UNIilERSI'I'Y OF NORTH CAROIJINA
Department of Statistics
Chapel Hill, N. C.
AUTOREGRESSIVE; HODELS AND TESTING OF HYPOTHESES
A[3S0CIAT:E:D WITH THESE MODELS
by
.
John Hilliam Adams
June 1962
e
Contract No. AF(638)-213
Some tests of the ~~othesis of independence
are given. The alternative hypotheses are
autoregressive schemes. In some cases the
non-null distributions of the test statistics
are given in closed form; in other cases approximations are given which are as accurate as one
pleases.
Thia research vias fmpported by the Mathematics Division of the
Air Force Office of Scientific Research.
••
Institute of' Statistics
Mimeo Series No. 327
ii
ACKNOVJLEDGll1ENTS
My largest debt of gratitude is due to Professor S. N. Roy.
He proposed this problem} and without his suggestions and encouragement this thesis would never have been finished.
P. Ie Bhattacharyya should be thanked also.
Dr.
He spent long hours
listening patiently vnlile I described the progress of the work}
and he made valuable suggestions.
Dr. "V'l. C. Clatvrorthy and the vTestinghouse Electric Company made
it possible for me to come to Chapel Hill in the first place.
.
e
grateful to them •
Finally I wish to thank Mrs. Doris Gardner and Miss Martha
Jordan.
They had the tedious task of converting the manuscript
to its fj.nished form .
..
-4
I am
iii
TABLE OF CONTENTS
CHAPTER
I
PAGE
INTRODUCTION - - - - - - - - - - - - - - -
v
SOME RESuLTS IN lI1ATRICES -
1
1.
Introduction - - - - -
1
2.
A Result in Matrices - -
1
3. A Property of the Tshebysheff
Orthogonal Polynomials _
5
4. The Characteristic Roots of a Matrix -
.
e
II
8
5· Identification of Roots
16
6. The Inverse of a Matrix
17
7. Some Identities- - - -
22
SOME TESTS OF INDEPENDENCE
24
1.
Introduction
- - - -
24
2.
An Intersection Test -
24
3. A Second Test of Independence
32
4. Non-Central Distribution of G (when
5.
~
= 0)
A Test of Independence with a Circular Model
37
6. The Non-Central Distribution of G
42
7. The Consistency of These Tests -
47
ANALYSIS OF VARIANCE \mEN THERE IS NO INTERACTION- -
53
1.
Introduction
53
2.
Testing for Treatment Effects Hithout
Interaction
- - - - - - - - - - -
3
III
34
- - - - - - - - - - - - - - - - -
3· Comparison of the Power Functions
53
58
iv
CHAPTER
IV
.
e
.
PAGE
ANALYSIS OF VARI.ANCE
INTERACTION - - - -
1.
Introduction-
2.
The Analysis - -
nr THE PRESENCE OF
- - - - - - - - -
_ 64
64
- - 64
INTRODUCTION
The general outline of this thesis is the
follovnng~
Cr~pter
I
contains some results in matrices and one or two miscellaneous resuIts; Chapter II contains some tests of independence, the non-null
distributions of some of the test statistics (in one case an approximation), and the proof of the consistency of some of the tests; Chapter III contains a comparison of the power functions of two possible
tests on data from a t,rO-v1aY classification (no interaction) when a
circular model is assumed;
Chapter IV contains an approximate
analysis of variance for a multi-way classification with interaction
.
e
when a circular model is assumed.
These problems arise out of the
follovdng context.
Consider a set of
r
jointly normally distributed random vari-
••• , Y •
r
Let the
e2~ectations
of these random
variables be known linear functions of some unkno1m constants; and
2
2
is an unknown scalar
let their covariance matrix be Vo- ,where 0multiplier.
When V is kno'tm to be the identity matrix I, the above assumptions constitute what is sometimes called the univariate linear hypothesis.
The problems usually considered in this situation are those
of hypothesis testing and confidence intervals.
These problems and
their solutions are well known and need not be elaborated here.
If, however,
V is not known to be
I, then the problems of test-
ing and confidence intervals are made more difficult, and some altogether new problems come up.
A few of the many possibilities are the
vi
following:
(i) the matrix V is known except for the scalar multi-
plier ~2; (ii) the matrix V is diagonal with unknown elements;
(iii) the matrix V is a patterned matrix vdth unknown elements.
Case (i) is the case of quasi-independence, and leads to no
new problems because a linear transformation can be found which transforms the random variables Y , Y , ••• , Y into -the random variables
2
l
r
X whose covariance watrix if I~2, whose expectations
r
are linear functions of the unknown constants, and whose distribution
Xl' X2 ,
'00'
is multivariate normal.
ali~Ys
Thus Case (i) can
be reduced to the to-
tally independent case.
Case (ii) arises when the variance components model is assurned.
The variance components model will not be considered in what follo·ws.
Case (iii) arises when the random variables or their distributions satisfy certain restrictions.
For example, assume that
Yl , Y ,
2
••• , Yare a sequence of random variables, with zero means, which
r
satisfy the stochastic difference equation,
,
where
/ p
I
i
= 1,
2, ..• , r
,
< 1, and the €i are independen-tly and normally distributed
with zero means and variance ~2.
It is assumed that Y is normally dis1
tributed with mean zero and variance ~2 and independent of the €., then
~
v ..
.
~J
= P Ii -J0/
(1 _
2 -1
p )
,
which is a patterned matrix depending on only one unknown parameter.
A
more general pattern results when the random variables are assumed to
be a segment of a stationary time series.
Then
V is a positive definite
vii
matrix with elements constant on all diagonals.
Some of the problems which arise out of Case (iii) are the
following:
(iv)
under a model such that V is the patterned matrix
which results when the random variables satisfy the first-order
stochastic difference equation, shown above, V depends on one unknown
parameter, i.e., on P, and the problem is to devise a test for the
hypothesis that p
(v)
=
0 against the alternative hypothesis that p
f
0;
test the hypothesis that V is a particular patterned matrix (not
the identity matrix) against the alternative hypothesis that it is
some other patterned matrix;
(vi)
analysis of variance, including
tests of hypotheses and confidence intervals, under a model in which
the random variables associated with the experiment have as covariance
matrix a particular patterned matrix.
Problem (iv) has received considerable attention in the literature.
A few of these papers are mentioned below.
Koopmans ~12_71
proposed the first-order serial correlation coefficient as a test
statistic.
He found the null distribution of this statistic, but its
form was not suitable for numerical computation, and he was forced to
make use of approximations.
The circular serial correlation coeffi-
cient of first order was suggested as a possible test statistic, and
R. L. Anderson [-2_1 found its null distribution.
Siddiqui
Dixon~9_1 and
L-15, 16_.7 investigated apprOXimations to the distribution of
the serial correlation coefficient.
T. W. Anderson~5_7 proved that
there is no uniformly most powerful test of the hypothesis p
1
Numbers in square brackets refer to bibliography.
= 0,
viii
even against one-sided alternatives, unless the model is modified to a
circular type, in which case uniformly most powerful tests do exist for
the one-sided alternatives.
The appropriate test statistics for these
cases are the circular serial correlation coefficients of the first
order.
For the two-sided alternative, he found a test which is uni-
formly most powerful within a certain sub-class of tests, and the test
statistics are again the circular serial correlation coefficients of
Hannan ~11_7 described an exact test first proposed by
first order.
Ogawa ~13_7.
Chapter II contains results which it is hoped bear on this
problem.
kind.
Section 2.2 describes an intersection test of a particular
This test may be useful against more general alternatives than
specified in
(iv).
The non-null distribution will, however, be diffi-
cult to find, and it is hard to say ag8inst just what alternatives the
test will perform best.
Section 2.3 describes a second test, and in
section 2.4 it is shown how an approximation to the non-null distribution can be found.
alternatives.
This test may also be useful against more general
In section 2.5 a circular model is considered, and a
test is proposed whose exact non-null distribution is derived in section
2.6, and whose consistency is proved in section 2.7.
Problem
(v)
is the most difficult of the three.
It includes
such problems as testing the hypothesis that the observations derive
from an autoregressive scheme of order q agai.nst the alternative hypothesis that they derive froIn a scheme of order qr.
thesis sheds any light on this problem.
Nothing in this
However, a result proved in
ix
section 1.2 may be useful in deriving a test for a hypothesis slightly
different from the one specified in
(v).
Suppose that it is desired
2
to test the hypothesis that V is doubly
tive hypothesis that it is not.
the following:
symmetric against the alterna-
The result proved in section 1.2 is
if V is doubly symmetric and of even dimension r,
then there exists an orthogonal matrix H, not depending on V, such that
HVH I
=
where VI and V2 are both (r/2, r/2) matrices;
and if V is not doubly
symmetric, then H does not transform V into the form shown above.
So
if there is available more than one realization of the random variables,
the methods of multivariate analysis could be used to test the hypothesis that the transfor.m: of V is of the form shown above against the
alternative hypothesis that it is not.
If this hypothesis 'Vlere reject..
ed, then one would be in effect rejecting the hypothesis that the observations vlere deriving from a stationary process.
It vlould seem
that such a test could be useful.
A solution to problem (vi) was the original goal of this work.
It was desired to find a way of analyzing data which arise from multivariate normal populations subject to a multi-way classification.
For
example in the balanced two-way classification there are b blocks, t
treatments, and r observations in each treatment-block combination
2
Doubly symmetric means symmetric about both d.iagonals.
x
which occurs in the design, and it is assumed that the observations
are generated by independent random variables except with in cells
(a cell is a particular treatment-block combination) where the r random
variables satisfy a stochastic difference equation of the form previously mentioned.
It was hoped that procedures could be found which
depend on statistics whose central distributions are known and tabulated.
Such classifications arise in the following types of situations.
Suppose there are bt identical machines which produce objects in a
sequence and that these objects each have a common measurable characteristic.
If b lots of homogeneous (Within lots) raw material are
available, and if t different adjustments of each machine are possible,
,
e
then it may be required to study the effect of the machine adjustment
vrhile eliminating the disturbing effect of heterogeneity in the rat'!
material (between lots) and the possible interaction between the
effects of raw material and machine adjustment.
It would not be un-
usual if the observations generated by such a process did not behave
like the relaizations of a sequence of independent random variables.
In such cases one is obliged to assume a more complex model, and the
one chosen here is the stochastic difference equation of first order
(in actuality the model assumed is circular, but it is hoped that the
effects of this will be negligible when the number of observations in
a cell is moderate).
Such a model is relatively simple, and it seems
reasonable in light of the empirical fact that objects produced one
after the other are usually more nearly alike than those objects widely
separated in the seuqence of production.
xi
An approximate analysis of a particular case is described in
section 4.2.
If the observations within a cell derive from a model
which T. W. Anderson ~5_7 has called modified circular, then it is
possible, when r is odd, to find two linear combinations of the observations within each cell (there will be as many such pairs of linear
combinations as there are cells) such that the corresponding pairs of
linear combinations of random variables are independent, have non-zero
expectations, and have variances which are equal except for a quantity
which vanishes as r increases.
And if r is even (see section
4.3), it
is possible to find two linear combinations of observations within
each cell such that the linear combinations of the corresponding random
variables have non-zero expectations, have variances which are equal,
,
e
and have covariance which is zero except for a quantity which vanishes
as r increases.
Thus, supposing the small quantities mentioned above
to be zero, it is possible to analyze the transformed observations as
if they were generated by independent, homoscedastic random variables
with two random variables associated with each cell.
The effects of
blocks and interaction can be eliminated by means of the usual analysis.
CHAPTER I
SOME RESULTS IN MATRICES
1.1 Introduction.
This chapter contains the proofs of some
of the results which will be used in later chapters.
