Gertrude M. Cox
CONFIDENCB BOUNDS ON CANONICAL REGRESSIONS 1
by
S. N. ROY
INSTITUTE OF STATISTICS; UNIVERSITY OF NORTH CAROLINA
Institute of Statistics
Series No. 9A
Mj~eograph
March 11, 1954
1
work sponsored by the Office of Naval Research under Contract No.
N7-onr-28402.
CONFIDENCE BOUNDS ON CANONICAL REGRESSIONS
l
BY
S. N. ROY
Institute of Statistics, University of North Carolina
1.
(p
Summary.
S q)
This paper starts from a (p+q) -variate normal population
with a p.d. dispersion matrix consisting of submatrices
E1l(P x p), L22 (q x q), ~12(P x q) which stand respectively for the
dispersion matrix of the p-set, the q-set and that between the p-set
and the q-set, and then defines, in a natural manner, the matrix of
regression of the p-set on the q-set, in the form E L-1 • This matrix
12 22
is denoted by
~(p
x q) and a bilinear function
is considered where
~l(P
each of unit modulus.
such bilinear compounds
x 1) and
~2(q
~i(l
x
p)~(p
x
q)~2(q
x 1)
x 1) are two arbitrary vectors,
Simultaneous confidence bounds are given on all
~1~~2
with a joint confidence coefficient greater
than or equal to a preassigned level.
For this purpose certain results
and techniques are used which were discussed in previous papers ~1,2,3._7
2.
Introduction.
We recall the confidence statement ~l, (6.1.4)_7,
with a confidence coefficient 1 (2.1)
t a (n-2)
b -
2 ~
(1 - r )
/ n-2
where
~
~:
s
-l < ~ < b
s2 -
-
\In-2)
/ n-2
(Which is now a scalar) stands for the population regression
of Xl on x
2
(Where Xl and x
2
have a bivariate normal distribution),
b for the sample regression (in a random sample of size n
1
2 ~
(1 - r )
+
~
3), r
Work sponsored by the Office of Naval Research under Contract No.
N7-onr-28402.
-2for the sample correlation, sl and s2 for the two sample standard deviations
and t~ for the ~pper Q/2-point of the t-distribution with D.F.(n-2).
We also note that
(2.2)
where P, 01' and 02 stand respectively for the population correlation
coefficient and the two standard deviations.
Denoting by C(M) the characteristic root of a p x p matrix (whose
elements are real or complex numbers) we recall also ~2, (1.2)-7 that,
if A(p x q) and B(q x p) are two such matrices, then
(2.3)
C(AB)
= C(BA),
meaning thereby that any non-zero root Oft"1.B) is also a non-zero root of
(BA) and vice versa.
we also note that
(2.4)
tr (AB)
= tr
(BA).
We further recall ~2, (2.2.4)_7 that if A and B are two p x p hermitian
matrices, one of which, say A, is p.d. and the other, i.e., B, at least
p.s.d., then, denoting by c
and c . the largest and the smallest
ml.n
max
characteristic roots, we have
c i (A) c . (B) < all c (AB) < c
(h) c
(B).
mn
ml.n
- max
max
We next recall ~3, first paragraph of section
(2.6)
IIIf E
l'
then E II
2
~
-7'"
II
3-7
that
-3~
We now start ~l, section 6.2-7 with a random sample of size n (
> p+q;
p ~
q)
from a (p+q)-variate normal population, and next reduce for the means and set
n-l
=
(n-l)
,
q
where 8
11
n-l
q
p
, 8
22
and 8
stand respectively for the sample dispersion sub-
12
matrices of the p-set, the q-set and that between the p-set and
th~
q-set
and where Y and Y have the p.d.f.
l
2
L.
const. exp. [
-~
)_1
12
tr
L.
22
We next recall ~l, section 6.2_7 that there exist non-singular ~l(P x p)
and ~2(q x q) such that
p
where
~
q-p
stands for a diagonal matrix the squares of whose diagonal
elements are the (all non-negative) characteristic roots of the matrix
,-I"
(
L.-1"
~12~22~12 i.e., the squares of the population canonical correlations
l1
between the p-set and the q-set).
As in ~l, section 6.2_7, denoting by
I{m) an m x m identity matrix, we have
-4-
L
12
~l ° 1.,="I_(P--:)--lr--(_~--:0~'t""0_)
=
) -1
I q
2..
22
o
Ii -9
=
o
~'1
-1
0
o
D
~1
-1
2
-(:~/1'~
o
~'2
J8/1 -8
J 1/1 -0
o
I(q)
o
-( D
D
Jl/l -0
=
0
I ('1.)
~'1
0)
I(q)
Going back to (2.7) and using (2.4) we have now
(::) (Y~
11
(2.10 )
tr
C12
L
D
= tr
(
~I
1 -1
x
°
- (D
Jl 1 -8
0
0.
/81-0
D
Ji/l - -
Y2 )
l
0) (:i
I(q)
°
~2
r ,r(q) r
~:-
b /~-l-.
