,
ON SOlVJE DIS1RIBUTION PROBLEMS IN MULTIVARIATE ANALYSIS
by
K. C, Sreedharan Pillai
Special report to the Ford Foundation of
work at Chapel Hill under Ford Foundation
grant to the Institute of Statistics,
Special Fund Code Number 324-For 1(10)
for research in the development of multivariate analysis of variance techniques
into a form useful for the social (behaviorial) sciences.
r
Institute of Statistics
Mimeograph Series NOl88
January, 1954
li
ACKNOWLEDGMENT
I wish to place on record my deep feeling of
gratitude to Professor S. N. Roy for his help and
guidance during the preparation of this work.
I
am also indebted to Professor Harold Hotolling for
his active interest and valuable suggestions all
through the progress of this piece of study.
Acknowledgment is also made to the Institute
of Statistics and the Ford Foundation for all
£acilities including financial assistance which have
made this work possible.
,I
iii
TABLE OF CONTENTS
PA,JE
CHAPTER
ACKNO!'JLEDGEl'IEi'ifT
•
INTRODUCTION
I
•
•
•
•
ii
vi
l\lATHEJ.IIATICAL PR:3L]"lINARIES
1
1.1 Introduction
1
1.2 Vandermondels Determinant
•
1
1.3 A Special Function and the CorresDonding
Determinant
2
1.4 Integral of f(x l ,x 2 , ••• , x k ) over the Domain
o ~ Xl :5 x 2
3
1.5 Certain Properties of I Functions
11
1.6 Integration of f(x l ,x2 ,x ) .
3
1.7 Integration of f(x l ,x 2, ••• ,xk )
19
1.8
II
~ • • ~ ~ ~ x ~ 1
THE
•
Integral of g(Yl'Y2"."Yk)' a Second Type of
Function' , Over the Domain O~Y15".~Y~~o
LARG~~ST
~1ULTI\n.RIATE
2.1
••
CHARACTERISTIC ROOT OF A l-iATRIX IN
ANALYSIS
• • •
Introduction
•
•
2.2 Distribution of the Roots
•
•
•
•
34
•
42
•
42
.' .
43
•
• •
Cumulative Distribution Function of the Largest
•
Root ; •
•
III
APPRoxn~TION
LARGSST ROOT
TO THE PROBABILITY INTEGRAL OF THE
• •
3.1 Introduction
•
25
45
61
61
iv
CHAPTER
PAGE
3.3
IV
Approximate Expressions for Obtaining
Upper Percentage Points (5 '10 or less)
of the Largest Root.
••
• •
61
Percentage Points for the Largest Root
66
3.4 The Distributions of the Largest and Smallest
Roots for 8=2 as Hypergeometric Series
73
OF THE-SUM OF THE CHARACTERISTIC ROOTS
OF A MATRIX
• •
• • •
•
76
M~1ENTS
4.1 Introduction
4.2
v(s~ in general
4.3 The Homents of V(2)
79
•
81
•
4.4 The Moments of V(3)
85
4.5 The Moments of V(4)
90
J_~. 6 The l10ments of V( s)
V
76
•
•
The Moments of
•
•
95
ASyf:tIPTOTIC AND APPROXTI1ATE DISTRIBUTIONS OF V( s)
97
5.1
97
Introduction
••
•
5.2 The Asymptotic Distribution of V(s) as a Gamma
Function
•
97
5.3
99
An
Approximation to the Distribution of V(s) •
5.4 Comparison of the ~IS from the Exact and
Approximate Distributions
•
VI
104
5.5 Upper Bound to the Error of Approximation
•
108
THE MOMENTS OF THE DISTRIBUTION OF U( 8)
•
116
6.1
Introduction
6.2
The Distribution of
116
•
AIS
•
116
v
PAG~Z
~~H.APTER
6.3
The fJoments of U(s) , in general
6.4 The Homents of U(2)
6.5 The Homents of u(3)
•
6.6 The Moments ot U(4)
•
6.7
VII
•
AN
•
•
•
•
The Moments of u(s)
lPPROX~\TION
7.1 Introduction
•
•
118
120
•
121
•
•
• •
•
7.2 Approximation to the distribution of U( s)
7.3
•
119
TO THE DISTRIBUTION OF u(s)
•
117
•
•
•
123
123
•
•
123
Comparison of the Approximate and Exact Cumulative Distribution Functions for 8=2
• •
127
7.4 General Behavior of V(s) and U(s)
•
BIBLIOGRAPHY
•
•
133
•
•
135
INTRODUCTION
This dissertation presents sorre results on distribution
lems in multivariate analysis.
p~ob-
IVT.ost of the important distribution
problems in multivariate analysis can be reduced to those connected
with the characteristic roots of a matrix whose elements are functions
of the observations, possibly also involving some constants specified
by the hypothesis.
Tests of hypothesis of the equality of dispersion
matrices of two multivariate normal populations or of the equality of
mean vectors of
1 multivariate
norrr.al populations or of the inde-
pendence of t1'10 sets of variates having a joint multivariate normal
distribution, turn out to be expressible in terms of statistics which
are functions of the characteristic roots of certain matrices (different for the different hypotheses) obtained from the samples.
Tests
based on the largest and the smallest arr..ong these roots have been
- -7 in various situations; H. Hotelling
suggested by S. N. Roy /-20
L-r,
8, 9_7;has proposed the
SUIT.
of the roots as the statistic for
tests of certair. types of hypotheses while S. S. l'!ilks
L-21_7
advocates, in some cases, tests based on statistics which are found
to be the product of the characteristic roots of certain matrices.