In addition,
some more or less well-known identities, which are quite frequently
used throughout the thesis, are stated without proof.
1.2.
A Result in Matrices.
metric matrix, then, when
r
If V is an (rxr), doubly sym-
is even, there is an (rxr) orthogonal
matrix H such that
e
'.
(1.2.1)
=
HVH'
where Vl
and V are (r/2 x r/2) matrices.
2
always be written in the form
(1.2.2)
H
A
The matrix H can
I~
;-r;
_.
where A and Bare arbitrary, (r/2
~::
r /2), orthogonal matrices,
and K is the (r/2 x r/2) matrix vlith elements
(1.2.3)
k ij
=
r:
if
i + j
r + 1
= "2
,.
otherwise
and if V is not doubly symmetric, then there is no matrix H, of
the form shown in (1.2.2), which reduces
(1.2.1).
V to the form shown in
2
That this result is true can be shoiVll in the following
v~y:
first the matrix H is orthogonal.
This can be verified by a
simple multiplication.
HH' is
(1.2.~.)
but KK'
A
1
---
I AK
A
I
I AK
-;T; \ff ~ =
V2
= I,
The product
1
2
A(l + EX' )A'
I A(l-KK' )B~
B(l -
I B(I+RK' )B'
K1('
)A'
and A and B are orthogonal by hypothesis.
Second, suppose that
V is an arbitrary symmetric matrix.
Then V can be partiticned,
,
'_
where
I
(1.2.6)
VII
vII
v
v 12
v
12
vI;
v l ,r/2
22
v 2;
v 2 ,r/2
=
v(r+2)/2,(r+2)/2
v(r+2)/2,(r+4)/2
v(r+2)/2,r
v(r+2)/2,(r+4)/2
v(r+4)/2,(r+4)/2
... v(r+4)/2,r
v
rr
3
and
v 1 ,(r+4)/2
v 1 ,(r+2)/2
v 2 ,(r+4)/2
(1.2.8)
V =
12
vr / 2 ,(r+2)/2
vr / 2,(r+4)/2'"
vr / 2 ,r
The transform of V is
(1.2.9) HVH' =
1 A(V11-lICV12+V12K'+K'V2tt' )A'
'2 . --.-------.. .
B(V11-KV12+v12K ' -KV22K I )A'
Since A and B are both of full
ditions that HVH'
(1.2.10)
rarll~}
be of the form shown in (1.2.1) are that
V11 - KV
12
+ Vl~' -. KV2~'
=
0,
These equations are satisfied if and only if
(1. 2.12)
and
KV12
B(V11-KV12-V1~' +KV2iC' )B'
necessary and sufficient con-
and that
(1. 2.13)
A(Vll+KV12-V1~'-KV22K' )B'
-~--------,--.<
= V1~'
"
4
The (i, j) element of KV K' is v . 1
. l' for
22
r-J+ ,r-J.+
i ~ 1,2, ... , r/2, j?- i, and is v . 1
. l' for j = 1, 2, .•• ,
r-J.+ ,r-J+
r/2, i ~ j; the (i, j) element of KV1'2. is v j ,r-J.+
. l' for i, j = 1,
and the (i, j) element of V12K'
i,j = 1,2, ... , r/2 .
2, ..• , r/2;
If
is
v.~,r-J+
. l' for
(1.2.10) and (1.2.11) both hold, then (1.2.12) and (1.2.13)
both hold, and it fol10vlS that
when
i
= 1,2, •.. , r/2,
j ~
i
,
(1. 2 .14)
when
j
= 1,2, ..• , r/2, i
~ j
and
i,
j :;::
1, 2, ... , r/2
Equation (1.2.15) implies that V
is symmetric about its secon12
dary diagonal and similarly for Vi2' Equation (1.2.14) implies that
the rna trix V* ,
(1. 2.16)
Vo/.·
~
::l~
o
I V22
,
is symmetric about its secondary diagonal.
Therefore, (1.2.10) and
(L 2.11») taken together} imIJly that V is sy:m:metric a.bout its
secondary diagonal; and, since V is
amounts to double symmetry.
If V is doubly
s~etric,
then
s~nmetric
by hypothesis, this
5
V:-G 0-G V~12 }K
I
Vll_i~2]
(1.2.17)
=
Vi2
or
(1.2.18)
K
V22
:::
V12
V22
0
Vi2
V22
KV2~'
KVi~'
KV12Kf
KVIIK'
K
0
,
from which it follows that
and that
f
-)(,
KV12
-- V1z-
(1.2.20)
This proves the result.
A Property of the Tshebysheff Orthogonal Polynomials.
1. 2
Tshebysheff orthogonal polynomials are a set of
(1.2.1)
when
E
~
where the
(1.2.2)
i
P. (x) :::
c ij
j:::o
C
iJ.
if
(,E
J=o
~~)
Pik
i
Pik
,
J
i
= 0,
1, 2, •• "
r-l ,
c ifJ'
~~)
= 0
and l~ = (2k-r-l)!2, k = 1, 2, ' •• , r.
The numbers
(1.2.3)
j
Pi(x),
are so determined that
r
i
E (E
k=l j=o
i ~ it
ci ' x
r polynomials
The
=
,E
J=o
'
,
c i . ~,
k
= 1, 2, ••• , r, i = 1, 2, ••• , r-l ,
J
are tabulated, (see R. L. Anderson and E. E. Houseman
L.~7,
and
6
Fisher and Yates ~1~7), for various values of
Suppose
r
is even.
1
1
The (rxr) matrix P,
1
P
(1.2.4) P .-
P r~2,1
p
P
ll
P12
P
31
;J?32
P
P
e
I
,..,
,I;"
r~l,l
r.
P
r-2,2
2,r
r~2,r
1,r
'D
1:)
'" r-1,2
J.
3,r
r-l,r I
is of the form
p
-
,
where A and Bare (r/2 x r/2) matrices, with the rows in each
matrix
mutt~lly
orthogonal, and K is the (r/2 x r/2) matrix de-
fined in (1.2.3) .
That this statement is true may be seen in the following way:
The values of Pik , according to R. L. Anderson ~~7, are
7
:1.+1
2
2j-l
2j-l '\:
r.
j=l
(1. 2 • 6)
when
i
is odd
when
i
is even
Pik =
i
"2
2'
c.~,J
2' u.. J
.K
L;
j=o
W'Pen
l'
is even,
u.
oK
= -u1'- k +I'
k = I, 2, •.• , 1'/2.
.
Thus vThen
is odd P1' k = - P.
k I' k = I, 2, ••. , r/2; and when 1 1s
1,1'- +
even P'1k = P.1,r-k+l' k = 1, 2, .•. , r/2i and from this fact it
i
follows that
P is of the form shovm in (1.2.5).
That the rows
of A are mutually orthogonal follows the properties of the orthogonal polynomials and simiJar1y for
B.
The matrix A is
,
e
1
1
1
P2,r/2
,
P1'-2,1
and the
IDa trix
r-2,2
P r-2j1'/2
B is
P31
P12
P
32
P
p
PIJ.
(1.2.8) B =
P
1'-1,1 - 1'-1,2
P1 ,r/2
P3 ,r/2
8
1.3 The Characteristic Roots of a Matrix.
It is required
to find the characteristic roots of the (rxr) matrix V,
2
r .. l
I
P
P
p
P
1
P
p
(1.3.1) V ::;
r .. 2
1
2
1 - P
p
p
'VThere
e
vThere
I
p
r ... 2
is a real mUllber
This matrix
.
r-l
p
r-3
IpI
<
1
1 .
is positive definite, and its inverse is
is the (rxr) identity matrix,
with all elements zero except
. for
B
is the (rxr) matrix
::; brr ::; 1, and C is the
11
(rxr) matrix with all elements zero except for c .. ::; 1 if
lJ
b
li-j 1=1.
If
~,k::;
-1
1, 2, .•. , r , are the roots of V ,then
-1
~
are the roots of V.
The problem is therefore equivalent to that
of finding values of
~
I(I
(1. 3.3 )
2
+ P -
~) I
so that
-
lB - pC I ::; 0
•
Let 6 r (AJP) be the left-hand side of (1.3.3).
by minors yields
Expanding 6 r CA, p)
where D (A,p) is the determinant of the (rxr) matrix V* ,
r
9
v* =
(1,3.5)
Expanding D
r
2
(1 + P - A)I - pC •
(A,p) by minors leads to the difference equation,
and it is knovm, (see Siddiqui ~1,-7, Hildebrand ~1~7), that this
difference equation has the solution,
h
were
1 + P2 - A
= 2p
cos
A
~,
Substituting (1.3.7) in (1.3.4) and equating
6r (A,p) to zero
results in the equation,
r
P
sin(r-3)O
sin Q
= o.
If equation (1.3.8) is multiplied by sin G, ( sin 0
=0
is not a
solution), and if the complex representation of the sine function
is used, equation (1.3.8), after a few manipulations, becomes
Thus
i{r-3)G
(1,3.10)
e
---2--
«1_A)e
2'0
-i(r-3)G
-
p)
=+
e
~ «l_A)e- iG _ p) ;
10
and this can be expressed in the form
(I-A) sin (r-l)
2'""" G :::
(1.3.11)
p sin
(r-3)
-2
g
,
and
(1-",) cos (r-l)
Q
2
or in the form
(2 cos Q - p) sin
and
(1.3.14)
(2 cos Q _ p) cos (r;~) Q :::
cos (r;3) Q ,
since 1 + p2 - ~ ::: 2 cos Q.
F~na1ly, since
(1.3.15) 2 cos G sin (r;l) G
:::
n (r+l)",
._- ':>'
+ sin (r-3)
Q
cos (r+l)
- - g + cos (r-3)
-
Q
Si
2
2
and
(1.3.16)
2
cos
Q
cos (r-l)
...._. g
::=
2 2 2
,
are 1fe1l··known identities) (1. 3 .13) and (1. 3.14) can be reduced to
(1 . 3 . 17)
. (r+l)
-2 '"'e'
s~n
:::
p
.
S:ln
(r-l)
--2- '"'e'
)
and
(1.3.18)
cos (!+1) G :::
2
p cos (r-l) G
2
The problem of the roots is solved when values of
G, say G ,
k
are found which are solutions to (1.3.17) and (1.3.18). The roots
-1
2
Of V
will be Ale::: 1 - 2p cos Gk + P •
11
Apparently there is no closed
fOl~
for (1.3.18) which holds in general.
solution for (1.3.17) or
It is possible, however, to
find approximate solutions which are as accurate as one cares to
make then.
An iterative solution is proposed here, and this solu-
tion is shown to converge.
The numerical solution of (1.3.17) proceeds in the following
way:
if
Q,
K
if
::::
P
(1.3.20)
and if
the solutions are
p~l,
::::
(2k-l)rc
0, the solutions are
Q
k
21\:1(
r+l
:::
1,2" ••• , r/2,
,
k
l,2, .•. ,r/2,
l,2, ••• ,(r-l)!2# r odd
be an open interval on the
~:
where
,
k:::: 1,2,
(r-1)/2
if
r
"'J
if
r/2
is odd.
_._-- ... .....2
_.-
~-
~~.-
sin (E-~)Q
2
G
2k:n:)
r+l
r
€ ~\:'
and
Q-axis,
,
is even, and k::: I, 2,
The function
sin (r+1)Q
is continuous for
r even
k :::
r'
~
,.
1,,2, ••• ,(r~1)/2# r odd
2kJt
(1.3.21)
r even
:::
the solutions are
p ::: -1,
Let
,.
k --
,
r
f(G),
••• J
12
1
(1.3.24 )
r
,.