°-1Xl
=
tr
Z2
(z'1
z'2 )
(Y'
1
Y2
,
Y' )
2
-1
-5where
-1
(2.11)
-1
~2
~1
Thus it is easy to check from (2.7), (2.10) and (2.11) that (Zl' Z2)
have the p.d.f.
(2.12)
conat. expo
-!
tr (::
)(Zi Z2)'
Consider now, for any two arbitrary non-null vectors ~l(P x 1) and
~2(q
x 1) and for a fixed positive QO' the statement
which can be written in terms of Y1 and Y as
2
where
(2.15)
Q = D
Now putting
fIJi - 8
~ -ly
1
_/D
1
\/G /1
0) ~-ly
_ (1) 2 2
-6-
(2.16)
£i(l x p)
~~l and £2(1 x q) = !2~;1
= !i D
1
Ji/l -8
and using (2.8) and (2.6), we check that (2.14) reduces to
or
2
~£i(S12 - ~S22)£2-7
<
Q
O
,
where
( 2.17 )
f3 ( p x q ) =
~l( ~
-1
0) ~2
.
-1
= L121.22
f3 defined by (2.17) can be appropriately called the matrix of population
regression of the p-set on the q-set and it ia the only set of population
parameters that occurs in the statement (2.17).
3. Confidence bounds on the regression matrix f3.
It is well known ~1-7 that the statement (2.17), for all arbitrary
non-null £1 and £2 1 is exactly equivalent to
where Qila (i
= 1, 2,· ••• ,
Pi
0
the determinantal equation in Q:
~
Ql
~
•••
~
Qp
~
1) are the roots of
-7Now put ~
= Q/l
equation in
- Q, so that we have from (3.2), the determinantal
~
The statement (3.1) can now be replaced by the statement that
(3.4) the largest characteristic root
~ QO/l - QO'
i.e. ,
where
,
(3.6)
which may be appropriately called the matrix of sample regression of
the p-set on the q-set.
We note that (3.5)
==:> .(3.1} ==:>( (2.13),
so ,that
Q
p
is the largest
characteristic root of the matrix
(Z l Zi)-1(Z l Z2)(Z2Z2)-1(Z2Zi), where
(Zl' Z2) have the p.d.f. (2.12).
The joint distribution of these central
Qils, and also of the largest root Q being known, all that we have to do
p
to make (3.5), i.e., (3.1), i.e., (2.13), a simultaneous confidence statement with a Joint confidence coefficient 1 -
0
is to choose Q = Qa(p,q,n-l)
O
where the quantity on the right hand side is defined by
P (central Q > Q )
p- O
= a.
SUbstituting now Qa(p, q, n-l) (to be sometimes denoted more simply
by
Q~)
for QO in (3.5), we have a simultaneous confidence statement with a
joint confidence coefficient 1 - a:
-8Now applying 42.3), (2.5) and (2.6) (in the same manner as in
~3-7), we have from (3.5), now with a joint confidence coefficient
> 1 -
the following simultaneous confidence statement
~,
Using these, we check that (3.8) can be replaced by the following (with
a confidence coefficient> 1 - a):
We next recall the following two well-known results (repeatedly used in ~1-7):
(3.10)
~
all c(M)
~i
< g (for a p x p real matrix M with real roots)
-
~<.Y
(1 x p) M(p x p) ~l(P x 1) (for all arbitrary unit vectors ~l)Jand
(3.11) !' (1 x
qJ !(q x 1) ~ h( > 1)
(for all arbitrary unit vectors
:=:;> I!2(l x q) ~2(q x 1)\ s {T1
~2).
Applying (3.10) and (3.11) to (3.9) we have (with a joint confidence
coefficient> 1 - a() the following simultaneous confidence statement ,(for
- 9 -
»)
all arbitrary unit vectors ~l(P x 1) and ~2(q x 1)
.
~
(3.12) ~i (B - ~) ~21 S ~right hand side of (3.9)_~
I
,
or ultimately
,
d t Bd
-1
-2 where
A set of simultaneous confidence bounds on just the elements
~-matrix
would be a subset of the bounds on the total set
worthwhile to check that if p
(2.1).
~1so
if p
= 1,
= q = 1, (3.13)
~ij
of the
~i~~2.
It is
reduces, as it should, to
we should have another special case of (3.13) giving
a set of simultaneous confidence bounds on all linear functions of the partial
regressions of one variate on several others.
Thus, in several ways, (3.13)
seems to be an appropriate generalization of (2.1).
References
1.
S. N. Roy and R. C. Bose, "Simultaneous confidence interval estimation,"
Annals of Mathematical Statistics, Vol. 24 (1953), pp. 513-36.
2.
S. N. Roy, "A useful theorem in matrix algebra," mimeographed paper.
3. s. N. Roy, "Slme further results in simultaneous confidence interval
estimation," mimeographed paper.