l'
SOME FURTHER RESULTS IN STI'lULTANEOUS
CONFIDENCE INTERVAL
EST~\TION
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by
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.~
'
.
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S. N. Roy
Institute of Statistics
University of North Carolina
....
Institute of Statistics
Himeograph Series Nc>. 86
December, 1953
SOME FURTHER RESULTS IN STI,.w:rJTANEOUS
CONFIDENCE INTERVAL ESTn1ATION
by
S. N. Roy
Institute of Statistics, University of North Carolina
1.
Summary.
In this paper confidence bounds on the characteristic roots
of ~ and of EI Z
21 are
one
~-variate,
given, where Z stands for the dispersion matrix of
and Zl and
~2
for the dispersion matrices of two p-variate
normal populations, and where the confidence coefficient is to be greater
than or equal to a preassigned level.
statements from a previous paper
then
To this end certain confidence
L-l_7 serve
as the starting point and
certain results in matrix algebra, taken over from another paper
L-2_0and
certain further results stated and proved here, are used to ob-
tain the bounds of this paper.
2.
2.1.
Introduction.
Statement ~ ~ Problems.
We take over fromL-l_7 the two confi-
dence statements (5.1.5) and (5. 2.4) and renumber them as
and
n
(::>.1.2)
n~
81a (p,nl'n2 ) !z." IS2E.
~ E,1(j..LDl /
vG j..L-ISlj..Lt-~l/ J(;j IJ.I) So
The statements (2.1.1) and (2.1.2) are supposed to hold respectively for
all non-null a (p x 1) and ~ (p x 1), and each with a confidence coefficiont
In (2.1.1) 8 stands for the sample dispersion matrix, n + 1 for
1 - a.
·~hf':
\
sample size, (:)'s for the characteristic roots of
matrix given by ~
= rD,-,r'
~,
and
61a (p,n)
and
~,
82a (p,n)
r
is an orthogonal
are subject to the
0nly restriction
peela-< 81 -< 6p< 8
2a \
wh~re
~)
=
1 -
~
,
8 1 and 8 p are the smallest and the largest charaGteristic rcots ,'"If
and 6 2a are otherwise, for the moment, left flexible, unlike
was jane in the previous paper ;-1 7.
nSf
8 la
-
wta~
-
In (?1.2) 8 and 8 stand for the tWQ sample dispersion matrices,
2
1
nl+l and D 2+1 for the t"lrJO sample sizes, r::..) IS for the characteristic rootl:
elJ.
' and ~2 = IJ.IJ.I and
of L: 1 l;2-1 IJ. is a non-s~.ngu1ar matrix given by ~1 = IJ.D
9 (p,n ,n2 ) and e (p, n ,n 2 ) are subject to the only restriction
2a
l
1a
1
,
where 61 and Sp are the smallest and the largest characteristic roots of
(nl/n2) 81S~1.
61a and 8 2a are otherwise, for the moment, left free, unlike the development of the previous paper L- 1 _7 •
Let us denote by c(M) any characteristic root of the matrix.N. Than
3
it is \V'ell known thAt trw statements(2.Ll),md(2.1 0 2)a.re respectively
equivalent to
n
~ e
<
- n
Notice that(::\
= ci(Z)
l
(p,n ,n )
l 2
2a
in (2°L5) and
::0
ci(Zlz~l)
•
in(2.1.• 6)
It is now our purpose to try to obtain confidence bounds on
(i=l, ... ,p).
G.1. IS
(or
their functions) in t'orms of c. (S) 's (or their functions) in the case of
1
.
(?1.5) and of c,(Sl),o.(8 ) (or their functions) i.n the case of (2.1.6).
1 21 . 0
For c. (Z)
1
's
the confidence bounds are given by (3.1. 3) and (J .1. 6) and
for c (ZlZ;l) by p.2.8).
i
To derive these vie need the following results
°
in matrix algebra.
2.2.
Some :Juxiliary iilatrix res111 ts.
