-.
fA.
A NOTE ON CONFIDENCE BOUNDS CONNECTED WITH THE HYPOTHESIS
OF EQUALITY OF TWO DISPERSION MATRICES
"w
by
s. N. Roy
University of North Carolina and University of Minnesota
•
This research was supported by the United
States Air Force through the Air Force
Office of Scientific Research of the Air
Research and Development Command, under
Contract No. AF 18(600)-83. Reproduction
in whole or in part is permitted for any
purpose of the United states Government.
I~stitute of Statistics
Mimeograph Series No. 190
'February, 1958
•.
......
,.
~::- ~ :-~; .:.:
. .i'~:.~
'
•
A NOTE ON CONFIDENCE BOUNDS CONNECTED WITH THE HYPOTHESIS
OF EQUALITY OF TWO DISPERSION MATRICES
by
S. N. Roy
1.
Summary and introduction.
In some previous work by the author and
R. Gnanadesikan ~l,2~, among other results, confidence bounds with a
confidence coefficient greater than or
tic roots
equa~
to 1 - a, on the characteris-
c(~l ~~l),s of ~l and ~2 fromN(ll' ~l) and N(12' ~2) were
given in the form (numbered (2.6) in ~l~)
where (for i
•
= 1,2)
Si denotes the dispersion matrix of a random sample of
size ni+l from N(~i'~i)'
cmax and cmi n denote the largest and smallest
characteristic roots of Sl S2 1 and Al and A are two constants depending
2
upon 0, n1 , n2 , p (the number of variates) and the (known) joint d1stribution of the smallest and largest roots of 8 1S2-1 under Ho : ~1 = ~2'
Furthermore, with anyone variate i (1 = l,2, ••• ,p) cut out and the (truncated) sample and population dispersion matrices being denoted by sii),
(i)
(i)
(i)
S2 ' ~l ' ~2 ' or with any two variates i and j (i 1 j = 1,2, ••. ,p) cut
out and the (truncated) sample and population dispersion matrices being
denoted by s(i,j) S(i,j)
~(i,j) and ~(i,j) and so on the following
1
' 2
'
1
2'
,
types of confidence bounds (numbered (2.7) and (2.8) in ~l ~) were given:
(1.2)
and
Al c
max
(S(i) S(i)-l)
1
2
> all c(~(i) ~(i)-l) > A
-
. 1
2
-
c
2 min
(S(i) S(i)-l)
1
2
- 2 -
(1.3)
~1
6(i,J) 6(i,j)-1)
cmax ( 1
2
>
>
all c(E(i,j) E(i,j)-l)
-
1
h c
(6ii,j)
2 min
2
s~i,j)-l),
with obvious extensions to truncations of·a higher order.
However, in ob-
taining the truncated confidence bounds like (1.2) and (1.3) (numbered
(2.7) and (2.8) in
flJ ) from (2.3) of f1J certain steps were omitted,
thus rendering the derivation abrupt.
It is the purpose of the present
note to supply the missing steps and make the derivation complete.
2. Derivation of the confidence bounds on the roots connected with the
truncated matrices. We go back to £1] and note that it is perfectly clear
how the passage from (2.1) to (2.3) and from (2.3) on to (2.6), which is
the confidence statement on the total matrices, are made.
Now starting
from (2.1) or rather from the step just back of (2.1), we shall deduce a
result on truncated matrices which is exactly analogous to and would be
implied by (2.1) on the total matrices.
The passage from this analogous
result to the analogous of (2.3) and from there on to (2.7) and (2.8) of
£1] will obviously be exactly identical with the passage from (2.1) to
(2.3) and from there on to (2.6).
Back in (2.1) of £1], we observe that
the step back of (2.1) of £1] is the distribution of Yl(p x n ) and
1
Y2 (p x n2 ) given by
•
where Y1Y = n16l and Y2Y = n262 , and where 61 , 62 , E and E have been
l
2
already defined. Put El = T1T and E2 = T T where T andT are lower
2
l
2
1
2
1
2,
- 3 triangular matrices and notice that
r -1
,-1
-1
,-1,
r .I.. T1 81 T1 + T2 82 T2 -'.
=t
-1
1 2 )'s when E1 = E2 is
exactly the same as of cf Til 8 1 T (T 1 8 2 T{1-1 ]. s when El :f E2 •
Thus given a, we can find ~ and ~2 and make , with a probability 1 - a,
As before 1 we note that the distribution of c ( 8 8
{1 2
the statement
(2.2)
which is exactly equivalent to (2.1) of £1] and also to
for all nonnull a.