=
f(G)
o
i.e.,
(2k-l):rc
=
G
=
G
is unity at the left end ~oint of each
f(G)
vanishes at the right end point of each
If it can be shown that
9
€ ~,
fl(G)
then it will follow that
~
•
is strictly negative for
f(G) takes on each value between
zero and one exactly once in each interval
It is to be proved that
R .
k
fl(G),
sin r G - r sin Q
,
2 sin2( (r~1)j2)'~
.
e
G €~.
is strictly negative for
sin G >
0
and sin
r
Q
<
0
~,and f(G)
for
From
G €
(1.3.22) it is clear that
R,.
.to
This
~roves
the result,
and it can be asserted that
(1.3.17) has exactly one solution in
each interval
that
~,provided
0 < P < 1.
To find the solution for a given value of
following iterative procedure is proposed:
Put
p, rand k, the
(1.3.17) in the fonn,
. -1
(1.3.26)
s~n
Then choose a point, say Go' from the interval
this point
value of
Q
o
p and
for
Q
in the right hand side of
rare knovm so
value which can be calculated.
~,and
(1.3.26). The
(1.3.26) now has a numerical
This nu.11lber, say G , is a first
l
approximation of the true solution.
A second
ap~roximation,
2, is obtained by substituting Gl back in (1.3.26).
G
substitute
say
In this v~Y
13
a sequence of approximations is produced.
The conditions under which this sequence converges to a true
root are stated in the following theorem from Brand f§]:
If
€ R) where
R is an interval containing a root of the equation
o
g = g(G») and if g(g) is differentiable throughout R) and if
Q
Igf(G)/ < M <
I
for
G € R) then the sequence of iterative solu-
tion of the equation G = g(G») starting with the point
g , cono
verges to a root of the equation.
That the conditions of this theorem are satisfied in the case
of (1.3.26) is easily seen.
exactly one solution.
Go €~,
First
and
~
contains
Second) the derivative of the right-hand
side of (1.3.26) is
which exists for all G i'lhen
I
If
and
r-l
(1.3.2 )
when
Ip I < 1;
r-1
< r+l < 1,
r+I
p
I <1 .
-1 < P < 0, then the
~
become
"'1'2) .•. ) r/2
(
2krr
r+1 '
2krr )
r
'
k =
~
if
r
1)2, .•. , (r-1)/2 is
is even
r is odd
14
In this case,
(1.3.30)
f(G)
{
o if G
==
~~
if G
==
2k~
-1
f'(G) <
To prove that
°
r
for all G
€ ~,
it is necessary to show
that
sin r G - r sin G <
(1.3.31)
for
G €~.
sin G >
°,
This is true for the same reason as before, viz. that
° and sin r G < ° when
G
€ ~~.
So, exactly as before, it
turns out that there is e.x.a.ctly one solution in each interval
~.
The proposed iterative procedure still converges since it was proved
,
e
to converge for
Ipl < 1.
This completes the analysis of equation
Consider equation (1.3 .11i).
if
P ==
(1.3.33)
and if
(1.3.34)
p = 1, the solutions are
0,1,2, • •• , (r-2)/2 if
2k~
'ilk ==
If
r'
k ==
r
is even
{ 0,1,2, . •• , (r-l)/2 if r
is odd
0, the solutions are
G
k
P ==
(2k-1)~
== - - - - -
r+1
-1, the
,
k
f'2"" ,
r/2
==
if
r
1,2, ..• , (r+1)/2 if
sol~tions
,
k
is even
.
r
is odd
are
={.
1,2, .•• ,r/2 if r
is even
1,2, .•• , (r+1)/2 if r
is odd
15
For 0 < P < I, the intervals
~
(2k-l)1t)
r+l
'
are
1,2, .. .,r/2 if r is even
k -
{ 1,2, ... , (r+l)/2
it
r
is odd
and for
< P < 0, the intervals
-1
(1.3.36)
~
are
l,2, ••. ,r/2 if
R: «2k-l):rc
r+1'
-k
k
={ l,2, ..• ,(r-l)/2
r
is even
is
r
is
odd
To prove that
(1.3.37)
,
e
cos (r+l)
. - 1'1'='
2
r-l
cos"2 Q
d
d g
is always negative
that
sin
Q
>
0
for
Q
>
sin r Q
g -
2 cos
1\'
Q €
and sin r
-r sin
=
2 (r-l)
1'1
"2
~
it is only necessary to observe
for
0
there is exactly one solt .tion in each
1
Q € F •
k
This proves that
The proposed iterative
I~.
solution will converge if
( 1.3·311") dd1'1'<:>' (2
r+1
exists for
Q €
cos
-1(
r-l»
p cos -2 g
r-l
""-J
r+_
/1
.
Sln
2
-p
1'1
(r-.1)
2
""
cos
2(r-l)
1'1
"2 ""
and is less in modulus than some number which
~
is less than one.
vllien
'pl
sin
(¥)
< 1, the derivative exists for all G,
and
r-1
r+l
p
p
g
<
/1···
-p
2
cos
2 (r-1JI'I
2
'='
r-l
r+l
16
This completes the analysis of equation (1.3.18).
1.4 A Problem of Identification.
For
r
even, consider the
(rxr) matrix,
where
±
1
(1.4.1)
V2
-1
V
I
-1
V
-K
I
K
~-K
1
V2
=
~l ~
o
,
V
2
is the (rxr) rnatrix ShO"iID in (1.3.2), and K is the
(r/2 x r/2) matrix defined in (1.2.3).
It is required to find the
characteristic roots of VI and V .
2
The roots of V-I can be calculated by the method of the preof VI and V are also the
2
roots of V-I, the problem consists of identifying which roots of
vious section; and, since the
-1
V
ro~ts
are also the roots of either VI or V ·
2
Let 6r/2(~'P) be the determinant of the matrix VI - AI ,
where C 1s the (r/2 x r/2) matrix with all elements zero except
for
c ..
J.J
=1
if
Ii -
jl = 1,
and E is the (r/2 x r/2) matrix
Expanding by minors yields
with all elements zero except
2
.
(l-A)(l-~p -I.. ) D _ / (A,p)
r 4 2
2 2 4
-p (2-~rp -2~)Dr_6/2(~'P)+P Dr _8/2(AJP)
where
Dr (A,p) satisfies the difference equation (1.3.6).
tion of (1.3.6) is given by (1.3.7).
recalling that
1 + p2 - ~
= 2p
The solu-
substituting (1.3.7) in (1.4.3),
cos G, and setting 6 / (A,p) equal
r 2
17
to zero, leads, after a little algebra, to
(1.4.4)
(2 cos G - p)(2 cos G - 1) sin
_..(4 cos G - P -1 )
.
Sln
(r;2) G
. (r-6)
2'" G + Sln
-2-
(r-4)
Q
=0
.
Repeated application of the trignometric identity,
yields the equation,
=
(1. 4.6)
O.
-1
Thus it is seen that the roots of VI are those roots of V
which
are derived from equation (1.3.18). The roots of V are, of course,
2
1
which are derived from equation (1.3.17).
the roots of V1.5
The Inverse of a
~~trix.
It is required to find the in-
verse of the (rxr) matrix V)
vlhere C is the same matrix that appears in (1.3.2).
Professor Rotelling) in his lectures on least squares) gives a
method for finding the inverse of a sUb-matrix of a matrix if the
inverse of the matrix is known.
of the ({r+2)x(r+2)
where C is an
In this case V is a sUb-matrix
matrix VI '
«r+2) x (r+2)) matrix
of the same form as appears
in (1.5.1), and B is the «r+2) x (r+2»
zero except for b
(1.5.3) V~l
11
:::: b
= 1. The inverse of Vl is
r+2, r+2
1
p
2
p
p
1
p
p
1
2
p
::::
matrix inth all elements
r+l
P
...
...
r
p
p
-1-2
r-l
1 - P
p
r+l
p
r
p
r-l
1
By applying Professor Hotelling 1 s method, the inverse of the «r+l)
x (r+l»
matrix V ' which is Vl
2
with the
(r+2)-nd colunm deleted, can be found.
(r+2)-nd row and the
By applying the method a
second time the inverse of the (rxr) matrix, vlhich is
V2 with
the first row and the first column deleted, can be found. This
last inverse is the inverse of V.
The method is this:
v..
J.J
If
are the elements of
v1-1
and
are the elements of V··1 ' then
2
v.J.,r+ 2 v r+ 2 ,J.
v r+2,r+2
In this case
,
::::
i
= 1,2,
.•• , r+2,
so
v*
..
J.J
=-.- 11 - p
2
(p Ii-j I -
p
2r+4-i- j )
,
j ::::
1,2, ••. ,r+2 ;
19
= 1,
i
2, .•• , r+1,
j::: 1 , 2, ••• , r+1
0
If v** are the elements of V-1 ,then
ij
or
** =
v ij
1
2
1 - P
(ti-jl
P
2r-i-j+4
p
-
i-1
2r-i+3)( j-l
- ( f!
- P.
-A
-
p
2r- j +3)
2r+2
1 - P
i = 2, 3, ..
0
r+l, j
= 2,
3, ... , r+l ,
or
1
=----2
( I i-j I
2r-i-j+2
- P
p
1 - P
2r.'i+2) ( i
i
-(p - p
2r-j+2)
,
p-p
2r+2
1 - p
i
= 1,
2,
"0'
r,
j
= 1,
2, ••. , r
0
It is required to find the value of the quantity,
r
= z:
~,{.-K"
V
ij
where
1
ij
,
is the (rxl) matrix vdth all elements one.
From
(1.5.9),
it is seen that
r
(1. 5.11) I:
ij
1
=-"'"'---;::
2
2r-i-j+2
p
1 - P
i
2r-i.+2)( p j ... p2r- j +2)
- ( E....._:...£
__
I - p
2r+2
,
20
or that
r
(1.5.12) ,~
.....
**
v ij
~.
1
1 - p
~ pli- j !
2
p2r-i- j +2
ij
ij
r
1
E
- 1- P2 ij
r
.E
_
I
\p
i+..:1
-P
..
2r-i+j+2
-p
4r-i- j +4)
2r-lti,..j+2
; +P
1 _ p2r+2 .
.
Since
( 1.5.13 )
;
t-J
2r-1-j+2 _ 2
P
- P
1j
.E
p
i+j
.E
=
2r-i+j+2
p
r+3
= p
ij
(1.5.16)
.E
r
2r+i-j+2
ij
p
4r-i-j+4
= Pr+3
p2(1_ r)2
p
= ~'''''':-~-'--
j=l
r
.E
r-i
p
i=l
p
r-J'
P
.E
(1-p
)2
r
r
j-l
2
i-I
p .E P
.E P
j=l
i=l
ij
(1.5.15)
r-i
P
1=1
r
(1. 5.14)
;
t-J
(1_ pr)2
p2(1_r/)2
=
.
(l-p)
r
. 1
.E pJ- =
j=l
2
,
.- ,.,
r+3 (1- pLo)C
P
2
,
(1_p)2
2r+4 r
r-i r r-J'
2r+4 (1- r)2
= pEp
E P
=p
P
1=1
j=l
(1_p)2
it turns out that
2
(1_pr)2
(1. 5.18) .E v**
.. ::: _1_ ~ p1i- jl - 2- P 2 - - 2
lJ
1 2 ..
ij
~
p
lJ
I-p
(l-p)
Finally
r
(1.5.19)
.E
ij
p
ti-jl
r-l
:::
r + 2
,
(l-p)
.E
i=l
(r-i) p
i
)
1
r+l
I+p
)
21
but
(1.5.20)
1"'-1
r;
i
(r-i) p
i=l
=
r-j
1"'-1
r;
( r;
j=l
i=l
So
( 1.5.21)
~
6
pli- jl
=
ij
2
r+l
2p - rp + 2p
(1_p)2
1'" -
The final result is
(1.5.22)
r(l_p)(l+{+l) _ 2e(1-pr)
==
(l-p»)
(l+pr+l)
It is interesting to note that this result can be arrived at
in another v18.y.
where
It is true that
H is the (rxr)
diagonal form.
orthogonal matrix which reduces
Thus
where A are the characteristic roots of V-1 •
k
the characteristic roots of V- l are
2p cos
k~
h'
~K
=~
1"'+1
(sin k1£,_
1"'+1
It is knOvffi that
2
~
1"'+1 + P
and that the k-th column of H'
(1.5.26)
V-I to
1
is
vlhere
~1
sin 2k1~ ,
1"'+
.