Lot us denot(-; by A' the transpose of
A, and shorten positive definite into p.d o and positive semi-definite into
p.s.d.
Also let c . (M) and c
(M) denote the smallest and the largest
mID
max
characteristic root of a p.d. matrix M and, if any matrix B is p x p, let
tr (B) (s=l, ••• , p) stand for the sum of all s-th order principal minors
s
of B.
It is well known that
P
6
tr (B)
s
c.
illi210 •• lis=1 ~l
(8)c. (B).,.c. (B)
~2
~s
,
and, in particular, that
p
p
p
trl(B)= ,6 Ci(B)~ Z b .. and tr (B)=
ci(B)=/Bl,
~=l
i=l ~~
P
i=l
Jt
-7 = c -/-B(p x p)A(p x p)-7 and
c /-A(p x p)B(p x p)
-
the product of two p.d. matrices is p.d. and if A(p x q) i-rank
r <
min (p,Q)_7 is a
D~trix
with real elements, then AAI is p.s.d. of rank
r.
We take over from
L-2_7
the following:
c . (A)c . (B) < all c(AB) < c
(A)c
(B)
rrn.n
rrn.n
- max
max
,
where A and B are tHO symmetric matrices of which one is p.d. and the other
at least p.s.d.
The generalization to the product of a finite number of
matrices is obvious and is also given in L-~7.
1~Je
also take oven from
L-2_7,
the followinc: result:
c . (MMf) -< c 2(M) < c
rlnn
-
-
max
(MMf)
,
where M is a square matrix tdth real characteristic roots.
From (2.2.4)
5
it is easy to see, by replacing A by AB-
C
•
nun
I
(if B is non-singular») that
(AB-I)c . (B) < all c(A) < c
(AB-I)c
(B)
mln
- max
max
•
Next, we establish that
where A and B are two p x P p.d. matrices and d
and d
1
numbers such that d ~ d
1
a sufficient)condition
2 •
any two positive
Notice that (b) is a necessary (though not
for (a).
I
It is easy to check that L-d < all c(.U3- )
1
Proof.
2
•
•
It ,
7
p) is p.d •
(,vhere .\t~·dlBt is a submatrix formed by the intersection of any t
rOHS of .i~-dlB Hitl: t columns bearing the same numbers)
d
l
< all
c(AtB~I).
Now, if all
cCAtB~I)
~
> d , one consegl ence is that
1
,
6
For a given t, summing over different possible submatrices we have
(??8)
•
Using the same kind of argument for the other ':31f of the inequality and
remembering that t
=>
1, 2, ••• , p, and combininc, we hwe the result that
By a slight rephrasing (Which is obviously permissible here) we have
from (2.2.9) the result (2.2.6).
It is well known that "If El' then E
2 11
==;;s
Il
E is a necessary condition for E "
2
l
=::::;;s
'tE ( E2 " ,
l
-->
P(E l ):: P(E ), the last one being a r:ecessary (though not a
2
sufficient) condition for the other statements. This will be used in the
derivation of the confidence bounds.
3.1. Bounds on c(Z)ts. Starting from (2.1.5) and noting that
-
7
,
we have, with a confidence coefficient I-a, the confidence bounds:
(3.1.2)
ln
Q
la
(p,n) < all c (S~-l) <
-
1 92a (p,n),
or
-n
n 9 - 1 (p,n) > all c (~S-l) > n 9 - 1 (p,n).
2a
1a
-
From (2.2.5) we observe that (3.1.2) - > the following:
(3.1.3)
nQl-l(p,n) C (S) > all c(~) > n9-2l (p,n) c . (S),
a
max
a
mlU
which is thus also another set of simultaneous confidence bounds with
a confidence coefficient > I-a.
From «~,,~!,6) we also observe that (3,1.2) - - > the fCll1owing:
which is thus a sot of simultaneous confidence bounds with a confidence
coefficient> l-a.
Notice that, using (2.2.1), trt(S) and trt(~) are
easily calculated in terms of 9i 's and C:>i's.
3.2.
Bounds on c(~1~;1),s.
