UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report No.
r)J
60-984
ON CERTAIN ALTERNATIVE HYPOTHESES ON
DISPERSION MATRICES
by
S. N. Roy, University of North Carolina
and
R. Gnanaqesikan, Bell Te]~phone Laboratories
August, 1960
Contract No·, 49(638)-213
Qualified requestors may obtain copies of this report from the ASTIA
Document Service Center, Arlington Hall Station, Arlington 12, .Virginia.
Department of Defense· contractors must be established for ASTIA services,
or have their " need-to-know" certified by the cognizant military agency
of their project or contract.
Institute of Statistics
Mimeograph -Series No. 261
ON CERTAIN ALTERNATIVE HYPOTHESES ON
DISPERSION MATRICES
b;y
S. N. Roy, University of North Carolina
and
R. Gnanadesikan, Bell Telephone Laboratories
Murray Hill, New Jersey
ABSTRACT
For two multivariate nonsingular normal distributions,
the familiar null hypothesis of equal dispersion matrices is
considered against various alternatives stated in terms of certain characteristic roots.
Based on two independent random
samples from the two distributions, similar region tests are
proposed for the null hypothesis against each of the alternative hypotheses.
Also, for each case, conservative confidence
bounds are obtained on one or more parametric functions which
measure departure from the null hypothesis in the direction of
the corresPonding alternative.
Finally, a physical interpreta··
tion is given for the alternative hypotheses considered.
ON CERTAIN ALTERNATIVE HYPOTHESES ON
DISPERSION MATRICES
by
S. N. Roy, University of North Carolina
and
Ro Gnanadesikan, Bell Telephone Laboratories
Murray Hill, New Jersey
1.
INTRODUCTION
For two nonsingular p-variate normal distributions,
N[~~l]
is
and
N[~2'~2]'
c~sideredo
the familiar null hypothesis
Ho~
L1 = L2
Ho can be stated also ip the form,
= 1, where c (A) denotes the characteristic
Ho :' all c(L:: l ~;l)
roots of the square matrix A.
following are considered:
As alternatives, however, the
(i) HI:' all
c( L::l L;l) > 1;
(ii) H2 : all c(L::l L::;l) < 1; (iii) H3 :'all c( Ll L::;l) > 1 or < 1;
(iv) H4 ::at least one c( LIL::;l) > 1; (v) H :at least one
5
l
c(LIL::;l) < 1;, (vi) H6 :'atleast one C(LI L ) > 1 and at least
one (LIZ;l) <: 1; (vii) lir:'at least one
ZlL::~/) > 1 or < 1.
2
<
It may be noted that (iii) is the union of (i) and (ii), (vi)
is the intersection of (iv) and (v), (vii) is the union of (iv)
and (v), and (vi) is the complement of (iii).
Also, while each
of the alternatives is mutually exclusive with Ho ' yet, only
(vii) is the complement of Ho ' and it is only (vii) that has
attracted attention heretofore [2,5,10].
The relations in
logical structure between the various alternatives may be useful
in understanding the forms of the tests proposed in Section 2 of
- 2 -
this paper for Ho against each of the alternatives. Section 3
presents the confidence bounds associated with each of the
tests.
Section 4, the final one, aside from other things, dis-
cusses certain possible physical meanings of the alternatives
considered here.
2.
TESTS FOR Ho AGAINST EACH OF THE ALTERNATIVES OF SECTION 1
Let Sl andS 2 be the two (pxp) sample dispersion
matrices based, respectively, on idependent random samples of
sizes (nl+l) and (n 2 +1) from the two populations. We assume
thatp ~ the smaller of n l and n 2 , so that, Sl and S2 are
positive definite almost everywhere. Let cM(A) and cm(A) denote,
respectively, the largest and the smallest c(A).