,
"'1
sin rk 1t')
1"'+l
1
22
k
= 1,
2, ••• , r.
Thus
r
ok
sin l...E)
r+1
(.E
-1
2
r
2
(1.5.28) - 1 . E
r+
k=l
p::::: 0
r
1-2p cos r+1 + p
jk:n )2
r+l
r(r+1)
:::
2
First, (see Titchmarsh ~117, and
1.6 Some Identities.
.L11:.7) ,
1- P2 _ _
2
1~2p cos Q + P
00
1 + 2
n
cos n
p
.E
n=l
and
(1.6.2)
for all
p
00
sin 9
=
2
1-2p cos 9 + P
g and
Ip
1<
cos k x
::
k=l
.E
k=l
sin
.
Sln
Sln
sin k x
:::::
nx
-2 cos
x
--.
n
(1.6.4)
sin n G ,
n=l
L__67) ,
n
.E
r
p
.E
1
Second, ( see Brand
(1.6.3)
,
2
sin jk+TC)
. 1
r 1
J=
k:n: ' - 2
1~2" cos __ + p
r+l
k=l j=l
(1.6.1)
2
r
(.E
sin
.E (.E
Hildebrand
k:n:
(1.5.28) reduces to the identity,
r
(1.5.29)
j=l
r+1 k=l
which proves that
when
r
.E
1 = --
l' V
2
"2
nx
"2
---x
sin -
2
(n+1) x
"2
sin (n+~) x
2
,
,
g ,
23
n
I;
cos (2k-l)x
==
k==l
n
(1.6.6)
I;
k=l
for all
x
sin (2k-l)x ==
sin 2 n x
2 sin x
1 .. cos 2 n x
2 sin x
,
,
such that the right-hand side of each formula exists.
CHA¥mR II
SOlvIE TESTS OF INDEPENDENCE
2.1 Introduction.
pendence.
Thi~
chapter contains some tests of inde-
The first test is an intersection type test suggested by
Professor S. N. Roy.
It is based on a set of statistics which are
independently distributed as beta random variables under the null
hypothesis.
The others are based on statistics which have the F dis-
trj.bution under the null hypothesis;
anCl., for these, it is possible,
in some cases, to determine the non-central distributions.
In other
cases the non-central distributions can be approximated with any desired degree of accuracy.
2.2 An Intersection Test.
Consider a set of normally distri-
buted random variables, say Y , Y , "', Y ,
2
r
l
,
(2.2.1)
1{
:=:
with
,
1, 2, "', r
and
1
1
(2.2.2)
Var(~) :::
v
v
r-l
r-2
...
v
...
v4-2
1
v
v r- 3'"
1
r-l
r-3
2
0"
A set of observations, say Yl , Y2' "', Yr is given and it is required
to test the hypothesis that all the v. 's are zero,
1
First suppose that
null
h~~othesis,
~k
= 0,
k
the statistics B ,
h
= 1,
2, "', r.
Then, under the
25
h ::: 1, 2, ... , r-l
)
have beta distributions, with parameters (1/2, (r-l'i)/2) and the B
h
are mutually independent in the statistical sense.
That the B have beta distributions is obvious; that they are
h
independently distributed is proved in the following way: Make the
transformation
Yl :: X co::.; " 1
sin 0.) cos
(2.2.4)
1.
Cd
,
k
Ie ::: 2,
3, .•. , r-l ,
r-l
Y
r
:::
xlT
sin O.
J.
i:::l
which has the Jacobian J ,
r
r-2
IT"
i=l
•
1"'\) r- i-I
( Sln
"".
J.
That (2.2.5) is the Jacobian is verified by induction.
= 2,
When r
-x sin "1
(2.2.6)
::: X
sin 01
and this is the same as (2.2.5) when r - 2.
In general
,
26
oYl
t oYl
-oX
(2.2.7) J r
0(;)1
OY2
'oX-
oY2
-dG
1
oYr
oYr
=:
-oX
Look at
.
e
oY
.·•
-
l
-'-
oG r-l
OY2
. · · -dG _
,
r l
•
0(;)1
oYr
·•
(2.2.4). It is clear that
OG;_l
X
is a factor of the last (r-l)
columns of J , that sin G is a factor of the last (r-2) columns,
r
l
that sin 9 is a factor of the last (r-3) columns, etc., and finally
2
that sin G ~ is a factor of the next to last column. Thus
r-~
r-2
TT,
(2.2. )
(sin G. )r-i-l 6.
where 6
r
is the determinant whose
r-l
cn
r-1
sin
1=1
•••
Q.)
J.
cos Gl
r
J.
i=l
ni=2
,
r- th rOvT is
r-l
sin 9 '
sin G., cos 9 2
1
J.
J.=3
0
cos Gr- 2 sin Gr- l' cos Qr- 1) .
The same factorization of J
]
r+ .
yields
27
where b. 1 is the determinant whose
1.'+
1.'-1
(2.2.11)
TI
(cos G
1.'
i=1
r-th row is
1.'-1
sin Gi , cos G1 cos Gr 1=~ sin
Qi
'
1.'-1
cos G cos
2
Q
n. 3sin
G., ••• , cos GIcOS G ,- sin
:J..
1.'1.'
1.' :J..=
Q )
1.'
whose (r+1)-st row is
(2.2.12)
1.'
1.'
(r-T sin G., cos G ~ sin G., cos
1 i=2
i=l
:J..
:J..
...
and whose (r+1)-st
(0, 0,
c
i=3
sin G. ,
:J..
cos G 1 sin G , cos Q ) )
1.'1.'
1.'
co1v~
..., 0,
is
-sin G , cos G ) •
1.'
1.'
Compare (2.2.11) with (2.2.9).
The first
in each except for the constant factor
(2.2.11).
1.'
G~ ~
1.' elements are the same
cos Q
Compare (2.2.12) with (2.2.9).
r
which appears in
The first
1.' elements
sin Q which
1.'
1.' elements of the first
are the same in each except for the constant factor
appears in (2.2.12).
Finally, the first
(1.'-1) rows of b. are the same as the first 1.' elements of the
1.'
first (1.'-1) rows of 6 l' This can be seen by looking at (2.2.7)
1.'+
and (2.2.4).
So
,
o
o
o
-sin Q
r
cos G
r
(2.2.14) 6
r+
1::: sin G cos G
r
r
,
cos .1.'
G
sin G
x
r
where
x
is the last row of 6;
1.'
and, therefore
sin Q
_--::;~1.'
cos Q
)
:::
6
1.'
Since
~:::
1.'
.
1, the result is proved.
Since
(2.2.16)
the joint probability density function can be written,
exp
1
X2
-2 2(J'
Xr -
1 1.'-2
.
1
J.:::
. 1
(sin Q.)r-~~
This proves that the random variables, X, G1 , G2 , .•. ,Qr_l' are distributed independently. It is easy to verify that
r
L;
(2.2.18)
k=h+l
h ::: 1, 2, ••• , 1.'-1 ,
29
and this proves that the B.h are distributed independently.
The proposed intersection test is the follo1nng:
¢(z):: 1 and accept it otherwise, w'here
hypothesis when
(2.2.19) ¢(z) -""
The
Reject the
Ol
{
other1'lise
b are
h
, k :: 1, 2, ••• , r-l ,
v1h and v2h are determined from the relations
and the numbers
1
(2.2.21)
P (v < B < v ) :: (l_a)r-l
2h
r 1h
h
Because of the independence of the B
h
r-l
(2.2.22)
Pr(¢(.!):: 0)::
IT Pre v].h < Bh < v2h ) :: 1 - ex ;
h=l
a.
i.e., the size of the test is
If ~:: ~
f
0, k :: 1, 2, .•. , r, then the B are no longer
h
beta distributed under the null hypothesis, and the test (2.2.19)
is not applicable.
For this case consider the following test
which can be performed when
Let
r
is even.
H be the (rxr) matrix,
30
H =
1
\f2
A
AK
~
where A and B are the matrices
,
(1.2.7) and (1.2.8), except
that both A and B have been made orthogonal by normalizing their
rows, and K
vector
~
=
is the matrix defined by (1.2.3).
H X,
Then the random
has the multivariate normal distribution; and,
fram the properties of the orthogonal polynomials, it follows that
and, from what
v~s
proved in section (1.2), it follows that
°
(2.2.25) Var(~) = H V li' (J'2 =
(J' 2
°
where V is the covariance matrix of
(2.2.26)
V
=
Vll
I
X,
V12
~i~r-;ll
Therefore, the r.a.ndom vector
*.. .
E(~) ~
(0, 0, ... , 0) ,
and
(2.2.28)
*
Var(~) =
(J' 2
,
31
where
(A(Vl1 + Vlt(')A')*
is the matrix A(Vl1 + V12 K')A'
the first row and first column deleted.
The statistics
with
Bh '
r
~
Z
k=h+l
1"
h
==
2, 3, .•. , r-1 ,
2
Xk
1:
k=h
are mutually inlependently distributed as beta random variables,
with parameters
(1/2, (r-h)/2), under the null hypothesis; and
test (2.2.19) can be performed on the transfonaed observations.
2
( *) :: I cr.
The hypothesis tested is that Var!
'e
B(V II - Vl 2K')B'
(2.2.30)
Clearly,
:: I if and only if VII - V12K' :: I, and
1
'Ill
VI
1
v
v
...
...
v
1"-2/2
v
r-4/2
VII-VIi<' ==
v
1"-2/2
v
r-l
v
v
1"-2
1
r-4/2
1"-2
vr / 2
1"-3
Yr-2/2
::
if and only if all the v' s are zero.
i
before is tested.
I
Thus the same hypothesis as
32
2.3
A
Second
as in (2.2); i.e.
T~I?~
.o_"f Il19-ependence.
Consider the same model
Yl , Y2, ... , Yr are multivariate normal with
k ::: 1, 2, .•. , r,
and
...
...
1
v
:B'or
r
(j
r-2
2
1
r··2
-+-_.
I
1
K
);
I
v
r-l
even, let II be the (rxr) matrix,
=:-
where
v
r-l
v
,
~K
and K are both
r, then the random vector
(r/2 x r/2).
If
~
=:
0, k
= 1,
2, ••• ,
X::: H Y has the multivariate normal
distribution with,
(2. 3 .l~)
E(~J )
=
(0, 0, ... , 0) ,
and
(2.3·5)
Var(~)
V11 + VliC'
0
:::
(j
0
2
V1l - V12K'
where V1I and V
are the (r/2 x r/2) partitions of (2.).2).
12
Under the hypothesis of independence, the statistic G,
33
G :::
,
r
l;
k:::r+2/2
~
has the F distribution with (r/2, r/2) degrees of freedom.
