A NOTE ON THE CONFIDENCE BOUNDS
FOR THE CHARACTERISTIC ROOTS
OF DISPERSION MATRICES OF NORMAL VARIATES
C. G. Khatri
Institute of Statistics
Mimeograph Series No. 406
September 1964
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
C:'lapel Hill, N. C.
A NOTE ON THE CONFIDENCE BCUNDS FOR THE CHARACTERISTIC ROOTS
OF DISPERSION MA.TRICES OF NORMAL VARIATES
C. G. Khatri
September 1964
Contract No. AF-AFOSR-84-63
This research wo~· 8Uppo~$edby tbel4athematics Division of
the Air Force Office of Scientific Research.
Institute of Statistics
Mimeo Series No. 406
A NO'IE ON THE CONFIDENCE BaJNDS FOR THE CHARACTERISTIC ROOTS
OF DISPERSION MATRICES OF NORMAL VARIATES
by
C. G. Khatri
University of North Carolina and GUjarat University
IntroducitmUl.
1.
Roy and Gnanadesikan [6]
have considered certain types of
different alternatives in tile case of two dispersion matrices, of
~1ich
first
three are strong alternatives, in the sense that all characteristic roots of
E:t~
-1 (.
J.. e.
-1)
cn k~2
are less than or equal to 1, or all ch .E- £:1
,f;ll~c.
union of these two alternatives.
>
-
1, or
In this note, we derive the shorter confidence
bounds tuan t:tlose given by Roy and Gnanadesikan [6] with probability ~ (l-a ), when
the alternative is true.
Moreover" we show that when the alternative is complement
of the null hypothesis, then we come across the same types of confidence
as derived by Anderson [1] by a different approach.
types of alternatives for more than
tv~
bounds
Keeping in view the certain
population dispersion matrices, we give
shorter confidence bounds tJ.lan those derived by Gnanadesikan [2,3], wi tn probability
~ (l-a).
All our confidence bounds are based on either
X2 or
F distributions.
The confidence bounds t'nat vIe derive have not the purpose in mind to give confidence
bounds on tile partials and J.1ence, for the partials we cannot say that our confidence
bounds 'rill a.t all be suitable and this is true for Anderson's results [1] too.
Hence, vIe shall not try to give or compare any types of partials from them.
Without loss of generality; we shall assume that
pendently distributed as Wishart denoted by
wnose density function is given by
~(i=1,2, ••• ,~
W(lJ.' n , p, ~i
i
L
are inde-
(i=l, 2, •.• , ~)
2.
We snaIl denote cn. A the j-th maximum cnaracteristic (max. cn. ) root of
J .....
~:pxp.
Le. cil ~ ~
1
g" tt, and
by
~
Ch
>
A ~ •••
2
p ~.
vIT1en
Ch
k = I, g,l' D and ~
l
will be denoted
respectively.
2. Simultaneous confidence bounds on
chp
'f'
~
-or chI I:
..... :
Let ~:px~ be symmetric positive definite and
! = (a ij ),
Then
Proof:-
by definition
chp A <
€'
A
for non-null vectors
~
t"'IItI
_
#"IlItI
"""I
€
§.'
.s
< conI 11
___..---
~
__
a.tL
1\
: pXl.
Let
~
have zero elements except at the
i-th place
and so we get
.
. A-I < a jj
cn
1y
Si~lar
p""
< cfllA
-
-1
Since
".J
-1
ch.A
J ....
positive definite, we get the lemma 1.
~.
where
~
If
g,*
=(
= diag.
be distributed as
( '1 , ..• , 'Y ) , then
p
1
X2
distributed as
S*if)
with
nand
(
W(g,*; n,p,~)
A
s *i i''1)
ian.,...
(
'Yj
. d
dent1y
S *jj)-l a re J.n
epen
n-p+l degrees of freedom respectively when i
!
j.
This lemma is a special case of Bartlett's decomposition t:neorem (e.g. see
Kshirsagar [4]), and hence the proof is omitted.
Since
~
is symmetric positive definite, we can find an orthogonal matrix
such that A
E A'
,.., ,.".....
tion of g,*
(~-t.l.)
=~
ljII
D , a diagonal matrix with '1 -> ..• -> 'Y.p > O.
1
_tV '1
~ ~t
is
~
Then the distribu-
W(~*; n,p)~ ).
Let us obtain the confidence brond on '11
sidering the pair of hypotheses
Ho('Yl=l)
with probability ~ (l-CX ) by con-
and Hl('Y ~
l
1).