Then, using a
heuristic argument similar to that in (5], the following tests
are proposed, wherein w(H) denotes the acceptance region for
the hypothesis Hand w(I), where it occurs, denotes the region
of indecision or no discrimination between the two hypotheses
in
question~
(i)
W(Ho)~
w(I)
(ii)
~
W(Ho ) :
w( I) :
~
1
c,,"SlS2 )
1
cm( sl8 2 )
~
cm( Sl s~/)
1
Cm(SlS2 )
~
<
AI; w( HI) :
Al
< C~SlS;I)
A2J W(H2 ) ~
A2
Cm(SlS2~) >
< CM( SlS2
AI;
,
C~SlS21) < A2 ;
(2.1)
1
) ,
(Cont'd)
- 3 -
(iii)
oo( HO )
:
oo( H ) :
3
oo( I) :
A3
< Cm(SlS2
cm( SlS2-1)
l
Cm(SlS2 )
1
cm( SlS2
)
>
l
<
)
A3, ,
,
l
CM( SlS2 )< A3 ;
l
Cr;( SlS2 )
or,
< A3 < Crv(SlS;;l)
<
,
A3
and/or
< CriSlS;;l) ,
(iv)
oo(Ho ) :
l
CJ..S lS2 )
< A4'
oo(H4 ) :
1
Cr/ SlS2 )
(v)
oo(Ho ):
l
Cm(SlS2 )
> A5 ,
oo( H ) :
5
Cm(SlS2 )
. (vi)
oo(Ho ) :
A6
, oo(H6) :
oo(I)~
-1)
< cm( .SlS2·
l
Cm(SlS2 )
l
Cm(SlS2 )
l
cm(SlS2 )
(Vii)
< A6
~
1
>
A4'
<
A,
5
-1) . < A6;
,
Cr/ SlS2.·
1
Cr;(S lS2 )
and
,
>
A6 ;
< A6
and
,
1
cJ: 3 13 2 ) \:: A6 '
L
and
1
CJ...SlS2 )
A6
oo(H o ) :
-1)
A7 ~ cm( SlS2.
ro(H ) :
1
cm( Sl S2 )
7
< A3 ;
<
"7
or,
,
>
A6 '
-1) < A, ;
< cJ.. SlS2.
7
and/or
J: S -1)
cS
l 2
>
A,
7
.
. (2.1)
For Case (i), given Al the probabilities assigned to
the three regions, oo(H o )' oo(Hl ) and oo(I), under Ho can be determined. Likewise, given the probability assigned to the
region oo(Ho ) under Ho ' Al can be determined by the methods
described in [3,4], and hence the probabilities assigned to
OO(H ) and OO(I) under Ho may be determined. It should be noted
1
that the method of evaluating the probability assigned to the
region oo(I) under Ho ' for a given AI' has not been explicitly
- 4 -
considered.
The authors, however, feel that this will not
present any essentially new difficulty and that the methods of
[3,4] will be applicable to this problem also.
Similar remarks hold concerning the determination of
the other "rs, in Cases (ii)-(vii), under (2.1).
For Cases (iii),
(vi) and (vii), where we have two constants to determine since
the tests are two sided in each of these cases, in addition to
the conditions of a given probability for W(Ho ) under Ho ' we may
impose the condition of local unbiasedness of each of these tests.
These two conditions taken together will enable us to determine
both constants involved uniquely.
As discussed in [3,5,9], for
Case (vii), the condition of local unbiasedness implies certain
optimum power properties of the test for this case.
For the
other two cases, however, such implications of the condition of
local unbiasedness are yet to be established.
all the
"IS
Further, regarding
in (2.1), it should be noted that, in addition to
depending on the conditions discussed above, they are also functions of p, n l and n 2 •
Case (vii), as noted in Section 1, with the test given
under (vii) of (2.1), is the one that has been considered in
great detail elsewhere [5,6,7,8] and is included here merely for
completeness.
Finally, it can be seen that all the tests proposed
under (2.1) are similar region tests .
........
--~~_._.~
..
-
- 5 -
3.
