* This researd.
t'J9S
partially suppcrted ty the C. S. I. P.
!\SY1LPTQTIC DISTRIBUTION .EDE TH~ j
OF THE GENERf.iLISED HOTELLING I S BETf.\
~N
-t~_sf
~lAToIX
Devartment of statistics
University
North C~roZina at C~peZ HilZ
of
and
University of the 0range Free state
Insti tute of Statistics Fir;eo c:e-ries No. %6
!IovsDl.er, 1""7"
B10MATIiEMATICS TRAINING PROGRAM
An Asymptotic Distribution for the j-th esf
of the Generalised Hotelling's Beta Matrix
D. J. de Waal*
Vnivepsity of Nopth CapoZina
University of the Orange Fpee State
INTRODIJCTIOi~. Let
1.
i
= 1,
... ,r.
-(i)
X
ex
xCi) (p x 1) ,
ex
= 1,
... ,N. , be Li.cL
N(ll(i), r) ,
1
Let
N.
1
= L
x(i)/N.
ex
ex=l
N.
1
1
A.
=
1
L
ex=l
r
A
= L
i=l
X*
=
r
r
X=
A. ,
1
(~ (i(l)
l
N.x(i)/N
II
1
i=l
= l
N.ll(i)/N
1
i=1
-
- XL
B = x*X*'
M*
Let
M(p x q)
= (~
be a transform of
it follows that for
independent of
p
~
r
, IN
q (ll
. (r) _ ~)) .
ii) ,
(jJ(l)
~
N,
f,';*
n = E-1 MM' ,
A is distributed
B which is distributed
tv
let
and
-1
W(E,q,n)
W(L,n)
Let
-1
= B2A- l B2 ,
* This research was partially supported by the C.S.l.R.
q
=r
n
- 1
=N
- q
then
-2I
2
B being the symmetric square root of
T;
Hotelling's generalised
H: 1.I
against
l.I (i)
;t
/j)
(i)
B, then
ntrV
is the well known
statistic for testing the hypothesis
C)
= l.I J
for some
i,j ,
fo r all
i ,j
i;t j
Fujikoshi (1970) has derived
the asynptotic distribution for T~ in the noncentra1 case to order terms
-2
n
, which also holds if q < p ~ N
Tne equivalent of V in the case
q < p
where
~
N is defined as
Z(p
x
q) - N(M,
I
@I ) .
q
For this case
Q
is defined as
·-1
Q
= M'l:
We shall consider here the asymptotic distribution of the j -th esf of
tr.F
The result reduces to the result given by Fujikoshi if
J
shall refer to
j
= 1. We
V and F as Hotelling1s generalised beta matrices, since
they are distributed as multivariate beta distributions and Hote1ling's
generalised
2.
T~ are related to them.
PRELIMINARY RESULTS.
Lemma 2.1
(de Waa1 (1974))
atr.r
-,--..,..o<-J- = trA. E*
dO*
J rs
2.1
rs
where
aE
rs - ao*rs
atr.E
E*
A.
J
= _"",,=,J_ = (_l)j-l
aE
H
-3and
(dO~rs J = (.!.(l
2
Lelmla 2.2
+ 15
rs
)
_d_J
dO = d
r,s
rs
= 1,
... ,q .
Let
then
2.2
I
.
etr(Bd)exp(v 1 -JAtr.E).
-1 = etr(ABr.)exp(vAtr.R
J
J
J
l:=vR
-1
O(v
){l +
-1
)}.
Proof
0_tr(Ba)exp(v 1 -J. Atr .E)
J
I
a exp(v 1-J. Atr .E)
-1 = L b ~
R
r,s
rsoo
J
~=v
sr
~
I
~
~=v
R-
1
I
1
= v - jA exp(vAtr.A.)L
b trA.E*
1
J J r,s rs
J sr E=vR-
Also
.
tr 2 (Bd)exp(v 1 -JAtr.E)
J
I
-1 ~ L L b b
R
~=v
m,n r,s mn rs
~
- \
-
n
"
Lm
\
Lr
s
b
b
2
a
I-j
ao* ao* exp(v
Atr.r)
nm
sr
_d_ {
mn rs ao*
nm
I
J
E=vR
-1
}-j exo(}-j Atr.E)
.