We note by lemma 2,
e
3.
that
8*11/1'1
and
H (I'1 ~ 1)
against
H (I'l=l)
o
s*ll,-l are distributed as
(1'1
c
J
vlith
nand
n-p+1 d.f.
is tested by the critical region
1
* ~
s 11
X2
constant
J
or ( s-lHl ) -1 ~
constant, Ql.nd on account
CJ
of
s*
11
'
we shall choose the critical region
S~l ~
(1)
where
c
c
is determined from
~~ (X~
(2)
s ~l
Since
2= c)
= 1-0
.
depends on the nuisance parameters of ~
, we cannot carry out
the exact test given by (1), but we shall obtain the confidence bound on 1'1
probability
2= (l-Ct).
"Ii th
We note from (2) that
(3)
but by lemma 1,
ch..S
.1'"
*>
= ch-S
-J.N-
Hence (3) gives us the following confidence
bound on 1'1
(4)
wi th probability
(w.)
>
be.
(1-0 ) and it will less than
-,
"
N
C11 S
1
< c •.
_
2= (1-0 ) by considering the pair of hypotheses HIo (r.p
H (r. > 1) as
2 pl
r.p> b-
(5)
(7)
wnen
Applying arguments similar to (2.1), we obtain the confidence bounds on
wi th probability
"There
1
b
ch S
p'"
is determined from
P-t
(X2
n-p+l
~ b)
= l-<l¥
= 1)
I'
p
and
4.
(~)
NOvT let us consider t:ne pair of 11ypotheses
H (1. =l::!l)and
o p
H = HI U H
2
3
[7], tue critical region is
By union-intersection principle
(8)
where, using lemma 2,
b
1
and c
are obtained from
l
The test procedure (8) cannot be carried out in practi ceo
find the simultaneous confidence bounds on 1.
1
1 -< c 1
(10)
where
1.
b
and
1
c
1
-1
ch
S or
1 .....
a~e given by
or 1.
p
Hence, noting lemma 1, ~
with probability ~ (1-0: ) as
-1
1. > b
ch S.
P - l
P ,.,
(9).
He note that the confidence bounds given by (4), (5) and (10) are shorter than
those which can be derived by Roy and Gnanadesikan's technique [0].
(~....!t,)
~
Now, we note that Ander son's confidence bound [1] on all ch
can be
derived fr:>m the following considerations.
(1-0:)
= Pt.
(X2
1< b )
n-p+ - 1
p~(X2n -> c1 ) = P'"
[( 1.
,~
p
s*:pp) -1_< b ] P't (
1
S*PP,-l < all cll (t ) < b- 1 s *11 ] by lemma 2,
..... - 1
and so using lermna 1, we have
(11)
~.
1-0:
g,.
!
Let
let 0 and
....1
roots of
0
""P
A.
tV
:s ci.1
-1
P~ [b1 C'i1p ,j,:S all ch(~)
Simultaneous confidence bounds on
~mma
en.
:s
;t! be tvro Px:P
!Yld
cnp
-1
~J ~
2!
Chl~.
-1
chI ~~ 2
sYmmetric positive definite matrices, and
be two unit c'n. vectors corresponding to the max. and minimum (min.)
~ = (a.;;),)
...
If ,_
,~-1
= (a ij )
,.
,
t
- 0' 1l 0
i - "":i. fa "":i.
and
t
i
o.~ B- ...... tV
-
1
0
"'i
5.
and
c ~llAB -1
J?J'PR..t.
pxp
~~
' t
1
'/ a j j
, ( t ajj)-l] •
1
be an ortJ.logona1 matrix whose first column is
u. ,=t.
~
and tde last
..v
lemma 5, for the other part can similarly be proved.
Cd
~1
such that ~'~ =~ is III diagonal matrix. Let
= A'M. Then
ii
i
and u = t
f'or i=l, p. Now, we shall only prove tile f'irst part of tile
~
column is
l.~
~~
-
~:
Let
I
[ a .. t • , a.. t
>
..,J~
AB-
p_N
l
= ch
(D~z
p"'"
U
"V
-~~z ) <
""
[r:. t P ,
P
_
~.pI tP]
We note that
by using lemma 1-
Moreever; by lemma 1,
j ,
=
z < [a.. , a J ]
P n
and so the f'irst part of' the lemma 3 is proved, and this proves the lemma.
ch A
p,.,
Since ~ (i=1,2) are symmetric positive definite, tllere exists a non-sin6~lar
matrix C
'"
SUCil that
~£' = J2.I' ,a diagonal matrix, and
(12)
w'nere
1'1 ~ 1'2 ~ ... ~
>
'Yp
0
J
'Yi
* and
g,1
Then
= C!\~
*
~
~
-1
~£' =
Let
•
*' )
*
g,l = S~l~'
and
* = ~Q,'.
