*This research was partially supported by the C.S.l.R.
AN ASVr1PTOTIC DISTRIBUTION FOR THE
j-th esf OF THE CHARACTERISTIC ROOTS
OF THE NON-CENTRAL WISHART r~TRIX
D.J. de I'faal *
University of North Carolina
University of the Orange Free State
Institute of Statistias 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
~Jaa 1*
University of North Carolina
University of the Orange Free State
1.
n
Introduction.
=;
r-ll~'
and
Let
M
nS
=E(X).
= nXX' for X(pXn) be distributed W(E,n,n) where
The asymptotic distribution for
considered by Fujikoshi (1968) if n
Q=
O(n)
w.r.t.n.
for
tr R-lS,
and
n
of
lsi
R any real fixed
n
has been
and by Sugiura and Nagao (1970) if
Fujikoshi (1970) has derived the asymptotic distribution
= O(n). Fuyikoshi
for
= 0(1)
lsi
= 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 Q = 0(1) and Q = O(n) = ne say. Since the
characteristic roots of S are invariant under an orthogonal transformation
S
-+-
HSH'
H
€
O(p) ,
* This research was partially supported by the C.S.I.R.
2
Q
can. be considered as diagonal under the assumption
E
= Ip.
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 (pxp)
dtr.QF
J
2.1.
dF
Lemma 2.2
Let
dE
=-*dO
and
d = (l(l
2
rs
+
)_d )
rs dOrs = (-4-)
dO
rs
0
being the Kronecker's delta, then
trA.E *
J rs
2.2
where
2.3
Proof:
Since
* )
= tr(A.E
J rs
Corollary 2.1
dtr.R -1 E
2.4
J
dO *
rs
and
*
=V1 -J. trr J. Ers
3
atr.R
2.5
aolrs
-1
E
IE
= v l-J' trr *. E*
1
J
= -(1+20)
v
rs
where
2.6
and
1+20
Proof:
These results follow directly from Lemma 2.2.
Lemma 2.3
Let
T
2
2
= 2trr.
J
2
and
*2
n = Ztrr.
J
then
222
e- t tra
2.8
IT
trr2a exp(v j-1 trjR -1 E)IE =!1
1 2 v P
--t
-1
2 trr2r.
= exp(v -1 tr.R)e
J
J
+
O(v -1 ).
2.9
_t 2tra 2/n 2 3
j-1-1
e
tra (1+68)exp(v
trjR E)IE = !(I+Z8)
1 2 v
-1
-1
-It trr.*3 (1+28) + O(v -1 ) .
= exp(v tr.R (I+Z0))e
2.10
J
Proof:
From 2.2 it is clear that
J
4
trna exp(v
j-I
trjR
-1
= !I
E)IE
v P
= vj-Iexp(vj-ItroR-IE)E
J
I
w trA.E*
r,s rs
J sr E
= !I
v P
Hence
2
a
{j-l
j-l
-1
}
= Er ,s ~*:;:---:;:*v exp(v trJoR E)trMJo I~ = AI
aO' 0'
v
L.
rs sr
°
+ VJ -
1
°
1
1
exp(v J - tr.R- E)
dO'
J
= exp(v-ltroR-l)trr~trnr.
J
J
sr
IE
= !I
v
+ 0(V- 1 )
J
tr2a2trndexp(yj-ltrjR-IE)IE
atrnAo
* J }
= II
v P
a4
{ j-l
E -*.,.----:-*-*:---,..*v
exp(v j-1 tr.R -1 E)trM. }
m,n r,saO' aO' a d
J
J E
mn nm rs sr
I
= E
trA.E * trA.E * trQA·
1
J rs
J sr
J 1E =
vI +
= exp(v-ltroR-l)tr2r~trnr.
J
J
J
+
0(v- 1) .
1
= -I
V p
O(~~ 1 )
5
Continuing in the same manner it is quite clear that
and hence 2.8 follows.