In this dissertation, an attempt has been made to study the distribution problems posed by the tests of Roy and Hotelling and by
another test similar to Hotelling' s.
In the course of this study
convenient approximations to some already known distributions and
the lower order moments of an.d suitable ap:;:.roximations to
vii
certain unknown distributions have been obtained, the aim everywhere
being to make the above tests feasible for the experimenter.
In
some cases an idea of the upper bound to the error in the cumulative
distribution functions, due to these approximations, has also been
sought to be obtained.
In multivariate analysis (especially for the
situations rr.entioned above) where exact tests are necessarily com-plicated, no practical and safe procedure can be laid down without
such approximative attempts (which also keep in view the error com-.
mitted).
So far as the above tests are concerned, the present study
is believed to have made, among other things, an attempt in this
direction, leading to certain re sults which are expected to be useful in this area.
Chapter I is purely mathematical and serves as a foundation
for the statistical developments of the later chapters.
Certain im-
portant recurrence relations are obtained in this chapter which are
repeatedly used for handling the distribution problems discussed in
the sequel.
The recurrence relations obtained are of two types in the
main, of which the second is a generalization of the one given by
S. N. Roy L~O-T.
To get th6se relations, use is made of certain pro-
perties of the Vandermonds1s determinant.
In Chapter II, using one of the recurrence relations given in
the first chapter, a general method of obtaining the exact cumulative
distribution function of the largest (or smallest) root is offered
viii
which is a considerable simplification of a sliri1ar general method
introduced by S. N. Roy ~2r_7.
So far as the explicit expressions
for the cumulative distribution function are concerned, S.N. Roy
gives these for the number of r00ts going up to four and D. N.
Nanda L-12_7,up to five, while this chapter presents such expressions
for the number going up to eight.
Chapter III gives certain approximations which are useful for
computing the upper percentage points of tho largest root for small
values of one parameter and for the number of roots up to five.
These
expressions are very simple compared to those for the exact cumulative
distribution functions and such approximations alone could make the
computation feasible for large values of the number of roots.
illustration, upper percentage points
for the case of two roots.
(5
%
For
and 1 0/0) have been givon
Also, for this case, the distribution of
the largest and the smallest roots are given as hyporgeometric serios.
In Chapter IV, the lower ordor momonts of the sum of the roots
(denoted by v(s), all roots lying between zero and unity in this case)
have been obtained by using a result from Chaptor I.
It has been
possible to give tho first three moments for any number of roots but
the fourth moment has been obtained only for the number of roots up
to four.
Chapkr V gives (i) the asymptotic distribution of V(s) (for
ix
large values of one parameter connectod with same sample size)by
using moment generating functions, the derivation being much simplor
than the one developed by T. W. Anderson L-l_7 for a somewhat
similar asymptotic distribution problem; (ii) an approximation to
the probability density function of V(s) in the form of a beta
distribution which appears to be good for considerably smaller
values of the IIparameter ll than in case (i).
Chapter VI presents for the moments of the sum of the roots
(denoted by u(s), all roots lying between zero and infinity, which
is a constant times Hotelling's T;) a treatment which is parallel
to that of Chapter IV for v(s).
These moments have been obtained
by using a third rocurrence relation proved in Chapter I.
They are
also observed to be derivable from those of v(s) through a simple
relation.
In chapter VII, an apnroximation to
~he
distribution of U(s)
has been suggested which is of the form of an F distribution and
which, in the case of a single root, reduces to Hotelling1s T L-5_i.
For the case of two roots, the accuracy of the approximation haa
been discussed by comparison with the exact cumulative distribution
function obtained by Hotelling
L-9_7;
and, as in the case of V(S)
soaLso in the case of U(S), the approximation seems to be satisfactory, for most purposes.
CHAPTER I
MATHEMATICAL PRELIMINARIES
1.1
Introduction
In this chapter we shall obtain a few results which are
of a purely mathematical nature.
These results are required for the
distribution problems to be discussed in the next six chapters.
1.2
Vandermondels Determinant
It is well known that the function. )( (X. - x.) in k
i>j].
J
variables Xl' ••• , Xk can be thrown into the form of a determinant
(due to Vandermonde) as given below: -
(1.2.1)
k-l
Xk
k-2
Xk
• • • • • Xk
1
Xk-l
_
k l
Xk-2
_
k l •
1
•
Va
Q
k-l
X2
k-l
Xl
· · · • Xk _l
• • • · ·
• • · • · ·
k-2
X2
· • · · · X2
k-2
Xl
· • · · • Xl
•
•
1
1
The determinant Va in (1.2.1) has certain interesting
properties.
If we raise the index of the elements of the first
2
column of Vo by unity, the resulting determinant, VI (say), represents another function given by
(1.2.2)
k
It may be noted that
Z X. is the first elementary symmetric func-
. 1 J.
J.=
tion in the k variables Xl •.• X • Again, if the indices of the
k
elements of the first two columns of V are raised by unity we get
o
which shows that V2 is the product of the second elementary s~~etric
function and Vo. In general, if the indices of the elements of the
first j columns of V are raised by unity, the resulting determinant,
o
V , is the product of the j th elementary symmetric function and V '
O
j
1.3.
A special function and the corresponding determinant.
Let us consider a special function of the form
= '"kIT
i=l
( x~(l-x.)re tx.)
J.