~
are independently distributed as
and
(13)
...k.~J
Let
.€l
min. C~l. root of'
~
and
*
Sl=(
S
be tile unit ch. vectors corresponding to the max. una.
*ij),
and. let
~(~i~-1£1)(n2-p+1)/nl'Y1
Then
s
wnere
,§~-l = ( s *i j
),
are
and
*
~l
and
n,j {s
*
~
be distributed as (13).
*PP(~~.~)
'Yp(n1-p+l)}
,
ind~ndently distributet2 as F ,n.tn n1J n 2 -p+l and
6.
nl-p+l } n
. ~.
2
Let
and~)
d.f. respectively•
~
be an orthogonal matrix, ,{nose first and tile last c:)lumns are
*
~ '[l~
such tilat
Then it is easy to see that
is a diagonal matrix.
~
*
[1
1l. are independently distributed as
and ....€'S.,.€
""2.... =
~
W(i*l
; ~,p'~r) and W(~; n2'p,~)
vre note tilat if U- 1=(u ij ) and U = (u .. ) , u11
-
~
an d u pp =
IS* .€p'
.€pN2
(s*PPr p)-l
= (~~~*-1~,)
~-~
~
2
) ·1 t can be seen t'·llat ( u 11)-1, upp ,
X2 with respective d.f.
are independently distributed as
n2,~,11~P+1.
(U)
U·
s1ng 1 emma
1J
5*
11
/ r I'
n -p+l,
2
From this, lemma 4 is obvious.
Let us obtain the confidence bound on
pair of hypotheses
Ro(rl=l)
Hl(r~ 1).
and
r
= Chl~
-1
l~ 2
by considering the
l
Then, the critical region as in
section (2.1) is
*
sll
< a constant •
*
sl1
contains unknown parameters depending on r. 2 and so) we
*-1
....
~~:;
*
a function depending on ~
On account of (,§i ~
~l) ~ (~i~ ~l) 1\'
Here, we note that
req~ire
we shall choose the critical region
(14)
where
c
is to be determined from
(15)
We note that even though we cannot carry out the test procedure (14), but we can
make a confidence statement on r
(16)
(1-0: )
= P9z:.
[
l
vTitil probability ~ (.1-0:).
s~l (.€i~-l~
1)(n2-p+l)/nlrl
S ~~ [(Chl[1~1)(n2-p+l)/~lc~
~
By lemma 4, we have
cD
r 1 ] , by lemma 3 .
7.
?
Thus, (16) gives the confidence bound on 1'1 with probability
ing the pair
Ho ( 1'1=1)
(I-a) consider-
and HI (I'l~ 1) •
~) Similarly, ~ obtain t~le confidence bound on I'p = CI1pO~; 1 ~-l) with
probability
> (I-a)
-
by considering the pair of hypotheses
H' (I' =1)
0
p
and
H (r > 1) as
2 p-
(Chp~l~l)
(17)
where
b
(3.3).
n!b(nl-p+l)
I'p
is to be determined from
Here, we give the confidence bound on
~
by considering the hypotheses
c
1
and b
(20)
PL
'£'
with probability> (I-a)
p and ~ = H UH as
l 2
Ho(I'l=I'p=l)
-1._
-1,
...1
c l (cnl~l~ )(n2 -p+l)nl '
1'1 ~
(19)
where
~
1'1
or I'
1
i'1
o~ I'p ~ b~ (n1 :tP+l)- n2(Chp~1~'"
are given by
1
(F n -p+1 n
1
'2
:s
bl )
P.... (F
"l
1 > c1 ) = I-a
-
n l , n 2 -p+
The confidence bounds given by (16), (17) and (19) are silorter than those given
by Roy and Gnanadesikan [6].
(3.4)
Now, we note that Anderson's result [1] on all (ch.~~ 2-1) can be derived
fro m the fOllowing considerations.
(1- a ) = P.L
-L
= Pot [n,) {
= P
t
(F
n -p+1 ,n -< b l )
2
1
(nl-p+1hps*PP(~~~») ~
en! { (n:.._P+1)S*PP(,€
p~~)bl)
Pa (F
-~
n ,n -p+1 -> c l )
1 2
b l ] P-t.
:5 all
[sit(~i~-1~)(n2-p+l)/n11'1?
c1 ]
Ch(~~l):5 S~l(~i ~-1~1)(n2-p+l)/n1c l ]
8.
Hence, using lemma 3 we llave
where b
4.
and
l
c
wit~l
-
probability> (1-0: ) ,
are given by (20).
l
Simultaneous confidence bounds on
Since
~
. -...
(i=l, 2, ••• ,k; t=l,p) .
is symmetric positive definite, tnere exists an ortllOgonal matrix
~
~..1 E.6!
SUCll tilat
= D.