Consider
trd
3
=
exp(v
j -1
-1
trjR L)IL =!I
v P
L
r,s,e
*
3
d
*
* exp(v
dada dO
rs se er
-1
= exp(v -1 tr.R
)trr.3
J
J
j-l-l
tr.R E)I
J
1
E = -I
v
+ O(v- 1 )
It is therefore quite obvious that 2.8 is true.
Let
B = (I + 60) , then
3
trd B exp(v
=
L
j-l-1
trjR L)IL
r,s,e,
kb
rs dO
*
= ~(I+20)
3
d
*
*
dO kdOk
se e
r
exp(v
J' - 1 - 1
tr.R
J
L)l E = -v(I+20)
1
I
1
trA.E*
1
+0(V- )
J rs E = -(1+20)
v
= trr *3 (I+60)exp(tr R-1 (1+20)) + O(v -1 )
j
j
and hence 2.10 follows easily.
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~ - 2"etr( -0) etr [0(1 - 2?n -1] .
= I
Using the expansions
1
I - 2AB
I
and
Iii"
-
zn = etr(! r n l - i / 2 (2AB)i/i
+
0(n- 7 / 2))
2i =1
2 A__
'B~-1 Q = etr (6~ n- 1· /(2AB)lQ
2 · + O(n- 7/2)
etr ( I - __
) ,
vn.
i=O
3.1 can be written as
3.2
E etr(1n ABS)
= etr(1n AB)etr(iA2B2)(1
+
-i- A3trB 3
+
O(n-l))
31n
2
3
etr(2A Bn + (2A) B2Q + (2A) B3Q
In
n
nTn
Theorem 3.1
Let
nS
be distributed as
W(E,n,Q)
the asymptotic distribution of
Y
Iii"
-1
= --(tr.R
S
l'
J
- tr.R
is given by
3.3
where
.,.2
, = 2trr2J.
J
-1
E)
+
0(n- 2)) •
and let Q
= 0(1),
then
7
and
,'T j
~(r)(z) denotes the r-th derivative of the standard normal distribution
function
Proof:
~(z).
Without loss of generality assume
E = I.
Expanding
as a Taylor series at
itlJi,S= itltl I
T
P
T
i.e.
itlrl
)
j -1
-1
= etr (-T-(S-I)d exp(v trjR E) IE
= lr
\l
where
v =
T
itlil
the characteristic function of
In
-1
Y = --(tr.R S-tr.R
T
J
J
-1
)
can be written as
3.4
~y(t) = E e ity
itrn
= exp ( - - tr.R -1) E exp (itrn
--tr.R-1)
S
T
J
T
J
P
8
Hence using 2.8 and 2.9 ¢y(t) becomes
1 2
- ~
2it
3.5
¢y(t) = e
(1 + InT (trQr j
Inverting 3.5 gives the theorem
Theorem 3.2
Let
nS
be distributed as
W(L,n,Q)
and let
n
= O(n) = n0
say, then the asymptotic distribution of
is given by
per,; < z)
3.6
= ~(z)
-
4 trr *3
(3) (z)
. (I+60)~
J
31n0'3
where
0'
and
r~ = (_1)j-1
J
Proof:
2
L
= 2trr.*2
J
= I.
From 3.2 under the assumption
n = n0
, the characteristic function of
can be written as
-1
exp(-v tr.R
J
(1 -
v =
0'
itln
-1
2 2 2
(I+20))etr(- t a /0 )
4(it~3tra3eI+60)
3 Ina
where
O(n- 1)
fi=l (_1)j-iL1/2R-lLl/2«I+20)Ll/2R-lLl/2)i-l
Again we assume
¢r,;(t) =
+
+
Oen- 1))
r,;
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 Waal, O.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 noncentral case. J. Soi. Hiroshima Univ.
Ser. A-I, 32, 293-299.
Fujikoshi, Y. (1970):
variate analysis.
Asymptotic expansions of tests statistics in multiJ. Soi. 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.