1.
1.
'"IT
i>j
(x. - x.)
1.
J
,
3
where q, r > -1 and t is independent of the XIS.
It is clear
that-f(xl , x 2 ' ••• ,~) can be put in the form
II q+k-l(l
I xk
t
-~
)r ~
q+k-2(1
)r tXk
xk
-xk e
e
•• •
I
I
)r
_l
I x kq+k-l(l_
~-l e
tx
1<:-1
i
!
q+k-2(1
)r tXk_l
x k _l
-~-l
e
••
q (
)r t~_l\
x k _l 1-\_1 e
.
I
i
•
•
•
• • •
•
•
·..
•
• ••
xq+k-2(1
-x 2 )re
2
q+k-2(1
xl
-xl
)r
tx
tx
e
2
1
•
• • •
...
••• Xk ) has to be integrated over the domain:
o ~xl
~ ~ ~
•••
~ ~ ~
x, then the integral in question can be
!
I•
i
4
formally or sjrmbolically written in the form
x
(1.4.1)
f dx~ f
o
0
x
f
~-1'"
2
o
f(x1 , x 2 , •••J"k)dx 1
t
q+k-l(l_
xk
o
)r xk~
xk e
---k'
•
.. .
•
•
...
q+k-l(l_ )r
xl
xle
tx
Idx
1
=
..
•
X2
f
r
X
xk
•••
o
It has to be remembered that in expanding the determinant on the
right hand side of (1.4.1), one has to take care of the order of integration and hence we may call it a pseudo determinant.
For con-
venience the pseudo determinant in (1.4.1) may be denoted by
(1.4.2)
and when the r's are different, by
,
5
(1.4.3)
or more explicitly by
( 1.4.4)
U
qk' r k
• • • • ql' r l
qk' r k
• •
x;
•
· • ql'
·
r
l
,•
•
•
• • • •
qk' r k
• •
· · ql'
t
•
•
rl
Since the integral of f(x l , x 2 ' •••,xk ) involve the integrals
of the type
(1.4.5)
I(x'l q, r; F; t ) ·
i
Xl
q
x ( l_x)rF(x)etxdx
,
o
where F(x) is any function of x such that the integral in
(1.4.5)
exists, it will be proper to consider first the integral in
If F(x) is of the form
(1.4.5) •
6
the integral in (1.4.5) may be denoted by
(1.4.6)
Now consider I(x; q, r; F; t).
Integrating (1.4.5) by parts
we get
IO(X'; q,r+l;F;t?
I(x'; q,r; F; t) = - - - - - - - _.... + QI(x'; q-l,r;F;t)
q+r+l
q+r+l
+
tI(x;q,r+l;F;t)
q+r+l
+
x
where
I
)
1 ( x;q,r+l;F;t
= x q( I-x )r+1F(x)e tx
0
,
I
)
I(x;q,r+l;F;t
q+r+l
,
,
and F' = dF(x)
o
dx
It may be noted that the right hand side of (1.4.7) has been obtained
by integrating (l_x)r+ q and differentiating xq
,F(x) and e tx •
(1~)q
Using (1.4.7) let us consider the integration of the function in
(1.4.1) when k = 2. In this case (1.4.1) reduces to
7
(1.4.8)
r~
o
•
Using (1.4.7) we get
(1.4.10)
8
which, using the notations given above, can be written as
-
Ia(x;~,r+l;ql,r;t) + ~
( I(x;q2-1,r;ql,r;t) - I(x;Ql,r;Q2-1,r;t]
+ t \ I(x;Q2,r+l;Ql,r;t) - I(x;Ql'r;Q2,r+l;t)
J
9
The integration in (1.4.8) has been carried out with a view to 1owering the highest index by unity.
Hence
; t
;t
since
+ t
J
10
i.e.
(1.4.14)
=
(1. b.• 15) i. e. ""'
11
(1.4.16) i.e.
Notice that the reduction above does not depend on the relation q. =
1.
qi-l
Under this substitution the third term on the right hand
+ 1.
side of (1.4.12) will vanish.
,x
For the integration of f(x 1 ,x 2 , •• k ) in seneral, some further
results have to bc~ o'1tained, for which we do not h3ve to assune that
1.5.... Certain properties of I functions
~
Consider the function
(1.5.1)
=
•
12
•
Hence from (1.5.1) and (1.5.2) we get
Similarly it can be shown in general that
k
=
j( I(x;q.,r;t)
i=l
~
,
where (q~~ ••• ,ql) stands for any permutation of (qk' ••• , q1) and the
summation Z extends over all such possible permutations.
Again, consider the function
13
(1.5.5)
14
(1.5.6)
Now
=
The expression on the right-hand side of (1.5.6) combined with
3
I(X;Q2,r jQ ,r;Q1,r;t) gives
Again the right hand side of (1,5.7) combined with .
(1.5.9)
Hence
(1.5.5)
can be !~itten in the form
i
(1.5.10)
r(x;q.3,r;t)
u
q2,r
,. t
Xl\
Q2,r
NON
qpr
cunsider an expression of the form
ql'r
16
q" r
3'
(1. 5.11)
U
II
q2,r qn r
x; q3,r q2,r
Q3,r Q2,r Q1,r
+ U
x;
q3'r
q2,r
Q1,r
Q3'r
Q2,r
Q1,r
Qrr r
.3'
Q"2' r
qJ>r
q3'r
q2,r
q1,r
q3'r
II
q"2' r
qll r ;t
l'
Q3'r
Q2,r Q1,r
jt
•
By using the result (1.5.10) and :i milar results it can be shown that
(1.5.11) can be reduced to the form
•
17
'rhe result (1.;/~ .12) holds even when r is changed to r " and t to t" in
the rows where q is changed to q".