The test proposed is the following:
Reject the hypothesis
if ¢(E)::: 1, where
(2.3.7)
¢( x) ==
otherwise
1
v2 and VI are the appropriate percentage points of
The numbers
the
F
distribution and
r/2
g
k==l
:::
where the little
If
~k
==
~,
XIS
are the transformed observations.
k ::: 1, 2,
"0'
r, and r
is even, let
H be the
(rxr) matrix,
II
1
::: .----
'.12"
A
AK
B
-BK
-'--I
where A and B are the same as in (2.2.23).
tor
Then the random vec-
X == H Y has the multivariate normal distribution vnth
(2.3.10)
E(lS' )
= (0
ll, 0, 0, ... , 0) ,
and
A(V11 + V12Kt )A f
(2.3.1.1)
Var (~n
=
°
~
°
(J'
2
B(V11 - V1dC')B'
For the same reason as in the previous section the statistic
(2.3.12)
•
has the
G
,
=
F distribution with
(r-2/2, r/2) degrees of freedom if
and only if the hypothesis of independence is true.
The test pro-
posed in (2.3.7) is therefore appropriate.
Against specified alternatives, the
non-central distribution
of G cs,n be approximated i'lith any prescribed degree of accuracy.
An example will be given in the next section.
2. h Non- central Distribution of
G( when II
= 0).
The approxi-
mation may be calculated by a method due to Pachares or a method
due to Robbins £l~}.
since Pachares'
W01~
rrhe method due to Robbins is used here
is unpublished.
Consider the following alternative to the hypothesis of independence:
G,
35
Var(~)
(2.4.1)
=
r-l
p
r-2
1
p
p
1
P
'r-2
p
1
.r-l
p
2
2
l-p
(J
,
jpl < 1. This model arises in the case where the Y's
where
satisfy the stochastic difference equation,
,
(2.4.2)
and the
N( 0,
k
= 2,3, ••. )
r ,
€k are independently and identically distributed as
2
IT ).
The random vector
has the multivariate
(2.4.3)
E(~')
~
= H!
norw~l
,where H is the matrix (2.3.3),
distribution with
= (0) 0, ... ,0»)
and
(2.4.4)
Var(~)
=
where Vll and V12 are the
(2.4.l)
If
(2.4.5)
(J
2
(r/2 x r/2) partitions of the matrix
P is known, there is an (rxr), orthogonal matrix L,
L
=
,
= D(\;) and L2(Vl1 - Vl~')L2 =
D(Dk)j and from what was proved in section (1.4), the values of
such that
L1(V ll + Vltr')Li
Dk can be approximated, vQth any desired degree of
accuracy, vlhen p is known.
~
and
The random vector Z = H X has the nmltivariate normal distribution
(2.4.6)
l'1i th
E(~')
=
(0, 0, ... , 0)
and
Var(~)
D(~)
°
°
D( 11k)
=
G has the same distribution as the statis-
Thus the statistic
S,
tic
r/2
2:
s =
(2.4.~)
Tl , T , .•• , T
r
2
N(O, 1); and
where
as
(2.4.9)
PreS
k=l
2
"K
T
k
are independently and identically distributed
< c)
can be apprOXilTIated with any desired degree of accuracy by a method
due to Robbins
Il':!,7.
37
Robbins' Theorem
(2.4.10)
Pr
eA~resses
the probability,
,
< c
2
2
Xr+l -\- a r +2 Xr +2 + •..
a r +s
"There a, a l , .•• , a r , a r+ 2' .•• , a r+s are positive constants and
2'
a ? 1, k = 1,2, ..• , r, r+2, •.• , r+s, and the X s are indek
pendent chi-square variates, each with any number of degrees of
freedom whatever, as an infinite series; and he gives a formula
for calculating the maximum error committed by truncating the
series after any given mlmber of terms.
The maximum error tends
to zero as more and more terms are included, so any degree of accuracy
can be attai ned if a large enough number of terms is included.
It is clear that
PreS < c) can be approximated by the appli-
cation of Robbins' Theorem, since
r/2
(2.4.11)
S
=
2:
min(~)
k=l
min(111)
r
2:
~ T~
-mill~)
k=(r+2)/2
2
'
Tk 'l1 r -k+l
-'nlinG1cJ--
is the same form as the quantity which appears in (2.4.10).
The
A'S and the Tl's are, of course, positive since they are roots
of positive definite matrices, and the quantity (min(Ak))/(min(l1 ))
k
plays the role of a.
2.5 A Test
o~.Jndependence With
a Circular Model.
Suppose that
Y , Y , .•. , Y are a set of random variables with the multivariate
l
2
r
normal distribution and
38
(2.5.1)
E(~' )
::::
(0, 0, ... , 0),
Var(X)
::::
Vcr
and
(2.5.2)
2
,
where
-1
(2.5.3)
V
::::
2
1
«1 + p ) I - pC) ~
cr
,
with
(2.5.4)
0
1
0
0
0
1
1
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
1
0
0
0
1
0
C ::::
e
This is the so-called circular model (see T. W. Anderson ~_7).
The matrix V is (rxr), and ther is an (rxr), orthogonal
matrix H such that
vThere, 'YThen
is even,
r
--1_2 P cos
{
and, when r
is
1··2
odd.
P
~(k~l)~
+ p2 , k :::: 1,2, ... ,(r+2)/2
/2,(r+6) /
cos (2k-r-2)~
r
+ p2 , It:::: (
r+4)
2,'... ,r
39
1-2 P cos
(2.5.7)
~
If
r
2(k-l)rc
r
1
1-2 p cos (2k-rr
r
(2.5.9)
2
+ p , k
~~
r
:::
(r+3)/2,(r+5)/2, ••. ,r
H is
, .•. , cos (k-1) 2rrc)
r
II ,
Ifr
... , (r+2)/2, and
(sin(21~-r-2) !Sr
sin (2k-r-2) 2rc,
r
J
k ::: (r+4)/2, (r+6)/2, ••• ,r.
l/vr.
hr
is even, the k-th row of
= 2,3,
= 1,2, ..• ,(r+l)/2
:::
(2.5.8) (cos(k-l) 2rc , cos(k-l)
k
2
+ p , k
If
"'J
sin(2k-r-2)E2!
r
)-JF
Vr ,
The first row is a row of constants each
is odd, the k-th row of H is
r
Qrr , cos (k-l)
4 r-c , .•. , cos~k-1
t ) --2rrc)
(2.5.10) (cos(k-l) ~r
r
r
-2
,
r
k ~ 2,5, ... , (r+1)/2 , and
(2.5.11) (sin(2k-r-1) 2£ , sin (2k-r-1) 5..1! ,
r
r
l/vr.
--X:::
ri
1r
H Y has the multivariate normal (I.istri-
-
bution; and, since all rows of
and
r
These results are given by T. VI. Anderson 1-~7.
The random vector
(2.5.12)
sin(2k-r-l) rrr
The first rOvl is a row of constants
k = (r+5)/2, (r+5)/2, ... , r.
each
"'J
E(!')
=:
,0: ,
H
Slrrli to zero except the first,
0, 0, ... , 0) ;
40
If
r
(2.5.14)
G
has the
is even and
1
F
r/2
is even, the statistic
= r-2
,
r
distribution with
if and only if
= 0;
p
(r/2, (r-2)/2) degrees of freedom
and if
r/2
(r+2)/4 v2
~
L:
G2
k=2
T
=r-2
(r+2)/2
L:
k=( r+6) /lj.
has the
2
Xk
+
is odd, the statistic
(3r+2)/4 v2
L:
"11:
k=(r+4)/2
r
L:
+ k=(:3r+6) /4
I:
~
F distribution with «r-2)/2, r/2) degrees of freedom if
and only if
If
G ,
l
r
p
=0
.
is odd and
(r+l)/2
is even, the statistic
G '
3
,
has the
F distribution with
if and only if
p
«r+l)/2 (r-3)/2) degrees of freedom
= 0; and if (r+1)/2 is odd, the statistic G4 ,
41
(3r+l)/4
2
X
+.E
k::( r+3 )/2 ':k
(r+l)/2
.E
k::(r+7)/4
has tre
2
Xk
F distribution with
if and only if
p
+
r
.E
k::(3r+5)/4
,
~
«r-l)/2, (r-l)/2) degrees of freedom
= o.
Consequently, these statistics can be used to test the hypothesis
of independence, i.e., that
p:: O.
There is a uniformly most powerful test of the hypothesis p
for this model (see T. H. Anderson
-r57),
-'
but
j. t
=0
depends on the
serial correlation coefficient, and the percentage points of the distribution of the serial correlation coefficient are not generally
available.
Moreover, the non-central distribution of G and G
4
3
can be found, and the power function of these tests provides lower
bounds for the power f\mction of the uniformly most powerful test.
The power function of the uniformly most powerful test is, of course,
not knovffi.
The motivation for choosing the statistics
they were chosen should be made clear.
G , G , G , G as
2 3
l
4
In each case, the choice v~s
made in such a i-Jay that the variances of the
the numerator of the
variances of the
G'S
11k t S
appearing in
were all larger (or smaller) than the
Xkts appearing in the denominator, the larger
(or smaller) depending on
p > 0 (or p < 0).
It seemed that this
choice of all possible choices i·[ould have maximum power.
Also) for
42
this choice, it is possible to determine the exact non-central distributions of G and G in closed form.
4
3
2.6 The Non-central Distribution of G .
3
(2.6.1)
2
(r+5)/'+ ~~
(3r+!~)/4
Q = - - ( r,
~+ L:
1
r +1
k=2
k=(r+3)/2
Let
~),
and
(2.6.2)
2
CL=
~
(r+l)/2
--(
L:
2
3C
r - 3 k=(r+9)/4 -~
+
r
2
L:
3C)
k:::(3r+7)/4 .Ok
By checking (2.5.6) and (2.5.7) it is easy to see that
(2.6.3)
Var
(~)
:::
Var(}~
+ (r-l)/2)' k ::: 2, 3, ... , (r+5)/4, and
that
(2.6.4.)
Var()~)::: var(J;~
+ (r-l)/2)' k
:=
(r+9)/4, (r+13)/4, •••
(r+l)/2
•
Thus (r+l)Ql/2 has the same distribution as a weighted sum of independent chi-squares, each with two degrees of freedom, where the
'VTeights are
(2.6.5)
k ::: 2, 3,
weights
C ::: (1 - 2p cos(k-1)
k
"'J
2
~rr
2 -1
+ p)
1
~~
,
(J'
(r+5)/4, and (r-3)~/2 has a like distribution 1nth
(2.6.6)
k
:=
~~
C ; (1 - 2p cos(k-l)
k
(r+9)/4, (r+13)/4,
.. f'
+
2 -1
p)
1
2
'
(j
(r+l)!2
•
The distributions of the quantities (r+l)~/2 and
are qUite easily found.
Their characteristic functions are
(r+5)/4
(2.6.7)
'TT
¢l(u) =
(r-3)Q2/2
k=2
(1-2Ck uJ.'r- l
and
(2.6.8)
(r+l)/2
TT
k=(r+9)/4
¢2(u) =
(1 - 2 C
k
U
i)-l •
From a 'tvell kno'Wn theorem of complex variables (see Ah1fors
it follows that there exist b
k
f)})
such that
and
(2.6.10)
¢2(u)
=
(r+l)/2
E
k=(r+9)/4
bk
(
. )-1
1 - 2 Ck u ~
are identities in u for all complex numbers such that both sides of
the equations (2.6.9) and (2.6.10) exist.
To find the b an artifice described by Courant ~17 is emk
ployed. Multiply both sides of (2.6.9) and (2.6.10) by the quantity
(1 - 2 C ,
k
U
i).
They become
44
(r+5)/~·
(2.6.11)
IT
k=2
(r+5)/4
( 1 - 2 C u J.. )M1 -k
b k' + (1 - 2 Ck' u]..)
k~k'
E
k=2
bk
k~kr
and
(r+1)/2
+ (1~2 Ck'u i)
Z
b
k::(r+9)/4 k
(2.6.12)
k~kr
(1 - 2 C u 1)-1 •
k
Equations (2.6.11) and (2.6.12) are identities in uj hence choose
u so that
(1 - 2 C u i) = 0, and the result is
k
(r+5)/4
= TT
), k
= 2,
3, ••• J (r+5)/4 ,
k=2
k~kr
and
(r+1)/2
IT
k=(r+9)/4
Ckr
(.