ASSOCIATED CONFIDENCE BOUNDS
The confidence bounds obtained in this section are,
in each case, on one or more parametric functions which are
measures of departure from Ho in the direction of the corresponding alternative hypothesis. For instance, in Case (i),
o(ZIZ~~) > 1,
where the alternative hypothesis is that all
the
confidence bound sought is a lower bound on cm( Zl ~2 ). If, on
the other hand, one is interested in the alternative hypothesis
that all c(ZlZ;I)
>1
and bounded above, then, the appropriate
confidence bounds are a lower bound on
bound on cM(,ZlZ;l).
Cm(Zl~;l)
and an upper
It may be noted that these are the bounds
obtained in [8] associated with Case (vii),
a~d
that in the
spirit of the current attempt the appropriate confidence bounds
for Case (vii) are a lower bound on cMCzlz;l) and/or an upper
bound on cm(ZlZ;l) and not those obtained in [8J.
For Cases (i)-(iii) and
Ca~e
(vii) as considered here-
tofore [8J, the confidence bounds turn out to be in terms of the
appropriate characteristic root of
-1
~l Z2
.
Unfortunately, to
date, the authors have not been able to do the same for
Cases (iv)-{vi).
The results for these cases are in terms of
ratios of appropriate characteristic roots of
Case (i)
~l
and Z2.
Using the canonical form of the distribution
of the observations and proceeding exactly as in
Sect1ons5.1 and 5.2 of [6] and Sections 1 and 2 of [8],
we obtain with a preassigned probability I-a, the
statement
- 6 -
(~.l)
where Al is the constant under (2.1) such that
p
[Cr£:SlS21)
~
Al
I Ho] = I-a.
Also, Da denotes a
diagonal matrix whose diagonal elements are a l ,a 2 , ... ,
1
and Yl , •.. ,Yp are c(Zl
). Now (3.1) is equivalent
Z2
to
CM(DI A/7
SlDl1v!Y S11S1821)
~
Al which, in turn, is
equivalent to the set of simultaneous confidence interval statements
for all nonnull vectors
~(pxl),
with a confidence coefficient I-a. (3.2) may be rewritten
as
£or all nonnull
vectors~.
Choosing
~
successively so as
to maximize, one after the other, both sides of (3.3), it
follows that (3.3) implies that
(3.4)
- 7 We shall be using the phrase "choosing a successively
-
repeatedly.
•••
"
What this precisely implies is the following.
(3.3) and
Then it follows that (3.3)
Choose a so as to maximize the left side of
-
denote this value of a by a*.
implies
•
But the right side of the last inequality can be further
increased to AlcM(S2S1l), whence
(3.4) follows.
This
line of reasoning has been repeatedly used in [8,9].
turning to
Re-
(3.4) and writing 81 = TT', where T is a
triangular matrix with zeros above the diagonal, and
remembering that any nonzero c(AB)
obtain from
=a
nonzero c(BA), we
(3.4),
But if A is a real matrix with real c(A), then it is known
that
Hence, (3.5) implies that
- 8 or, equivalently,
where
~l
1
= ~.
(3.7) is thus a confidence interval
I
statement with a confidence coefficient
Case (ii)
~
I-a.
Our starting point here is the probability
statement, with a preassigned probability I-a,,'
(3.8)
where A2 lsthe constant under (2.1) such that
P
[Cm(SI
S21)
in obtaining
~
A2
I Ho]
= I-a.
Arguing again as we did
(3.3), we obtain that (3.8) is equivalent to
the set of simultaneous confidence interval statements
a'S S-la
- 2 I a'a
,
for all nonnull vectors !, with a confidence coefficient
~
I-a.
Choosing
~
successively so as to minimize,
one after the other, both sides of (3.9), it follows that
(3.9) implies that
- 9 -
As before, setting 8 1 = TTU and then using (3.6), we now
obtain that (3.10) implies
or,equivalently,
:::1:; .
where 1-L2
(3.11) is thus a confidence interval
2
statement with a confidence coefficient
Case (iii)
statement
j
I-a.