The meaning of the symbol "double
prime lt is evident from (1,5.11) as denoting a change in the index of
x.~ in a particular row of the pseudo determinant.
In general
(1.5.13 )
u
x;
q"k' r" , til
q"k-l' r" , t"
qk,r, t
qk_l,r,t
• •
, qk,r,t
•
•
•
...
....
•
~l,r,t
• • •
• • •
••
·.. ..
qk,r,t
q"l' r" , til
q1,r,t
• • •
• •
••
•
• • •
q ll til t"
• • •
• •
• • •
·.
1"
• • •
• U{
t_
\
J
- u
+ '. • • • •
,
..
18
qk~r,t
+(_l)k-l
•
U
•
qk_l,r,t
•
• •
• •
•
• •
•
·•·
• · •
q"k' r" , til q " k_l,r IJ , til
::: r(XjClk,rn;t ll ) u
- I(x;q~_l'rll;tll)
X;
U
X;
• •
• •
•
• •
II
II til
ql,r
,
• • •
ql,r
• • • •
• •
~_2,r
• • •
ql,r
qk,r
qk_2,r
• • •
ql,r
•
•
•
• • •
• • •
•
• • •
ql'!'
qk,r
• • • • •
•
qk_l,r
•
+
•
•
• •
qk_2,r
qk_l,r
ql,r,t
• • •
qk_2,r
;t
; t
•
.....
+ (_l)k-l I(x;ql,rll;tll) u
x·,
qk,r
• • • • q2,r
• •
• •
·• .•
qk,r
• •
·
~
• q2,r
,. t
•
19
It may be noted that in order to distinguish t
/I
from t, t
IS
have been
taken inside the inner paranthesis.
Let us USe integrat.ion by parts to reduce
(1.6.1)
; t
The integral (1.6.1) we shall write in the form
q3'r
(1.6.2)
u'
q2 j "r ql,r
Xj
+ U
Xj
;t
;t
•
It is to be understood that there are no elements in the pseudo de-
•
,
.
20
terminants where the spaces are left blank.
Using (1.4.7)
co
-Io(x;q3,r+1;t)
q2,r
q1,r
q2,r
ql,r
,. t
X;
U
"
Q3- 1 ,r
+ Q3 U
X;
Q2,r
Ql,r
q2,r
Q1,r
,. t
Q3,r+l
+tu
X;
Q2,r
Q1,r
Q2,r
ql'r
,•
J
j
21
x;
+ U
•
It may be noted that the integration in (1.p.3) has been carried out
with respect to x and in the last pseudo determinant on the right
3
hand side of (1.6.3) t has been introduced inside the inner paranthesis
to show that t is no longer the same in all cases.
Again integrating with respect to x we get
2
(1.6.4)
; t
=
U
x;
; t
22
+ tUx;
- U
; t
X;
•
Similarly integrating with respect to xl we get
I
(1.6.5)
q2,r
ql,r
q2,r
ql,r
J
ql'r, t
\
\
+ q3
q3- l ,r
J
J
qyr
=- U
;
q2,r
ql,r
q2,r
ql'r
;
23
+ t U
; t
XJ
•
Now combining (1.6.3), (1.6.4) and (1.6.5) we ~t
(
(1.6.6)
q3 +r+l)
U
x,·
q
3'
r
q
2'
r
q
l'
r
; t
; t
+
2 U
- 2 U
x;
x;
24
+
Q3 U
XJ
q3-1 ,r
q2,r
Ql,r
Q3;'1,r
Q2,r
Ql,r
Q3- 1 ,r
Q2,r
Ql'r
+ tUx;
,•
t]
; t
•
Using (1.5.13) and a result similar to (1.4.16) the right hand side
of (1.6.6) can be written as
+t U
q3,r+1
~,r
ql,r
X; q3,r+1
~,r
ql,r
q3,r+1
q2,r
q1,r
,. t
•
Notice, as before, that the above reduction does not depend on the relation, qi=~_1+1. If we use qk=qk_l+1, the fourth term of (1.6.7)
will vanish.
With the methods outlined in the previous sections we can now
proceed to integrate f(xl,~ •• ,xk)'
U
x;
The pseudo determinant
qk,r
qk~
l,r • • •
ql,r
~k~r
qk_l,r • , •
ql,r
~
• •
• •
qk,r
,
,
,•
,
•
• •
• •
qk_l,r • • •
•
~
•·
; t
•
ql,r
can be split up into the following pseudo determinants.
qk,r
(1.7.1) U x;
qk_l"r •• ql,r
qk_l,r • • ql,r
•• •
• •
qk_l,r • • ql'r
;t +U x;
qk,r
qk_l,r •• ql,r
;t
qk_l,r • • ql,r
,.,'
26
qk_l,r • • • q1,r
-+ • • • • -+
U
r
qk_1,r • • • q1,r
x.
r
;
qk_1,r • • • q1,r
J
J
qk,r
Using (1.4.7) each of the pseudo determinants in (1.7.1) gives rise to
four pseudo determinants as in the case of k = 3 given in (1. 6. 3) ,
(1.6.4) and (1.6.5) with an obvious limitation in the case of the last
pseudo determinant, which gives rise to only three.