J
}I
trA.E*
J sr r=vR- 1
(cont. on next page)
-4-
- Lm,n Lr,s
2
b b
mn rs
I
v 2(1-')
J A2exp(v 1-'JAtr.E)trA.E* tr A.E*
+ O(v- 1)
J
J nm
J sr E=VR-1
2
= A tr (Br.)exp(vAtr.r.) + O(v
-1
J J
J
) .
Continuing in this way, it is clear that the lemma holds.
lemma 2.3
Let
(Crowther (1974))
Z(p x q) ,.., N(M, I
density of S = Z'A-IZ
1
I
2.3
IAI
zi1
@
p
q < p , then for
I ) ,
q
A(p x p)
fixed, the
is given by
I
etr(-2n)
lSi
I
-2 (p-q-l)
where the expectation is taken w.r.t. the density
1
-zPq
2.4
TI
etr(-YY'),
yep x q)
Crowther (1974) proved this lemma for the case
lemma 2.4
~
I
P
(Fujikoshi (1970))
etr(_ln'n)r (l(n+q))
2.5
L
_ _2::....-~....Lp_2
I
(int)
Fq
E
II
Y
I
+
q
~(Y - !.!l)'
1:2
ltn
.
(v - ~2) I
1:2
_l(n+q)
2
rp (zn)
I
.
= (1 - 21t)
-2Pq
it,
etr(1_2it n n)
2q (pq - trn'n) +
I-Zit
[
I +
I
~pq(q
- p - 1)
'
2 (pq(p + q + 1) - 2(p + 2q + I )
trn'nJ
(I-Zit)
+ tr(n i n)2 + ----2--=3((P + q + l)trn'n - tr(n'n)2)
(I-2it)
(cont. on next page)
-5+
tr(n'n)2}
(1-2it)4
= g(tln'n)
+
~
~L
2
96n'
(1 a=O .
2it) -a B (I)}
,n., In
N
]
v.
, say
where the expectation is taken "". r. t. the density 2.4,
matrix and
+ O(n -3 )
Ba(~nln) ,
n (0 x q)
any fixed
a = 0,1, ... ,8 , given in Fujikoshi (1070) ~. 104.
Proof Fujikoshi (1970) showed that the characteristic function of Hotelling's
generalised
T~
= ntrF
is given by
2.6
C(t)
On t!le other 'hand for
= g(tIQ)
Z (:0 x 0.) ..., N(M,
.
I
r
@
I)
q
and
A(p
x :0) ...,
W(I,n)
independent of each other and using Lemma 2.3, the character function of
ntrRF,
2.7
R(O
li(
q)
fixed nonsingular
E etr(intRF)
A(Y - -41)) d.~dA
12
= etr(-~)rp(t(n+Q))
1
2P Q r p ("2n)
1
(int)
Equating 2.6 and 2.7 for
R
= Iq
proves the
lero~a.
-6-
Tire following theorem generalises the result given 1:-y Fuj ikoshi (lq71)):
Theorem 2.1
For
R(q
q)
x
fixed p.d.s. and
F(g
q)
x
Hotelling's generalised
beta matrix
- 2q(pq - trQR -1 )P (2
xf +2 (o 2')
+
2 ( (p +
~ +
l)trQR
-1
<
x) + ( pq(p
- tr(QR -1 ) 2) P ( xf2+ 6 (o 2)
<
x)
} :z{ I
+ ---
96n
p(::?
X
-
?
f +_a
1
.. - zPq
of -
variable with
f
(~2)
U
<
8
2
1
1
a=O
~2
,h.ere
u
= tr (lop-I)
2""
,
denotes the noncentral chi-sauare
degrees of freedom and noncentra1ity parameter
Proof From 2.7 and Lemma 2.4 the c.f. of ntrR- 1F is given
2.9
x)
<
Ba (2 QR - )
H
an.d
1)
+ q +
etr(-~21)rp (-21 (n+Q))
0
2
.
by
l ' -}
. _.!.. -.!..(n+o·
=
-----:--~-E
I
+
·;_·(V
--·4!R
._),
(V
-~1R
2) 2
.