C ) , k :: (r+9)/4,(r+13)/4, ...
Ckr - k
klk r
The inversion of ¢l(u) and ¢2(u)
and (2.6.10) is
~ediate)
(r+5)/1J,
(2.6.15) Pre (r+1)Q1/2 < c) = 1 - r:
k=2
(r+l)/2 •
in the form shown in (2.6.9)
(2.6.16) Pr«r-3)Q2/2 < c) = 1 -
(r+1)/2
-C/(2Ck )
Z
bk e
k=(r+9)/4
,
where c > 0 •
Finally,
(2 . 6 . 17)
Pr (G3 <c)
Q ~
1 <..
r+1
(
= P r"2
r-+1
r-3 Q)
'r-3 c""2
2
'
or
(2.6.18) pr(G < c)
3
=
(r+1)/2
b
k
k=( r+9) /4 2Ck
or
x
- 2ck
e
d x ,
co
~ f=2
(r+5)/4
Z
o
(r+l)/2
Z
k=(r+9)/4
d
e
x ,
or
(r+1)/2
Z
k=(r+9)/4
v1hich, of course, has meaning only v1hen c?: 0 •
.
e
The distribution of G
4 may be found in excat1y the same way•
It turns out to be
(2.6.21) Pr(G4<C)
(1'+1)/2
(1'+3)/4
=1
~
-
,
.E
k;:;(r+7)/4
(=2
c
>
0,
where
( 2.6.22) Ck
k
==
= (1 - 2
P
2
(r+9)/4, (r+13)/I+..... (r+l)/2,
(2.6.23) C
i
( = 2,
2 -1
cos (k-1) -.:1
+ p)
l'
2 -1
(1 - 2p cos (i-1) r~ + p )
3, .• 0' {r+3)/4.
1
2cr
'
The b's are
(r+1)/2
-'-I
k'=(r+7)/4
k' ~lc
tt
and
2
==
1
--2-'
C
k
(C-:C-- ), k
k
==
(r+7)/4, (r+11)/4, ••• ,
k'
(1'+1)/2 ,
and
(2.6.25)
), i
== 2,
3, ... (r+3 )/4 .
The non-central distributions of Gl and G2 are not so easy
to find. However, the modified statistics
and G J
Gi
2
47
(r+4)/4 2
Xk+
Z
(2.6.26)
(3r+4)/4
G' _ r-4 _1_c==2
1 r/2
-r
2
l:
k=(r+8)/~.
11k
~
l:
k.....;=(l..-r_+4)/2
+
>;:
2
r
Z
k=(3r+8)/4
,
11k
and
(:Jr+2)/4
E
(2.6.27)
2
G
k=(r+4)/2
==
+
~
,
2
r
L:
k=(3r+6)/4
Xk
have distributions ll'hich can be found by the method used to find the
distributions of G and G .
4
3
2.7 The Consistency of These Tests.
(2.7.1)
2 (r+4)/4 2
Ql:: - ( E
r
k=2
X,
11:
(3r+1.~)/4
+
L:
k=(r+4)/2
Consider Gl
and let
Y~),
and
By checking (2.5.6) and (2.5.7) it is easy to see that
40- 2
r
(r+4)/4
E
k=2
2
2 -1
(1 • 2p coS(k-l)..2! + p )
,
r
and that
+:2
•
e
r/2
2 : 2 -lj
Z
(1-2pcos(k-l) -2! + p) ..
k=(r+8)/4
r
48
Recall (1.6.1).
The limit of E(Ql) as
r
becomes large can
be written
(2.7.5)
lim
r->
1._2
= '~im
E(Ql)
r->
00
'tU
-
00
r
rib,
1 2 (1 + 2 '"' Pn cos 2nkrc)
- ,
k=l 1 _ P
n;l
r
'"
<f-,
<f-,
or
4<r 2
1
(2.7.6) -lim E(%.) = "=-2 ( "4
r-> 00
I-p
.
-I-
2 1m
r->oo
r1
r/4
~
I:
k=l n=l
n
2nkrc)
cos:r- •
P
Since the series on the right-hand side of (2.7.6) is absolutely convergent for
(2.7.7)
p
E(Ql) =
lim
r->
< 1, (2.7.6) can be rearranged to the form
lKr 2
1
n
r/4 1
2nkrc)
(To + 2 Z P lim l:: - cos - - •
1- p'. '+
n;l
r->OO k=1 r
r
---;s
00
Making use of formula (1.6.3), (2.7.7) becomes
(2 • 7 • 8)
..J.~m
-'l'
r->
E()
Q
1
00
2
40(1
--2
"4 + 2 '"
=
<f-,
Pn.:l..
..c1.m
n=l . r->oo
1-p
r1
SiuT
.
s~n
nrc
-r
r+4) nrc )
cos (-7
T
But
. nrc
lim
r-->
00
1
r
SUlrl+
.
s~n
nrc
-r
cos
CE#) .~1E
sin nrc
2
:::
,
2nrc
so that (2.7.8) becomes
1.
+ -
n
Z
rc n=1.
.e...n
nrc )
sin "2
•
.
The
S,
StUll
co
(2.7.11)
S
~
;:;
n=l
n
n1!
sin 2"
E.n
'
can be evaluated in the fo11ov;ing way.
Observe that
p
(~
So
~1!)
xn.. l sin
n=l
dx
n
..e_
;:;
n
,
sin ~
2
This is surely true since a power series converges uniformly within
its interval of convergence.
co
(2.7.13)
~
n=l
e
](1-1 sin --n1t
2
By formula (1.6.2) it turns out that
1
;:;
2
l+x
,
x < 1 •
Thus
co
(2.7.14)
~
n=l
n
.e...n
(J
n1t
sin -;::j
::
t.:.
~
dx
l+xt.:.
r)
::
tan
-1
p
(>
Therefore,
Next, by making use of (1.6.1) again,
(2.7.16)
lim E(Q2)::: 11m
r->co
r->
-12 ( 1 + 2
co
1-p
00
~
n=l
cos (r+4k) ~~ ) •
Inverting the order of summation and making use of (1.6.3) and
(1.6.4) leads to
n
P
50
(2.7.17)
lim E(Q2)
r->
00
cos 3nn) •
4
But
sin (~) nn
(2.7.18)
lim
r->
r~2
sin ~
00
sin ~
3nn
cos - 4
r
4
1
2
:= -
2nn
r
Thus
2
(
4~
2.7.19) . lim E(Q2) ~ -"2
r->OO
(1
"4 -
00
1
2::
1C
n=l
1-p
n
F_
.
n
n,r)
sJ.n'2
'
or, recalling (207.14),
(2.7.20)
1
lim E(Q2)
tan
1C
r->OO
... 1
p).
If
(2.7. 21)
1
2
2k
2 -2
1 r/4
J.1m .- (L: (l-2p cos --.!i + p) )
r-> 00 r k=l
r
=
0,
and
1
(2.7.22)
lim
r->
00
1
(r-4)/4
r
k=l
-- ( I:
(r+4k)n
(1-2p cos - 2.~-
r
+
2 -2 2
p)
)
=0
,
then Var(Q1) and Var(Q2) tend to zero and Tchebycheff's inequality
Q1 and ~2.
Making use of (1.6.1), (2.7.21)
applies to
becomes
r/4
00
00
(2.7.23) lim 1
L: (1 + 4 L: pn cos 2nkn + 4 ( l.: pn cos 2nkn )2)
2
r- >00 r
k=l
n=l
r
n=l
r
1
2
1-p
,.
51
but this limit is zero since
I
00
~
~
n=l
r > O.
pn
c o2nkrr
s-
I<
-!B..L. for
1- Ipl
r-
p < 1
and
The same reasoning on the case of (2.7.22) leads to the same
result.
This means that
and that
for every
€
> O.
There is a theorem in. Cramer
variables, say X
["§.7
'Which says that if ti'lO random
and Y, are converging in probability to points,
n
n
and C , then the ratio Xn/Y converges in probability to
n
2
the point C /C
if C ~ 0 •
2
1 2
Therefore,
say C
l
rr -Irr ..
4
4
tan-1
tan
-1
But
if and only if
rr -Irr -
4
p
=0
4
tan
tan
•
...1
-1
p
P
=
1
p
p
,
52
This proves the consistency of the test based on the statistic
G " The consistency of the other tests (those based on
l
will follow through s:lJ:nilar arguments"
G, G, G )
2 3 4
CHAPTER III
ANALYSIS OF VARIANCE WHEN THERE IS NO INTERACTION
3.1
Introduction.
This chapter contains the results of some
investigations into the problem of analyzing the data from a two'Way classification vThen the random errors are subject to certain
patterns of dependence.
For a circular model the pOvTer functions
of two possible analyses are compared.
3.2
~esting
!or Treatment Effects Without Interaction.
sider the following situation:
(rxl), i
= 1,
2, ••• , t,
j
The random variables
= 1,
Con-
Y ., Y.. is
- i J -~J
2, ••• , b, are independently and
identically distributed as multivariate normal vnth
E(Y .) = (IJ. +
- iJ
T.
~
+ t3.) 1 ,
J-
and
(3.2.2)
Var(Y.. ) =
all (i, j) ,
1
p
.•
•
p
1
.•
•
r .. l
P
r .. 8
p
-~J
2
-2
cr
l .. p
•
1
The parameters
A set of observations, say Y . ,
- iJ
(
p < 1) are unkno'Wll.
all (i, j), is given, and it is
required to test the hypothesis of no treatment effects.
One thing that can be done is as follows:
x. .,
~J
The random variables
=
are
1
-
11 Y · ,
-iJ
r -
indepen~distributed
as normal with
and
2
Var (Xij ) =
r-~p-rp
2
r+l
+ 2p __
2
(T
2
2r
(l-p ) (l-p)
The latter follows from (1.5.21) .
The
Xij'S
form a two-way classification vrlth one observation
per cell, each observation with the same variance; and the usual
analysis of 'variance can be performed, along with the usual tests.
In fact, this analysis Vli.ll be valid even if the structure of the
dependence is not the same as shovrn in (3.2.2) .
Another possibility arises if the model is changed slightly.
Suppose that
(3.2.6) Var(Y
.. ) = «1
-~J
212
+ p )I - pC)- (T
,
where C is the (rxr) matrix with all elements zero except for
c ij
=1
if
i-j
= 1.
This is a modification of the circular model
described by T. W. Anderson ~~7.
There exists an (rxr) orthogonal matrix H such that
The k-th row of H, say
Ek'
is (see T. W. Anderson ~~7):
55
k~
(3. 2 .8)
.
( sJ.n
k
= 1,
2, .•. , r,
k
:=
-r+I '
,
rk~
sin r+1
and
I, 2, ... , r .
-J.J
X.,
= H -J.
Y' j
,
(3.2.10)
E(X!.)
-J.J
==
Let
,2k~
sJ.n r+1 ' ••• ,
all (i, j).
Then,
(Il
+ T.J. + f3,)
__
J
{f¥
(~
(::1
sin
r
:E
K=l
f~l
r+
'
~
L>
K:=l
, 2f~
sJ.n - 1 ' , •. ,
r+
rf1! )
r+1
'
sin
and
(3.2.11) Var(X
.. )
-J.J
= D(~
) ~2
."k
•
Referring to (1.6.4), it is seen that
for k
Consider the ran clam variables
(X,J.J'1' X.J.J..I
'"A.'
-Zi'J
••• ,
even.
Z. j '
-J.
X.J.Jr~
'1)
The random vector Z.,
-J.J
r
even;
r
odd.