Denoting the sum
of the pseudo determinants in (1. 7.1) which correspond to the first
term on the right hand side of (1.4.7) by u(l), the sum which correspond to the second term by U(2)
and so on, we get
,
(1. 7.2)
where
•
•
••
•
; t
27
+ U
x,;
qk_1+ qk,2r+1,2t
Qk_2+ qk,2r+1,2t • • • • q1+qk' 2r+1,
Qk_1r ,t
Qk_2,r,t
•
Qk_1,r,t
· · . Q1,r,t
• •
•
Qk_2,r,t
2il
·•
1
•
• • • • Q1,r,t
.. .... . ..
x;
+ U
•
•
•
•
= - I O(x;Qk,r+1;t)
U
1
~
(-1
i=k-1
f'-J.-1.I(x';Qi+Qk,2r+1;2t)U
(by using (1.5.13) ) ,
~_l'r
•
x;
\
+
•
•
(
(1.7.4)
• • • •
•
Xj
•
•
qk~l':
•
•
•
•
• q1,r
•
• •
•
•
•
;t
•
• Q1,r
•
•
jt
28
· . • • ~l~r
• • · • • · . · • • •
..• • • · • • •
qk,r+l
· • · • ql,r
~lr+l
(1.7.6)
U(3)
=:
U
x;
• • •
;t
and
. qk_l+ qk,2r+l,2t
(1.7.7)
•
~_l,r,t
•
•
• • • • ql,r,t
qk_l,r,t
u(4) = U
•
• • • • ql+qk,2r+l,2t
•
•
·•
• • •
•
•
ql,r,t
•
29
qk_l,r~t
•
+ U
x;
•
•
•
•
•
•
•
•
ql,r,t
•
•
•
•
•
qk-l+qk' 2r+l, 2t
•
• ql+qk,2r+l,2t
•
In the last term here as well as in (1.7.3) the sign i.s positive or
negative as k is odd or even.
Now again using (1.5.13),
1E (-1) k-i-l I (xi q. +qk'2r+l;2t ) U
i=k-1
~
•
•
Xj
•
•
•
•
•••
• ••
•
•
•
•
Hence
•
qk_l,r
= -IO(x;qk,r+l;t) U
x;
•
• • •
• •
qk_1,r •
·
·•
•
;t
ql.,r
•
ql,r
;t
•
•
•
it
30
• qk_l,r • qi+l,r qi_l,r
+2
1Z (-1) k-i-l I ( x;qi+ qk,2r+l;2t) U
i=k-l
ql,r
•
•
x;
;t
qk_l,r • qi+l,r qi_l'r • ql,r
qk-1,r
+ qk' U
+ t
U
X;
X;
qk_l,r
•
•
ql':
•
•
•
qk-1,r
qk_l,r
ql,r
qk,r+l
qk_l?r
ql,r
•
•
qk,r+l
•
qk_l,r
•
•
; t
;
t
•
ql,r
Using a result similar to (1.4.16) the last pseudo determinant
on the right hand side of (1.7.9) can be written as the difference of
t.wo pseudo determinants as given below:
31
u
qk,r+l
qk_l,r
• • •
ql'~
• •
• •
• • •
•
• •
• •
• •
·
•
• • •
ql,r
x;
qk,r+l
qk_l,r
qk:l~r qk_l,r
(1.7.10)
= -u
X;
• •
•
.
qk+l,r qk_l,r
+ u
xl
\
qk,r
qk_l,r
• •
.•
• •
• •
qk,r
qk_1,r
•
•
··
• ·
ql~r
•
• • •
ql,r
• • •
ql,r
· .•
• · · • •
• · · ql,r
,. t
,. t
• •
; t
•
As before, notice that the abOve reduction does not depend on the relation q.~q. 1+1.
~
~-
Two special cases of (1.7.9) are worth noticing and
will lfe used in the succeeding chapters.
Case I: t = 0, qk = qk-l + 1.
32
(L7.11)
(qk+r +1 ) U
X;
I
qk,r • •
I
qk,r •
.
q1,r
°
•
j
\
Q1,r
..
o
I Qk_1,r • qi+l,r q.~-1,r •
+2
,
Case II; x = 1, qk
~
qk-1 + 1
I
f
(1. 7.12)
(qk+ r +1)
U
1;
qk,r • • • q1,r
..
•
qk,r •
•
ql,r
qk_l'r •
~
Q1'l1
1Z (-1 )1<:-i-1
.
I(x;q.+qk,2r+1;0 ) U x;1 ••
• • ,0
• •
• •
i=k':l
~ .
\I qk_1,r • qi+l,r q.~- 1,r • qpr
1
Z (-1 )k-i-l I ( 1,;oqi+~,,2r+l;2t ) U 1;
i=k-l
•
.
.•
~+1,r
• •
•
.
q. 1,r
~-
• •
qk_l,r • Qi+1,r q.:1..-1,r
. q1,r
•
•
. qpr
jt
33
+ tu
1;
qk,r+l
qk_l,r
• •
• •
qk,r+l
•
.•
ql':
,. t
• •
qk_l,r
• • •
•
ql,r
The result (1.7.11) will be used in the next two chGpters and (1.7.12)
in chapters iV and V.
(1. 7.13)
u
It may be further noted that when qk = qk_l+l
qk-l,r
qk_l,r
• • •
~l~r
• •
• •
• • •
•
.
