3.3
where
i
=
3
(2A) B3Q
nli1
2trr~
J
+
0(n- 2)) .
and let Q
= 0(1),
then
7
and
~(r)(z) denotes the r-th derivative of the standard normal distribution
function
Proof:
~(z).
r = I.
Without loss of generality assume
Expanding
as a Taylor series at
itvn. S = itlii" I
T
T
P
Le.
itln
)
j -1
-1
= etr (-T--(S-I)a
exp(v
trjR r)lr = !I
v P
where
v
=
T
itlil
the characteristic function of
y
In
-1
-1
= --(tr.R
S-tr.R )
T
J
J
can be written as
3.4
~y(t) = E e ity
itvn tr.R -1) E exp (itlil
= exp ( - ----tr.R-1)
S
T
J
T
J
= exp (etr. (-
itvn tr.R -1) E etr (itlil)
--- - sa
J
T
itlil.)
~
T
exp(v
T
j -1
tr.R
J
-1
r)\r _ 11
= exp(-v-ltr.R-l)etr(-t2a2/T2)(1
J
+
2(i~)2
tr~3)
3T
0
+
-
VP
_ 2it(tran
lilT
-1
j-l-l
O(n ,))exp(v trjR r)\r
= ~I
8
Hence using 2.8 and 2.9 ~y(t) becomes
1 2
- It
2it
3.5
~y(t) = e
(1 + InT (trnr j +
Inverting 3.5 gives the theorem
Theorem 3.2
Let
nS
W(~,n,n)
be distributed as
and let
n
= O(n) = n0
say, then the asymptotic distribution of
is given by
3.6
pel,;
< z)
= ~(z)
4
-
*3
3 trr. (I+60)~
31n0'
where
0'
r~ = (_1)j-1
and
1
(z) + O(n- )
J
= 2trr.J*2
f (_1)j-i~1/2R-1~1/2((I+20)~1/2R-1~1/2)i-1
i=l
J
Proof:
2
(3)
Again we assume
~
= I.
From 3.2 under the assumption n
= n0
, the characteristic function of
can be written as
',+,fir,; ( t )
1
28)) etr (t2a2/~2)
= exp (-1
-v tr j R- ( 1+v
(1 -
4(it~3tra3(I+60)
3 Ina
where
v =
0'
itv'n"
+
O(n- 1))
l,;
9
Using 2.10, the characteristic function becomes
3.7
Inverting 3.7 gives the theorem.
An attempt has been made to derive these two theorems to higher order
terms and will be given in a later communication.
It is interesting to note that if
j
= 1, the results coincide with the
results of Fujikoshi (1970).
References
Anderson, T.W. (1946): The non-central Wishart distribution and certain
problems of multivariate statistics. Ann. Math. Statist. 17, 409-431.
de Waa1, D.J. (1973): On the elementary symmetric functions of the Wishart
and correlation matrices. S. Afr. Statist. J., 7,41-60.
Fujikoshi, Y. (1968): Asymptotic expansion of the distribution of the
generalized variance in the noncentra1 case. J. Sci. Hiroshima Univ.
Ser. A-I, 32, 293-299.
Fujikoshi, Y. (1970):
variate analysis.
Asymptotic expansions of tests statistics in mu1tiJ. Sci. Hiroshima Univ. Ser. A-I, 34, 73-144.
Sugiura, N. and Nagao (1971): Asymptotic expansion of the distribution of
the generalized variance for noncentral Wishart matrix, when n = O(n).
Ann. Inst. Statist. Math. 23, 469-475.