E etr(itntrRF)
.I.pq'Y q
1TI t
12
12
(int) 2- r (!2n )
p .
I
I
1) etr (1znR -1) get IQR -1 )
= etr (-20.
Inverting 2.9 gives the theorem.
This result holds for
argument given by Fujikoshi (1970).
0
~ n
using the
-7-
3. THE ASYMPTOTIC DISTRIBUTION OF tr.F. We shall now consider the asymptotic
J
distribution of
= ntr. (FIT)
y
where
T is some fixed constant.
Theorem 3.1
t
J
Let
= n ( tr.(F/T)
J
+
R be any fixed p.d.s. matrix and
trR -1-r. - tr.R -1) , then
J
J
3.1
P(~ <
f
1
= -;pq
2- .
r.
= P ( Xf2 (8 2)
x)
-1)
o2 = Ttr (1-2nr.
. J
,
< x
) + n(n -1 )
T
,
= trQ/trQr.-1
~
. 1
= (-1)J-
L
i=l
J
Proof The characteristic function of
= ntr.J (FIT)
y
is given by
<Pet)
Expanding
exp(int trjP/T)
=E
exp(int tr. (F/'r)) .
J
as a Taylor series at
int F
--- =
T
for some p.d.s. matrix
R
-1
int R
J
and
-8-
i.e. for
v
= int
exp(v trj(F/T))
= exp (v 1 - j tr j (vP/T))
= etr(v(~/T
Let
3
1
R- )
= V(F/T
3.2
~(t)
=E
1 j
1
- R- )a)exp(v - tr.r)
J
IE=vR -1
~(t) can be written as
in Lemma 2.2, then
)
-1
etr ( V(F/T - R- 1
)f exp(vtrjR )
j
+
O(n -1 )
:; exp ( v(tr.R -1 - trR _1)
"T.) E etr (V)
- Ff. + O(n -1 )
J
where
f.
J
J
J
T
is given in Lemma 2.2.
Let
R
1
= -f.
T J
in 2.9, then using 2.5 the characteristic nlTIction becomes
3.3
¢(t)
= exp(v(tr.R- 1 - trR- 1f.)etr(- lo
z-,,·,)
J
1
-1)
etr (zrnfj
J
? •
(1 - .. It)
1
-'2"pq
( itT
-1 )
etr 1_2itnfj
Let
3.4
T
= trQltrnr.]-·1
then it fo11m1s from 3.3 that the characteristic function o.t:
~
can be written as
= n{ tr.(F/T)
J
+
trR
-I
_1}
r. - tr.R J
J
-9,., ()
"t
=
(1
-
1
--::-'Oq
t) 2-
?
_.1-
•
l
(ItT
etr l-=-Zit Dr-j ) + a(n -1 )
Droves the theorem.
whic~
Since
R is arbitrary, it way be useful to let
r. becomes (see Gradshteyn and Ryzhik
J
r.
3.6
J
=
Aj
IE=I q
= (_l)j-l
(1965))
j
L
(_l)j-i(.q.)I
)-1
i=1
= (_l)j-l
R
j -1
L
(-1) x
x=O
(Q) I
x
q
q
= (a-I)
.' 1 I
Jq
Tren
and
E;
becomes
The Iloncentrali ty parameter Hill t\en be
c/ = tr ~
In the special case
j
=1
t;;
it follows that
= ntrF
and the result by Fujikoshi (1970) follows immediately.
=T
In. this case
-10-
REFERENCES
Crowther,
n.
A. S. (197£1):
form in normal vectors.
de VIaal,
!).
J. (1974):
The exact non· central distribution of a quadratic
Technical report no. 87,
~tanforc1
An asymptotic distribution for the j -tJ: esf of the
characteristic roots of the non-central Wishart matrix.
Statistics
~imeo
!lniversity.
Institute of
Series no. 960.
Fujikoshi, Y. (1970): Asymptotic expansions of tests statistics in multivariate
analysis.
J. Sci. Hiroshima Univ. Ser. A-I,
Gradshteyn, I. S. and Ryzhik, I. M. (196.5):
Products.
Academic Press,
~.!. Y.
TabZe
:~~,
of
73-144.
IntegraZs:; 8ey>1:es and