•
• •
• • •
,
•
qk_l,r
• • •
ql,r
•
qk-l,r
t
=:
0
because the first two columns of the pseudo determinant will be the
same and this fact has been used to get the relations (1.7.11) and
(1. 7.12).
The method given above provides us with a recursion formula for
reducing a kth order pseudo determinant in terms of k-l and k-2 order
pseudo determinants,
This recursion relation will enable us to handle
the distribution problems to be discussed in the next four chapters.
34
1.8
Integral of g(Y1 'Y2' .,., Yk ), a second type of function, over the
domain 0 < y. <. •
-
1-
I
< Y. < Y <
- k-
00
Consider the function
k
1t
i=l
a
Yi e
-ty .
~
The integral of (1.8.1) can be written as in (1.4.1) in the
pseudo determinantal form
(1.8.2)
•
·e
••
•
••
•
••
••
••
•
•
•
•
•
•
35
Now if we denote by F(y;a,b;G;-t) the integral
I
f
(1.8.3)
Y
o
-ty
y%(y)e
(l+y)b
dy
T~here G(y) is any function of y such that (1.8.3) exists, then integration by parts gives
(b~a-1)F(y;a,b;G;-t)= -F (y;a,b-1;G;-t) + aF(y;a-1,b;G;-t)
(1.8.4)
O
-tF(y;a,b-1;G;-t) + F(y;a,b-1;Gl;-t)
where F (y';a,b-1;G;-t) and Gt have the same meanings as in (1.4.7).
o
The integration has been carried out by differentiating
a1
()
Y :£ G Y e
(l+y)s
-ty
and integrating
1 b
(l+y) -a
•
Again, denoting by F(y;c 2 ,b;c1 ,b;-t) the integral
(1.8.5)
(
o
and using (1.8.4), we have
,
(1.8.6)
=
_FO(Y;C2,b-1;C1 ,b;-t)+C 2 { F(Y;C2-1,b;C1,b;-t)-F(Y;C1,b;C2-1,b;-t~
-t{F(Y;C 2,b-1;C1,b;-t)-F(Y;C1 ,b;C 2,b-1;-t)} +2F(y;c 2+c1 ,2b-1;-2t).
Now
r
o
t
o
can be written in the form
r
o
37
:: W
y;
;-t
+W
;-t
y;
where Whas a meaning similar to that of U as in (1.4.l6).c Hence (1.8.7)
can be written as
;-t
(1.8.9)
38
c 2-l,b
+ c2
y;
1'11'
;-t
c 2 -l,b
-t
1/1f
c ,b
2
cl'b
-tW
y;
;-t
c 2 ,b
cl'b
y;
cl'b
cl'b
•
Following the arguments given in the previous sections and using results
-
.
similar to (1.$.4) and (1.$.13), the result (1.8.9) can be extended to
k variables and the following recurrence relation obtained;
(b-c k -1-)
W
. ". .
y;
•
•••
; -t
It
(1.8.10)
. ft. •-
.
ck_pb
+2. 1Z._ ( -1 )k-i-lF(y;01+ck,2b-l;-2t )w
i=k-1
.
; -t
....
•
y;
.. •
..°1+1 , b
ci_l,b •• cl,b
. ..
• •
• • ;-t
•
• •
•
•
•
ck_l,b • cC.~+ l,b ci_l,b •• c1 ,b
0
c
39
+ ok
lAJ
y;
°k~l,b
ck_l,b • • • cl'b
•
• •
· • ,. ...t
• · •
cl'b
•
•
•
°k_l,b • • •
ck-l,b
- t
w
Yl
I
°k,l.?-l
• • • • • • cl'b
,. -t
· · · · •• ••
· · · cl'b
· •
• •
•
• • •
•
ck,b-l • •
•
The last pseudo determinant on the right hand side of (1.8.10) can
further be expressed as the sum of two pseudo determDlants as in (1.8.8~
Two special cases of (1.8.10) are useful for statistical problems of
which the second one will be used in chapters VI and VII.
Case I.
•
•
• ••
• ••
•
•
; 0
(1.8.11)
•
.. ...
• • • ••
; 0
•
40
1Z (-1 )k-i-1F( y;c.+c ,2b-1;0 )W
~ k
. i=k-1
+2
.•
00;
y;
o
...
.
•••••••
•
•
•
•
.•
•
• •
• •
;0
i-t
(1.8.12)
. J.
k ' 1_
•
=2 Z (-1) -~-~(oo;c,+Ck,2b-1;-2t)W
i:k-1
~
Ck,~-l
-t irf
00;
•
00;
c _ ,b • • • c1,b
k 1
•
·•
•
• • • •
ck ,b-1 ck_pb.•
,. -t
· • cl'b
(
41
The recurrence relation (1.8.11) was obtained by . Roy
-/-20 7 and it is
-:"
suIt.
obvious that (1.8.10) is a generalization of his re- .
However (1.8.11) will not be used anywhere in our work while
(1.8.12) has an important application in. the last two chapters.
CRAPI'ER II
THB
L~RGEST Ctffil~CTERISTIC
ROOT
OF A ¥~TRIX
IN MULTIVARIATE ANALYSIS
2.1
Introduction
In trying to obtain tests for three different types of hy-
potheses in multivariate analysis, namely, (i) that of equality of
the dispersion matrices of two p-variate normal populations (ii) that
of equality of the p-dimonsional mean vectors for
f
p-variate normal
populations (which is mathematically identical with the general problem of multivariate analysis of variance of means) and (iii) that of
independence between a p-set and a q-sot (p ~ q) of variates in a
(p + q)-variate normal population, we run in each case into the characteristic roots of a matrix based on the sample observations.