Our starting point here is the probability
with preassigned probability I-a,
or
where "3
~
,
< "3
1
P [C m(SlS2 )
c~ DIN7
8 1 Dl /v0" 8;/)
~
"3 '
(3.12)
are constants such that
> ,,~,
or,
C~81S21) ~
"3
I Ho] =
I-a, and
further such that the test (iii) under (2.1) is locally
unbiased.
As before, we notice that (3.12) is eqUivalent
to the set of simultaneous confidence interval statements
~'(Dl/v0" Sl Dl ,0 8il)~
a'a
or
~'(DIA!Y 8 1DI A/l S;:l~
a'a
,
-1
"3~'S281 ~
~ •
a'a
--
(3.13)
-1
I
~
,
"3~'S2S1 ~
a'a
,
- 10 -
for all nonnull vect9rs
cient =
l-~.
~,
with a confidence coeffi-
Proceeding now exactly as in Case (i)
with the second inequality in (3.13) and as in Case (i1)
with the first inequality in (3.13), we obtain that
(3.13) implies
or
(3.14)
where
~3
=
1
~
3
,
1
and ~3.= --, so that ~3
'
> ~3'
(3.14) is
"3
thus a confidence statement with a confidence coefficient
> l,..~o
Case (iv)
Taking the approach of [1], for this case, we
*
,
write 8 1 = Dl;l~ A181AIDl;l~ and
*
8 2 = Dl
,
;ly172 A28~2Dl~~'
where 11ls are c(Zl)' 12 1s
are c(Z2) and A ,A 2 are orthogonal matrices defined by
1
,
the transformations Zi = AiD1iA i , (i = 1,2). We take as
our starting point the probability statement
\
where 1\ is such that p
[::~:~~
< 1\ I HoJ = 1-0:.
It is
known that if A is positive definite and B is at least
positive semidefinite, then
- 11 \
(3.16)
Using (3.16), we have
*
,
*
,
Cm(Sl) ~ Cm(Dl/Yl)Cm(~lSl~l) = Cm(Sl)/CM(~l)' and
c M(S2) ~ CM(Dl/Y2)CM(~2S~2) = CM(S2)/Cm(~2)·
Hence, (3.15)
implies
or,
equiv~lently,
where v = 1A.
(3.17) is thus a confidence interval state-
ment with a confidence
Case (v)
~oefficient ~
I-a.
Using the notation above for Case
(iv)~
we
take as our starting point the probability statement
. (3.18)
- 12 -
Again using (3.16) we have,
I
* ~ CM(Dl/rl)CM(AlSlAl) = cM(Sl)/Cm(L:l ), and
cM(Sl)
*
cm(S2)
,
~ Cm(Dl/r2)Cm(A2S~2) = Cm(S2)/C M(Z2)· ",Hence,
(3.18) implies
or, equivalently
where v'
= A~.
(3.19) is thus a confidence interval
statement with a confidence coefficient
Case (vi)
~
I-a.
Our starting point here is the probability
statement,
and
(3.20)
- 13 -
where A* < A*' are such that
-c (S )
c (S )
m
I
< A* < A*' -< c M(S2)
1
Now using
Ho] = I-a.
m
(3.16) and arguing exactly as in Cases (iv) and (v), we
p
1
c M(S2) -
obtain that (3.20) implies
v*'
,
and
(3.21)
t*, and v*'
where v* =
= A;' so that v* > v*'.
(3.21) is thus
a confidence statement with a confidence coefficient L 1,-0:.
For this case, we have from [8], the confidence
Case (vii)
statement
where
1-L
7
=
r'
1
7
'
1-L
7
(2.1) such that
P
1
= -, ,
,
<
A
7
7
and A
A
7
are constants under
[A7 < cm( SlS;/) < criS1 S~/) < A; I Ho]
= 1-0:, and further
e
such that the test (vii) in
(201) is locally unbiased.
The confidence coefficient of (3.22) is, of course,
> 1-0:.