Let us now partition T1 and
T
(2.4)
T
l =
where, of course ,
12
T
13
r
p-r
, T
T
0
ll
l1
are solid. A so put
,
T
13
T2
.-\ r
I
\
p-r
T
2
=
T
0
T
T23
21
I
I
--'
, T ,
2l
into
I
I
,
22
-r
T
23
.
r
I
i p-r
i
,
p-r
are lower triangular, and
T
12
and
T
22
where Tl and T2 are lower triangular matrices, and partition T and T into
l
2
- 4 -
e
Tll
(2.6)
T1
r
0
= T12
T
p-r
13
,
T2
=
p-r
r
T21
0
r
T22
T23
p-r
r
p-r
where, as in (2.4), T , T , T21 , T are lower triangular, and T12 and
ll 13
23
-1
-1
T
22 are solid. It is easy to check that Tl and T2 are also lower triangular matrices and can be expressed as
-1
I
I
(2.7)
T
-1
l
I
I-
;;
I
-1
-1
13 T12 1"11
T
I
L.
_.
,
r
T
,
I
-1
2 =
T
\
-
'-
-1
23
r
-1
13
T
r
p-r
-
and
p-r
-1
2l
T
-',
0
Tn
0
T
22
T
-1
2l
T
-1
23
p-r
\
r
I
I
i
p-r
\
-'
Using (2.5), (2.6) and (2.7) we can now rewrite (2.3) as
(2.8)
for all nonnu11
~,
where
0
I
r
I
-1 T
T
23 23
r
p-r
i
I
I
p-r
- 5 and
\-,..... ~
(2.10)
\.
T1-1 T1 =
I
.L
-1
'I'll T11
r
0
T-1 T
I
13 13
T-1 T T-1 T11 + T-1 T12
13 12 13
13
-I
r
p-r
p-r
The matrices T
..1
(2.9) and (2.10)
nd can be easily written in the analogous partitioned
2 T2
forms.
-1
and Ti Tiare the transposes of the right sides of
If we now specialize! so that the last p-r components are zero,
and denote by !* the r-dimensional vector with the first r components of
>
the specialized !, then it is easy to check that (2.8)
(2.11)
-lJ ! *
-1 T21
! *' ~r T21
-1 T11
! *' ~r Tl1
T'21 Tt
21
T'11 T,-l J
l1
!*
>
A.2 ,
for all nonnu11 (r-dimensional) !*. Furthermore, notice that
(2.12)
~(r+l, ••• ,p)
~2
_
-
T
T'
21 21 '
where the symbols on the left side of all the equations stand for the
truncated matrices obtained by cutting out the last p-r variates.
since (2.3) <-~:--;> (2.2) <:-.:-:> (2.1) of
£lJ <=;;>
(2.3) of £1]
Now
"¢
(2.6)
of £1], therefore, (2.11) <.::::-::> (2.13) analogous to (2.2), but on the
truncated matrices
<~>
(2.14) analogous to (2.1) of £1], but on the
truncated matrices <~> (2.15) analogous to (2.3) of £1], but on" the
truncated matrices - ..d> (2.16) analogous to (2.6) of
truncated matrices.
Now, since (2.8)
~?
£lJ,
(2.11) and (2.11)
but on the
:~
(2.16),
- 6 -
->
we have, with a confidence coefficient
(2.16)
A....
·'1
c
1~,
the confidence statement.
rS(r+1, ... ,p) S(r+l, ... ,p)M1
max 1.
I
:2
1
J
> all crt(r+l, ••• ,p) t(r+l, ••• ,p)-lJ
-
>A
-
:2
1.1
rS(r+l, ...,p) S(r+l, ... ,p)-lJ'
2
:2 cmini. I 1
It is clear that, starting from (2.1), we could have cut out any p-r
variates instead of the last (p-r) ones (with r
= 1"2,, ... ,p-l),
and thus
we have" with a confidence coefficient ~ l~" the set of (~) confidence
statements of the type (1.2) (with 1
i
1 j = 1"2, ••• "p),,
= 1,2, ••• "p)"
of the type (1.3) (with
and so on.
References
£1 J
Roy" S. N. and Gnandesikan" H." lIFurther contributions to multivariate confidence bounds,," North Carolina Institute of
Statistics Mimeograph Series No. 155.
£2 J
Roy" S. N. and Gnandesikan, R." "Further contributions to multivariate confidence bounds," Biometrica, Vol. 44 (1957),
pp. 399-410.