:=
(x,J.J'I' X,J.J'3' ... , X.,)
J.Jr
i\l'hether
r
,
is (r/2x1) or «r+l)/2 x 1) depending on
is even or odd, and
(3.2.14)
(!J.+'I'.+f).)
~
J
E(Zi'.) =.
- J
fr+l
(~
i=l
r+l
(=1
"'2
~
K=l
vrhere
p = r - 1 or
r
r
i:
in ,
sin
. -1
3fn , ... ,
r+
s~n
sin pf']( ) ,
r+l
clepending on whether
r is even or odd.
Also
(3.2.15)
Var(Z
.. )
-lJ
= D(~_)
~2 ,
.
~k
vrhere
(3.2.16)
h
= 1,
~
= (1 - 2P cos
2, .•. , r/2
if r
(2h-l)'](
2)-1
+ P
r+1
is even, h
= 1,
,
2, •.• , (r+l)/2
if
r
is
odd.
The random variables
Zijh are a
two-\~y
each h, with variance per observation "-h~2.
classification for
Thus the usual analy..
sis of variance can be performed for each value of h.
For each
analysis there is a function of the observations, say ¢(~h)' which
provides the criterion for rejecting or not rejecting the hypothesis;
i.e., the hypothesis is rejected when
¢(~h) =
1 and not
The following intersection test is proposed.
(3.2.17) ¢(zl' z2' ••• ,
where
q = r/2
Zq) =
or (r+l)/2
{
¢(~h)
o
if all
1
otherwise,
depending on whether
The hypothesis is rejected when ¢(zl' z2' ••. , z )
-
-
-q
othe~dse.
Let
=
0,
r
is
= 1.
even or odd.
57
The test can be made of size a
in the follo1ving way:
(3.2.18) pr(Ho rejected by h-th test
H ) = v;
o
then) since the tests are independerlt)
(3.2.19) PreHo not rej ected by any of g tests
·1
H )
0
= (1- v)g
;
and
(3·2.20) Pr(Ho rejected by at least one of g tests I H0 )
= l-(l-v)g.
Thus
(3. 2.31)
-e
a =
1 - (1 -
v)g )
or
(3.2.22)
v
=1
- (1 _
a)l/g .
Therefore, if each individual test is of size
by
v, where
v is given
3.2.22, then the intersection test is of size a. This method
of fixing the size of the test is; of course, arbitrary.
The guestion now arises as to how the power
f~unction
of the
pro~
posed intersection test compares vdth the power function of the test
on the means (by test on the means is meant the test on the
two-v~y
classification formed by averaging the observations in each cell).
For the modified circular model the test on the means would be
by the ordinary F test with variance per observation
l'
V 1
The formula (3.2.23) follows from (1.5.22).
3.3 Comparison of the Power Functions.
It will be shovffi that
the a-size test on the means is locally more powerful than the
a-size intersection test at least for most values of a.
It has,
however, been suggested by Professor Roy that the intersection test
is likely to be more powerful at other values of the non-centrality
parameter.
For small 6.2 and
0"
2 taken as one ( 0" 2 can be taken as one
without loss of generality since it cancels out in the comparison),
the power function of the test on the means is
where
d F ,
and 8
is a small number vanishing at the same rate as
Similarly, for small
t},
=
1 -
TI
11=1
f3h
•
the power function of the intersection
test is
f3*
4
6.
,
59
where
,
h
= 1,
2, ••• , q, and b
replaced by
F..
1/
(I-a)
at the same rate as
q4
is the same as
l
' and the
6.
0h
a
1
except that
F _ is
l a
are all small numbers vanishing
Therefore,
,
where
q
I
E-
E::
'e
h=l
and
0*
,
~
is a small number vanishing at the same rate as
6
4 .
It can be proved that
q-l
(3·3.8) (I-a)
vThere
ex
q
bl
~ a1
'
is any m:unber betw'een zero ana. one and
The quantities
a
1
and
b
is an integer.
have been defined, and
l
t-3
\=
1
b..",..-I---,..."..,-",........,~
B(_t':-_l , ('b-Ij"(t-l)
2
q
-
2
j
(~)
o
(1+ij:'I)
1-0("'"2
'F
b(t-l)
2
d F ,
60
o
F
(1 + b-l)
b(t.:TJ
2
d F
Make the transformation
Then
(3.3.8)
becomes
i
"A
v
f\(l-o{/
\)
q-l
(3.3.12) (l-a) ~
\
j
t-l
(b-l)(t-l)-2
X~ (l-X)
2
d X
>
o
'A\-d..
\
t-l
J X ~ (I-X)
(b-l)(t-l)-2
2
d X ;
o
and
(3.3.9)
and
(3.3.10)
(3.3.13) (I-a):=
become
1
_
B(~l , (b-l~( t-1»
o
a.nd
(3.3.14) (1_a)1/ q =
.1
___
B(t-1 , Jb-l)(t- 1 2)
2
2
o
d x
•
•
61
The integrals on either side of (3.3.12) can be evaluated
by parts.
The result is that
X~I_oI.. )
~
X t;1 (1-;;)
(b-l)~t-l)-2
d X
o
_ (1- ) v
B (t-l
(b-1)(t-1»_
a ''''l-a
2'
2
~
X1-0<.
u(x) d x ,
o
and that
X (\-01.) 1/'0
(!:1:
(b-l).( t-:l)-2
j X 2 (I-X) 2
(3.3.16)
d x
x (1_""')\
D
~ v(x)ax,
o
v1here
x
(3.3.17)
vex)
=
I
_~-3
(b-l)(t:.~)~_
2
2
Y (l-Y)
dy .
·0
Thus) since
X
1/ > X(l ) ' it follows that (3.3.12)
(1-0;) q -a
reduces to
q-l
.q-l
+ ( 1- ( I-a) q )
d x - (I-a) q
X(\ _ o() \,~
j vex) ax.:=
The function vex)
::<,-0(
is positive and monotone increasing.
by the 1av1 of the mean,
0
Therefore,
6e
and it follows that (3.3.18) is true whereby (3.3.12) is true.
NOvl assume that
q-l
«l-a) + b (l-a)-q E ~2 +
1
(3.3.20)
e/)
If K ~ E, then (3.3.20) is true when 6
and the fact that
4
6 .
,.,here
*
and
8
e
2
is small because of (3.3.12)
8 are vanishing at the same rate as
If K < E, then (3.3.20) can be put in the form
7(-*
8
is vanishing at the same rate as
6
4 ,and (3.3.21) vall
be true for sw.all ~2 if and only if
q-];
(3.3.22)
bl(l-a) q
E + (l-a) (K-E) - a l K ~ 0 .
The constant E is positive.
Then from (3.3.8), it follows that
(3.3.22) will hold when
From the Cauchy-Schwarz inequality it follows that
(3·3.24)
(l-a)
2
.
Thus (3.3.23) will hold when
,
which is true except for ex near one.
CHAPTER IV
ANALYSIS OF V.A.RIANCE IN THE PRESENCE OF INTERACTION
4.1 Introduction. This chapter contains the development of
a method for analyzing a two-way classification with interaction
when the within-cell errors have a modified circular dependence.
4.2 The Analysis.
Y.. , Y..
-lJ
-lJ
is
Suppose that the random variables
(rxl), i = 1, 2, .•. , t, j
=1,
2, ••• , b, are in-
dependently and identically distributed as multivariate norwal with
(4.2.1)
E(Y .. ) = (Il + ,,).. + t3 . +
-lJ
J
r lJ
.. )
1 ,
all (i, j),
and
(4.2.2)
where
Var ( Y. . )
-lJ
C is the
c ij = 1
Let
if
-Xi'J
by (3.2.8).
/i-j
=
( I - pC )
-1
2
(j,
(rxr) matrix with all elements zero except for
I=
1.
= H -lJ
Y•. , where
H is the orthogonal
Then
sin frc
r+l '
...
and
defined
~~trix
, ~
f=l
sin rfrc )
r+l
r
L:
f=l
.
2frc
r+l
Sln - - -
65
(4.2.4)
cr
2
,
where
,
1
(4.2.5)
k1(
= 1,
k
2, .•. , r .
2
1-2p cos r+l + p
Now consider the random variables
where
p=r
if
r
is odd and p
expectation of Z!.
_.~J
(4.2.y) E(Zi.) =
Zi'
,
-- J
= r-l
r
if
is even.
The
is
(~ + T. + ~. + y . . )
~J
Jr+l
- J
lJ
(cot ~ ~ , Cot 2 ~
2 r+l
2 r+1
2
• ••
J
Cot
.:2
2
tr
r+1
)
'
and
(4.2.8)
Var(zi')
- J
where
1
------- ,
1-2 P cos (2f-l)~ + p2
r+l
f = 1,
2,
•
(fI
•
,
(r+1)/2
if
r
f = 1,
is odd and
2, •.. , r/2
is
is even.
11
It is required to find (qx1) vectors
(4.2.10)
Var
(li ~ij)
=
Var
(12~ij)
,
and
12
such that
r
66
and
(4.2.11)
Cov (ll'
Z.. , l2' Z.. )
- -~J - -~J
The integer
q is
r/2
if
r
= o.
is even and {r+l}/2 if r
is
odd.
From (1.6.1) it follows that
(4.2.12)
AK*
2
:= - ; -
l-p
J +2
II
n=l
L;
pn
cos
n{2(-1)7f
r+1
If ~ is the (ruxl) vector whose j-th element is
1
pj-l, and W is
the (mxq) matrix whose (n,j) element is the coefficient of
pi-I in
the expansion (4.2.12) of A* _ , then the j-th element in the
2j l
(lxq) vector ~'W is the m-th order approxin~tion of A*2j _ .
l
Suppose that r is odd. Then the (r-l)/2 -th rOvT of W is
(4.2.13) ~~-l
=
r-l 7f
{cos '"""'"2
r+l'
r-l 37f
cos""2 r+l'
... , cos -r-l
2
""2
and
(4.2.14)
w' w
-n - r-l
=
""2
(r+l)/2
l:
l=1
cos (2(-1) E.1£- cos (2(-1) r~l .2!r+l
2 r+l
Making use of a well known trignometric identity, (4.2.14) becomes
(4.2.15)
w' 1T
-n .- r-l
2
=
1 (r+l)/2
-2
L;
(=1
(cos (2(-1)(n + .!::2:) ....2L.
2
r+1
+ cos (2(-1)(n r;1) r:l ) ,
which, upon application of (1.6.5), becomes
sin ( n + 2r-l)
(4.2.16)
1(
4 sin (n + r .. ~) ~
2
r+l
It is clear that the right-hand side of
+
sin ( n - -r-l) :rr
2
r-l
:rr
4 sin (n - ---)
--2
r+l
(4.2.16) is zero vdth the
exception of the cases where
(4.2.17)
n
=:
,( s(r+l)
- 2'
t(r+l)
.....r-l
_,
r-l
)
2
.
e
and for these cases
(4 .2.18)
wI
-n
w
- r-l
=
(r~l) (_l)s
2
Suppose
(4.2.19)
and
fl
and
12 are
taken as the following:
)
j
= 1,
If
q
is odd,
2, ... , (r+3)j4,
68
(4.2.20)
, j
= 1,2, ... ,
(r-l)/4;
, j
= 1,
, j
= 1,2, .•. ,(r+l)/4 .
11
That no ambigtlity can arise in this choice of
2, .•• , (r+l)/4,
and
12
is
seen as follows.
cos !-l (4j-3) ~
r+l
2
but sin(4j-3) -2~
=1
=
sin (4j-3) ~ sin (4j-3) ~ ,
2
r+l
and sin (4j-3) r+.
~l > 0 for
j
= 1, 2, ... ,(r+3)4.