As mentioned at the beginning of Section 3, in the spirit
of the present paper, (3.22) is not meaningful when the
a1ternativ~
is H .
7
It is meaningful when the
- 14 alternative is
all
~1(H2)
C(~1~2l) are
together with the specification that
bounded above (below).
On the other
hand, if we take the approach of Cases (iv)-(vi) above,
we obtain with a confidence coefficient
ing
confide~ce
~
I-a, the follow-
statement
,
and/or
(3.23)
where v
1
o
=~,
0
l '
v'
o = --p and AO < AO are constants such
A
O
4.
CONCLUDING REMARKS
While the parametric functions on which confidence
bounds are set in Cases (iv)-(vii) are also measures of departure from Ho in the direction of the appropriate alternative
hypothesis, yet they are not as neatly interpretable as those of
Cases (i)-(iii).
Also, the confidence regions for these cases
have not been obtained by "inverting II the acceptance regions of
the tests for these cases given in Section 2.
Hence, for these
cases, more satisfactory bounds are to be sought.
It may be
,
noted, however, that in each case the distribution problem in
obtaining the confidence bounds is one related to the null
hypothesis", this being a desirable feature.
- 15 We shall now consider one possible physical meaning of
the alternatives considered in this paper.
If
~l(pXl)
is p-
variate nonsingular N[~l,Zl] and ~2(pXl) is'p-variate nonsingular N[~2'Z2] and the elements (variates) of ~l are physically
of the same nature as those of
~2
(i.e., for e.g., the first
element in both is amount of steel produced, the second element
is total farm produce, etc.), then, if ~'(lxp) = (c l ,c 2 , ... ,c p )
is a vector of nonstochastic utilitarian "weights" that go with
the
~-variates,
the linear functions
utilitarian interest.
2'~1'
and
It is well known that
2'~2
is univariate
2'~1
N(2'~1'2'Z12) and 2'~2 is univariate N(2'~2'~'~2~).
known then a direct comparison of
values of
and
~l
~2'
and
2'~2'
If c' is
for observed
using the usual univariate techniques
would be qUite appropriate.
interested in
2'~1
are of
d~fferences
Thus, for instance, one may be
between the means
in the ratio of the variances, 2'~12/2'~22.
2'~1
and
2'~2'
or
For a known system
of utilitarian weights then, one may, for instance, wish to
test Ho:c'~12/2'Z22 = 1, against Hl~ c'l:12/2'Z2 2
is the well-known one-sided F-test.
> 1.
The test
But now, if c' is not
known or given, \.,then ohe may want to obtain a weight-free solution by protecting oneself against the worst possible set of
weights (in a sort of minimax sense) and pose the question as a
c'~ c
test of Ho:'~Zl~ = 1 for all 2, against Hl :c'Zlele'Z2 2 >1
2-
for all c.
This is exactly the null hypothesis of
- 16 Ho : all c( ~l~2l) = 1, against HI: 'all c( ~l~2l) > 1. Similarly,
the other alternatives considered in this paper may also be
interpreted as being physically meaningful; each for one
class of problems.
The tests proposed here are heuristic and investigations for the power properties such as unbiasedness and
monotonicity are underway and so, also, the explicit determination of the probabilities like the one assigned to ro(I) under
Ho for Case (i). Finally, the problem of "truncation" or
partial at~tements as considered in [8] and a generalization
of the results to the case of more than two dispersion matrices
are also being investigated by the authors.
,
<
REFERENCES
.e
J
..
.'
....
References (ContEd) - 2
[8]
S: N. Roy and R. Gnanadesikan, "Further Contributions
to Multivariate Confidence Bounds," Biometrika, 44 (1957),
pp. 399-410.
[9]
S. No Roy, Some Aspects of Multivariate Analysis,
John Wiley & Sons, New York (1958).
[10]
s.
S. Wilks, "Certain Generalizations in the Analysis
of Variance," Biometrika, 24 (1932), pp. 471-494.