In
case (i), this matrix is Sl(Sl+ S2)-1, where Sl and S2 denote the
usual product moment matrices and where both are almost everywhere
positivG definite, so that 8 (8 + 8 )-1 is almost everyNhere positive
1 1
2
definite, whence it follows that all the p characteristic roots are
greater than zero and less than unity.
In case (ii), this matrix is
8-1~(8-1~ + 8)-1 where 8-1~ denotes the ltbetwoen ll product moment matrix
of means weighted by the sample sizes and 8 denotes the ltwithin prod-
uct moment matrix (pooled from the product moment matrices of the
f
samples).
Then it is well known, that almost everywhere, 8 is
positive definite and S-l~ at least positive semi-definite of rank
43
s=min (p, f-l), so that almost everywhere s of the characteristic roots
~re
greater than zero and less than unity and the rest i.o. p-s are
zero.
In case (iii), this matrix is Sii S12 S22 sl2' where Sll is the
product moment matrix of the sample of observations on the P set of
variates, S22 that on the q set and S12 the product moment matrix betweon the observations on the p set and those on the q set.
known that if p
~
It is well
q < k, the sample size, then almost everywhere the p
characteristic roots of this matrix are gre3tor than zero and less than
unity
-;"6-7.
2.2 0istribution of the roots
In each caso, if the hypothesis to be tested is true, the nonzero roots (0 < 81 ~ 8 2 ~ ••• ~ Ss < 1; s ~ p) have the same joint dis-
- -7,
tribution, tho form of which was given by S. N. Roy ;-19
L-IO_7
and R. A. Fisher
L-4_7.
P. L. Hsu
The distribution can be written in the
form
° < 81 --< • •• --< es -< 1
,
44
where m and n have to be interpreted differently for the different
situations,
For example, in case (i)
n -p-2
l
m=-..---
2
,
and n ==
where n l and n2 are sample. sizes,
m ::::
In case (ii)
N-{-p-1
and n ==
where N is th<~ total of the sizes of
m :::: q-p-1
2
and n
we have
,
2
f
sa,np1es,
==
k-p-q-2
2
III ease (iii)
•
Following secti-:1n 1.3, (2.2.1) can be thrown into the de t'~rminanta1
f~mgiven
below:
em
+ s - 1 (1_8)n em+ s - 2 (1_e)n
em(l_e)n
s
s
s
s , •• S
S
•
{? ~ 2S)p.(llp'" ,es)=c( s"m,n
•
•
•
•
•
•
•
•
•
• • • • • •
•
•
• • •
am+s-2(1
e
)w
o~S-l(l~)n 1
- 1
•
91m(1-°1 )n
where
2.3 Cumulative distribution function of the largest
~ot
·The problem of obtaining the cumulative distribution of the
largest root has been already investigated by S. N. Roy
L-20._7 who
gave explicit expressions for the cumulative distribution for number
roots 2, 3, and
s
4.
D. N. Nanda ~12~ also gave such expression for
= 2,3,4 and 5. For obtaining the cumulative distribution Roy has
assumed the range of integration of the roots as zero to infinity because he works in terms of
A.=
].
6i
-.e ,(i = 1,
----1
2, ••• , s).
He makes
].
the cumulative d.istribution for s roots depend on that for s-l roots
and on incomplete beta functions.
However, it may be seen from the
results given toward the end of this chapter that the reduction can be
made much quicker and simpler in terms of certain functions of the incomplete beta functions rather t han in terms of the incomplete beta
functions themselves.
Moreover, if the computation is made for con-
secutive values of s, starting from 2, then these functions of the incomplete beta functions are such that most of the functions to be
~6
evaluated for a given value of s are already evaluated for the previous
values of s and only a few new expressions have to be evaluated at each
stage.
This method simplifies, to a large extent, the process of com-
putation.
While Roy gives a general method of obtaining the cumulative distribution of the largest root, Nanda expands the determinant given in
(?2.,) for each value of s
wp to , and obtains the cumulative distri-
bution by a method analogous to that of Roy.
Nandafs method of ex-
panding the determinant is less useful for handling the problem for
greater values of s.
The probability integral of the largest root is obtained by integrating (2.2.5) in the intervals 0 ~ 91 ~ ••• ~ Ss ~ x ~ 1.
get
(2.3.1) Fr(e
Hence we
< x)
s -
x
i
o
==
C(s,m,n)
•
..
•
•
em+s-2(1_e )ndS
s
•
s
s'
•
•
•
•
•
•
•
47
Comparing (2.3.1) with (1.4.1) it can be easily seen that if we multiply (1.4.1) by c(s,m,n) and put q=m,k=s,r=n,x=8 and t=O we get (2.3.1).
Hence it is obvious that the reduction formula (1.7.11) can be readily
applied to (2.3.1) to obtain the cumulative distribution for successive
values of s beginning from 2.
For s=2, app1yirlg (1.7.11) .with the
necessary changes indicated above to (2.3.1) we get
(2.3.2)
Pr(e~)= C(2,m,n)
[-I (X;m+l,n+l)I(X;m,n)+2 I (X;2m+l,2n+10
O
(m+n+2)
j
•
It may be noted that since t=Owe should have lvritten I(x;m,n;O)
instead of I(x;m,n).
But the
la~
notation is preferred for simplicity
and as no confusion is likely to arise.