Starting from (2.16)
confidence coefficient I-a, the confidence bounds:
we
have, with a
8
Using (2.2.2) and (2.2.,) we have
c. ( 8 2 ( ~ 1 )-1 D. ~ I 8 1-1 ~ D
mm
v'[""
Ire ~
-1) c
( -1)
min 81 '
where
In the same way we have
(3.2.4) c
(8 - 1 f)
max 1
c
(8 ) > all c(f)
max 1 -
Furthermore noting that
1
> c . (8 - f)
-
mm 1
c . (8 ).
mln 1
9
and using (2.2.4).we
have
)
i.e.,
> c . (t.)
-
mm
Combining (3.2.2), (3.2.4) and (3.2.6) we have
all0.
~
's ->
c . (S2(fJ.' )-~
fJ.'Sl-l~ .....J!-1) c . (S2-1 ) c . (Sl)'
nnn
10
10
lun
m~n
From this it is easy to check that (3.2.1)
n
-->
the following:
1
9-2a (p,n1 ,n ) cmm
. (Sl)'
. (S2-1 ) cmJ.n
2
2
-1
n
10
which is thus a set of simultaneous confidence bounds with a confidence
~~
coefficient> 1-a.
( -1
cmax 82 )
Notice that
= l/cmm
. (3 ) and c . (3- 1 ) = l/c
(3 ).
2
mID 2
max 2
Confidence bounds in terms of tr
could also be given as in
t
(J.1.4), but in this case the bounds would be more complicated and
do not appear to be so worthwhile as in the previous case.
3.3. Determination of the constants (Ola(p,n), Q2a(p,n» end
It has been stated in section 2 that the pair Q1a(p,n),
~2a(p,n)
second problem satisfy respectively the conditions (2.1.3) and (2.1.4),
but are otherwise free.
It is well known how the shortness (in the
sense of probability) of a confidence interval (or intervals) ties in
with the power of the associated test.
Let us consider the associated
tests, or rather, the acceptance regions of tho respective hypotheses
(3.3.1)
H(~
= ~O):
~l
a
(p,n) < ~l ~ ~
-
-
~ 9 (p,n)
p- 2a
and
11
In the first case it is possible
(;)0'
'-_.,,~
to~choose
gla and Q
(and this
2a
choice will be unique) as to let the second kind of error (which, aside
from p, n and a, depends only on the characteristic roots of ~ ~;1)
havo a (local) minimum, i.e., the power a local inaximum at
~
= ~O(~
r ~O
is supposed to be tho alternative).
It so happens in this
case that the resl11ting power function then monotonically increases as
each ci(~ ~;l) tends away from unity, provided that all are ~ 1 or ~ 1,
to begin with.
case, H(~
by
-1
~1~2
=
•
~ve
have an exactly similar situation in the second
~o) being replaced by H(~l = ~2) and ~ ~;l being replaced
The
j~pact
of this on tho shortness,in the probability
sense) of the resulting confidence bounds is obvious and need not be
discussed in detail.
The results just stated are proved in another
paper to be shortly submitted to the Annals of Mathematical Statistics.
It may be noticed, hOvTever, that for any pair (gla' 9 2a ) subject only
to (2.1.3) or (2.1.4), we are going to get an~Tay the confidence
bounds of subsections 3.1 and 3.2, with confidence coefficients> I-a,
the only difference being that they would not have the property of
"shortness II possessed by those that are based on (Qla' 9 2a ) determined
in the abovo way.
4. Concluding romarlm. In a later papor this technique will be used to
obtain the confidence bounds on "canonical ro[?;rcssions" discussed in
section 6 of
L-l_7 and
variate analysis.
on certain other types of parameters in multi-
12
REFERENCES
1..
S. N. Roy and R. G. Bose, "Simultaneous confidence intorval estirnation l1 ,
Annals of Hathematical Statistics, Vol. 24 (1953), pp.
2.
S. N. Roy, ".'l. useful theorem in matrix algebra", mimeographed papor.