~.LNJ.NJ.
f3.i , 1 -> t3.i , 2 -> '"
> f3.
:t,p
-
A
""J., I-'
a diagonal matrix, with diagonal ·elements
,
> 0 (i=1,2, ••• ,k).
Let
wtS.* ~ n.,p~..1
dependently distributed as
(4.1)
cht~
f'oIJ.
J.
A)'
~.L,I-'
S.* = 6.S.6 !.
Then
""'J.""l.""" J.
r-JJ.
* are inS.
NJ.
(i=1,2, ••• ,k).
Let us suppose that the k-th population is standard.
We shall try to obtain
~
with probability ~ (1-0:)
the confidence bounds on f3.
,,+8.
considering the hypotheses
k-l
H (t3 j = I\a- = 1,2, ••• ,k-1)
01
,
"p
(i=1,2, ••• ,k-l)
i,.I..'k,P
H1 =i~l H1 ,i (~,1:S !\,p),
Hi . (f3. 1 <
J.,
,i
-
We note t~lat for testing
by
and
Ho,J., (t3i ,l = ~,p) against
t~le critical region
8.
), we have
'k,p
(22)
where
S1*-1 = (s*ij )
"" \.
and d'
k
Ph
-e"
(F
is to be determined from
i
1 :> d ;)
n i , ~ -p+
-
].
= 1-0:
•
By the union intersection principle [7], the critical region for testing
against
H
l
is
k
(24)
H
a
w= U
. 1
i=
W.
J.
SUC~l tllat
Pi (x
N
€
w
I H0 ) =
Note that we cannot carry out t~le test procedure (24).
0:
•
Hence, we obtain the confi-
dence bounds on (13. 1 1f3k ) wit~l probability> (1-0: ) from (24) as
J.,
,P
-
-1 (.
)-1( nk-p+l ) n.-1 , ].=1
.
131 , 1 I8.
cH1S.i )(.
CJ.I Sk
'k, P -< d.i
P
]..
k-1
yhlere di, d , ••• / ~-l are to be calculated from
2
~ i=l,2, ••• , k-l) = l-a
(26)
P'l. (Fn.,n. -p+l 2: a i
J. K
(4.2)
Similarly, tile cQ)nfidence bounds on (~.
J., P
...............
(l-a) by considering the hypotheses
k-l
H2 = J.'!!l H2 ,J..t~.J.,p
2:
~.J., P Ip..
1>
....k, -
(27)
0
J.,p
"Ii t~l probability
>
=~k 1 ' i=1,2, ••• ,k) and
,
~k , 1)' can be given by
e:J. 1
(C~lp"'J.
S.)(ChlS~
)-1 n. (n._p+l)-l , i= 1 or 2 or ••••
f'llC
K
J.
are to be determined from
l'inere
T~le values of d.
J.
and
e.
J.
, (i=l, 2, ••• ,k-l)
tables [5,P.164] in some cases.
down for
(4.3)
H (~.
Ip..,
1)
....h.,
t~lis
can be determined from Nair's
The result similar to (2.3) and (3.3) can be written
case too, but ITe are not giving it, because it is very straightforward.
Let us consider
~
(29)
1-a
= P", [n1.
t..
...
1(
= Pt.'[,
s
(F
n.-p+1,~ < e. ~ i=1,2, .•• ,k-l) P..,.." (F
1> d.; i=l,2, ... ,k-1)
J.
J.
."
ni , ~c-p+ :'": J.
~Psl:
u(n. -p+l)e. } <~i I~l 1
J.
...,
J.
J. , P c,
,1=1, ••• , k-1
l
J
.,,
-I
( n -p+1)(n.d.) i=l 2 •.. k-1 :
p
J. J.'
.
,
:
<
13 ~~~.
........ 4,' I~I l .* S*"0') (n, -p+l)
J.
P <
J., 11 k,
;" .
A
~k 1 n.n.
J. J.
d
j
I
Hence using lemma 1, we Llave In td :~)robability
13· 1
- 13k , p
<~ <
-
~
{?-a) ,
(n k -p+1) cdl~i
d.n. ch 8
J.J.
1
'P"""
-'
'
. 1
J.=,
2
••• 1;:-1
lao
'Vldere
e
i
ai1d
d
i
(i=l 2 .•• ,1';: •. 1) are given b: (29).
He note taat tile confici.ence bounds given by (30)a.re sd:Jrter tdan t.l0se ~:..ven
by Gnanadesikan [2,3].
Our confidence bounds are not meant f:Jr deriving tdC confi-4,..
dence bounds on tlle partials.
TdC confidence bounds given b~T (30) are somesense
A
better
t.~n
taose given by Anderson [1].
11.
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