Similarly,
(4.2.24)
cos
(~:i)(4j-l) ~ =
but sin(4j-l) ~
= -1
sin (4j-l)
~
sin (4j-l) r:l
and sin (4j-l)r:l > 0 for j
= 1,
2, •.. ,(r+l)/4.
If ~l and ~2 are the (qxl) vectors whose j-th elements are
2
f lj
1/2-
and X2j' then
( 4.2.25)
(f- l'
Var
Z•. ) -
-~J
Var(
f ' z.. )
-2
-~J
H01orever,
,
(4.2.26)
and, therefore, it follows from (4.2.16) and (4.2.18) that
+ L:
8=0
r-l
s ( r+l )+ 2
(_l)s .Y_
r+l
,
or that
±im .9:?'W(~l - ~2):::
(4.2.28)
m->oo
r-l
2
(j
2
I-p
2
p """T
1 + P
r+l
r+l
1 - p
Therefore,
(4.2.29 )-lim (var(fi ~ij) - var({?, ~ij»
m-->oo
Since, for
0
•
q odd,
(r+;)/4
(4.2.;0)
;::
Var({l'
-
Z .. )
-~J
r:
j:::l
sin (4j-;) .!l
r+
1-2 P cos( 4j-;)
(j
1C
r+ l
+ p2
2
70
and
(4.2.31)
Var(f2' Z.. )
:=
- . -~J
and, for
q
(4.2.32)
var(x1 Z~.
4
----2
(r+1)
sin(4j-l) .-2!....
r+l
(r-1)/4
L:
j=l
(J
2
1-2 P cos(4j-l) 1(1 + p2
r+
even,
II'
)
:=
-~J
-
!~
-----2
(r+l)
(r+ 1) /4 sin(4j-3) r+1(1
L:
j=l
(J
2
it fo110vTS that Yare f l' Z,.) and Yare f 2' Zi') both tend to zero as
- -~J
- - J
r increases. However, it is clear from (4.2.28) that the difference
between the variances tends to zero at a faster rate; 1. e. ,
(4.2.34 )
Yare U , z.. ) - Yare f ' z,.)
Al -1J
- 2 -1J
---
lim
->
r
k
=1
and k
= 2.
(4.2.21) and (4.2.22)
(4.2.35)
cov
(4.2,36)
*
r
E(~ij) =
t
-:LJ
Finally, it follows from (4,2.19), (4.2.20),
that
(Ii ~ij' 12~ij)
Suppose that
o
Yare f.k' Z •. )
00
-
for
,
is even.
:=
0,
~~
Let --:LJ
Z .. = D(
(~ + T i + ~j +
r+l
2""
Cot 2.-2!....
0
s1'n 31(
r+l
2 r+l' , .•• ,
Yij )
k1(
sin -1)X:
r+ -1 j
,
Then
' - :rr co t -1 1( , 0 ,
( S1n
r+l
2 r+l
' (r-l):rr C t (r-l):rr 0'),
S1n r+l
0 2(r+l) ,
,
71
and
(4.2.3y) var(z*i')
- J
D(~
sin kr+nl ) a 2
'J{
==
From (1. 6.2) it folloVlS thet
(4.2.38) "k
~ sin krt
r+1
n-1
==
sin
p
Let ro be the (mxl) vector whose j-th element is
,;;.,;,..
the (mxr) matrix whose
in
. 1
pJ- ) and W be
(n,j) element is tlE coefficient of
pi-l
(4.2.38). Then~' Wa 2 is the (lxr) vector whose k-th element
is the m-th order approximation of' the k-th element in the diagonal
matrix on the right-hand side of
(4.2.37).
W is
.
e
r
w' w
-r -n
==
sin
1.::
k==l
rkn
sin r+l
'
which is the same as
1
(4.2.40) ~~
w· w
== -
(4.2.41)
1
==-2
yT' W
-r -n
r
1.::
2~
(.
(cos (n-r)
.
s~n
1 n-r
-2 -r+l n
n+r
r+l n
. 1 n+r
s~n -2 · - n
r+l
1
r+ l - cos (n+r) cnl ) ;
N
sin ~ -~:-~ n
_._-,-------
r
- sin "2
e
kn
cos (n-r)
'"2 n
cos (~)
2
~)
.
72
Finally, it turns out that
W'
=0
W
-r -n
vdth the exception of the
cases
(4.2.42)
n
,
s == 0, 1, 2, ... ,
,
S
=
1, 2, 3, ... ,
:=
and, in these cases,
n = r, 3r+2, 5r+4, .•. ,
(4.2.43)
W' W
-r -n
n :: r+2, 3r+4, 5r+6, .•.•
.
e
Let
K*1 ,2j-1
=2
(4.2.44)
t1,2j
::
K*2,2j-1
:=
sin (2j~1) .f~!
r+l
2
r
,
j
2,
... , "2r
,
j = 1, 2,
... , "2r
1;
since
= 1,
0
and
0
(4.2.45)
.
K*2 ,2j
=
2
2jr
-s~n r+l 1C2
r
There can be no ambiguity in this choice of
e
(4.2.46)
sin(2j-1) F1I.
r+ 1
:=
K*
-1
and
- sin(2j-l) -Er+1 cos (2j-1)"
,
,
73
cos(2j-l)~ = -1 and sin (2j-l)
which is positive since
for
j =1, 2, ... , r/2.
(4.2.47)
Similarly,
2'
, 2jrrc
sJ.n -r+l
- =-
sin ~
r+l
cos 2jrc
and the right-hand side is negative since
sin
;i~ > 0 for
j
r+~l > 0
= 1,
,
cos
2j~
= 1 and
2, "., r/2.
If ~l and ~2 are the (rxl) vectors whose j-th elements
are
•
,;*2
Klj
and
u*2
X2j'
then
Making use of (4.2.41) and (4,2.43), it folJ.Oi'lS that
2 ( Z p2s(r+l)+r
r
8=0
" p2s(r+l)-r) ,
s=l
-
~
or that
2
~im rot W(§l - ~2) =-r
(4,2.50)
m _.>
00
cr
2
i'" -I-
p
2r
p2r+2
Since
(4.2.51)
*' *
var(i1 ~ij)
4
=~
r
r/2
Z
k=l
sin (21t-l) rrc
r+ l
1-2p
cos(2k-I) _,!- + p2
r+l
cr
2
,
and
4
= 2
sin
r/2
l:
k=l
r
()
~
it follows that Var(! ~ij ) and Var(£.
r
~f'
*
~ij) and both tend to
2
However, it is clear from (4.2.50) that
J.
zero as
increases.
the difference between the
vari~nces
tends to zero at a faster rate;
i.e. ,
*',~~) - Var(£.2*' ~ij
*)
Var(ll
(4.2.53 )-lim
r->
-k
for k
=1
=
* )
Var(K*' Z..
00
and k
=2
0
-J.J
•
Let
1
U1ij
=
U2ij
= -2
"2
(4.2.54)
1
?~ f
*' Z..
* ) +
(%.1
(£2 ~ij»
-J.J
if
,
*' .'{-
.
~' ~ij)
* - (f2 ~~)
«-1
Then, since
(4.2.55)
where
~f
E(f~
-
oX-
Zoo) = C(l-J. + 1'i + f3 j +
-J.J
1'1) ,
Vr~l C is the product of the vector i~ and the vector
on the right-hand side of (4.2.36), and
(4.2.56)
,
2
1-2p cos(2k-l) ---1
+ P
r+
*
?:.,
2
75
it follo'VlS that
E(U 2J.J
· .)
= Q
(p.
2
+
T.
J.
+ (3. + y . . )
J
J.J
and that
Finally,
(4.2·59)
Cov(U1 .. , U2i .)
J.J
J
= ~if
(var(l*lt z.. )
-.
-J.J
-
var(f*2
-
z. .»
-J.J
.
The U
and U
are a set of l~ndom variables subject to
21j
lij
a two-way classification with two random variables per cell and
constant error variance.
If cov(U .. , U .. ) is negligible, then
l J.J
2J.J
the regular analysis of variance, which takes account of interaction
between blocks and treatments, may be performed.
For the case 'tV'hen
(4.2.60)
E(~l'
-
z.. ) =
-J.J
r
is odd,
C1 (ll + T. + (3. + y •. ) ,
.
J.
J
J.J
and
( 4.2.61)
E(f2T Z.. )
-
-J.J
=
C (ll +
f)
'-
T.
J.
+ (3. + y . • ) ,
J
J.J
where
C and C are constants detennined by forming the scalar
1
2
products of the vectors
and
with the vector on the right-hand
1
2
1
1
(1 Z.. and -~2 -J.J
Z.. are a set of independent
- -J.J
random variables subject to a two-'t~y classification with two random
side of (4.2. '1).
Thus
76
variables per cell.
var(K2'
-
z.. )
-~J
If the difference between Var(K' Z.. ) and
-1
-~J
is negligible, then a regular analysis of variance,
which takes account of the interaction between blocks and treatments, may be perfor.rned.
77
BIBLIOGRAPHY
~1_7
Ahlfors, Lars V., Complex Analysis, McGraw-Hill, New York,
195.3.
~2_7
Anderson, R. L., "Distribution of the Serial Correlation
Coefficient," Annals of Mathematical Statistics, Vol. 1.3
(1942), pp. 1-1.3.
[".3_7
Anderson, R. L. and Houseman, E. E., "Tables of Orthogonal
Polynomial Values Extended to N '" 104," Im-ra State College,
Agricultural Experimental Station Research Bulletin 297, 1942.
~~7
Anderson, R. L. and Bancroft, T. A., Statistical Theory in
Research,
15_7
Pir~~ Ed~~,
McGraw-Hill, New York, 1952.
Anderson, T. W., "On the Theory of Testing Serial Correlation,"
Ska~d.i~:i:..sk<Ak~:rietidskriftl. Vol. .31 (194/1), pp. 1111-116.
16..}
Brand, louis, Advanced Calculus, John i'Jiley and Sons, New York,
1955·
["7_7
Courant, R., Differential and Integral Calculus, Vol. 1,
First
~e_7
Editi~~,
Blackie and Sons, London, 19.37.
Cramer, Harold, Mathemati~al Methods of Statistics, Princeton
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19_7
Dixon, W. J., "Further Contributions to the Problem of Serial
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78
Fisher, R. A. and Yates, Frank, Statistical Tables for
..
Biological, Agricultural, and Medical Research, Third Edition,
Oliver and Boyd, London,
19~9.
Hannan, E. J., "Exact Tests for Serial Correlation,"
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£12_7 Koopmans, Tjalling, "Serial Correlation and Quadratic Forms
in Normal Variables," Annals of Mathematical Statistics,
Vol. 13 (1942), 1'1'. 14-23.
£13_7 Ogai'Ta, Masami, "A Note on the Test of Serial Correlation
Coefficients," Annals of Mathematical Statistics, Vol. 22
(1951),1'1'. 115-118.
Ll4.-7
Robbins, H. and Pitman, E. J. G., "Applications of the
Method of :Mixtures to Quadratic Forms in Normal Variates,"
Annals of Mathematical Statistics, Vol. 20 (1949), 1'1'. 552-560.
£15_7 Siddiqui, M. M., Distributions of Some Serial Correlation
Coefficients, Unpublished doctoral dissertation, University
of North Carolina, Department of Statistics, Chapel Hill,
North Carolina, 1957;
or Institute of Statistics Mimeograph
Series No. Jh4, University of North Carolina, Chapel Hill,
North Carolina, 1957.
L16_7
Siddiqui, M. M., "Distribution of a Serial Correlation
Coefficient near the Ends of the Range," Annals of Mathematical
Etatistics, Vol. 29 (1958), 1'1'. 852-861.
£17_7 Titchmarsh, E. C., The Theory of Functions, Second Edition,
Oxford University Press, London, 1939.