For further simplicity of
notation, the upper limit x of the largest root also will not be mentioned explicitly in the I-functions since in the present chapter and
chapter III no other limit is used.
pee3 -< x)= C{3,m,n)
{-I {m+2,n+1) I{m+l,m,n)
o
m+n+3
+2I{m,n) I{2m+3,2n+1)
-2I(m+l,n) I(2m+2,2n+l) }
48
For s
=4
[ -IO(m>3,n>1) I(m>2,m>1,m,n)
+2I(2m+5,2n+l) I(m+l,m,n)
-2I(2m+4,2n+l) I(m+2,m,n)
+2I(2m+3,2n>1) I(m>2,m>1,n) }
,
(2.3.5) whore I(m>2,m,n)- (-IQ(m>2,n+l)I(m,n)+2I(2m>2,2n+l
+(~2)I(m+l,m,n) J /
/ (m+n+3)
•
It may be observed that in getting the right hand side of (2.3.5) the
formula to be used is (1.7.10) with t
= 0 and not (1.7.11). On the
right hand side of (?3.4) all expressions except I(m+2,m,n) are already evaluated for s = 2 and 3 with the proper choice of m and n, and
the only new function to be computed is I(m+2,m,n).
No other ex-
pression need be reduced to incomplete beta functions at this stage.
Again
(2.3.6)
Pr(e
< x)= C(5,m,n)
5-
(m+n+5)
f
-Io(m>4,n>1) I(m+3,m>2,m>1,m,n)
+2I(2m+7,2n+l) I(m+2,m+l,m,n)
49
-2I(2m+6,2n+l) I(m+3,m+l,m,n)
+2I(2m+5,2n+l) I(m+3,m+2,m,n)
-2I( 2m+4, 2n+1) I(m+3,m+2 ,m+1,nJ '
(2.3.7) where I(m+3,m+l,m,n)=
{ -IO(m+3,n+l) I(m+l,m,n)
+2I(2m+4,2n+l) I(m,n)
-2I(2m+3,2n+l) I(m+l,n)
+(m+3) I(m+2,m+l,m,n)} /
/ (m+n+4)
(2.3.8) and I(m+3,m+2,m,n)=
t
-Io(m+3,n+l) I(m+2,m,n)
+2I(2m+5,2n+l) I(m,n)
-2I( 2m+3, 2n+l I(Ti1+2,n)] / .
/ (m+n+4)
•
(2.3.2), (2.3~3) and (2.J.4) were given by Roy £20_7 in a different
- .
.
-7
form and (2.3.2), (2.3.3),(2.3.4) and (2.3.6) by D. N. Nanda /-12
-
in almost the same form, the differences beine only in minor details.
The explicit expressions for s = 6,7 and 8 are given below for the first
time.
50
Pr(e
<
6-
x)= ~(6,m,n) [-Io(m+5,n+1) I(m+4,m+3,m+2,m+1,m,n)
(m+n+6)
+2I(2m+9,2n+l) I(m+3,m+2,m+l,m,n)
-2I(2m+8,2n+l) I(m+4,m+2,m+l,m,n)
+2I(2m+7,2n+l) I(m+4,m+3,m+l,m,n)
-2I(2m+6,2n+l) I(m+4,m+3,m+2,m, n)
+2I(2m+5,2n+1) I(m+ 4,m+ 3,m+2,m+1,m)] ,
where
-Io(m+4,n+l) I(m+2,m+l,m,n)
+2I(2m+6,2n+l) I(m+l,m,n)
-2I(2m+5,2n+l) I(m+2,m,n)
+2I(2m+4,2n+l) I(m+2,m+l,n)
+(m+4) I(m+3,m+2,m+l,m,n)
.
(2,3.11)
I(m+4,m+3,m+1,m,n)~
[-Ia(m+4,n+1) I(m+3,m+1,m,n)
+2I(2m+7,2n+l) I(m+l,m,n)
/
/(m+n+5)
,
-2I(2m+5,2n+1) I(m+3,m,n)
+2I(?m+4,2n+1) I(m+ 3,m+1,n)1. /
~/(m+n+5)
(2.3.12)
,
I{m+4,m+3,m+2,m,n)= { -IO{m+4,n+1) I(m+3,m+2,m,n)
+2I(2m+7,2n+1) I(m+2,m,n)
-2I(2m+6,2n+1) I(m+3,m,n)
+2I(?m+4,2n+1) I(m+ 3,m+2,n)} . /
/(m+n+5
and
(2.3.13)
I(m+3,m,n)z {-Io(m+3,n+1)I{m,n) + 2I(2m+3,2n+1)
+(m+3) I(m+2,m,n)} /
(2 3
• •
14)
(m+n+4)
•
Pr(e < x)= C(7,m,n) (
7m+n+7
1 -IO(m+6,n+1) I(m+5,m+4,m+3,m+2,m+l,m,n)
+2I(2m+ll,2n+l) I(m+4,m+3,m+2,m+l,m,n)
-2I(2m+lO,2n+l) I(m+5,m+3,m+2,m+l,m,n)
+2I(2m+9,2n+l) I(m+5,m+4,m+2,m+l,m,n)
52
-2I(?m+8,2n+l) I(m+5,m+4,m+3,m+l,m,n)
+2I(2m+7,2n+l) I(m+5,m+4,m+3,m+2,m,n)
-2I(2m+6,2n+1) I(m+5,m+4,m+3,m+2,m+1,nJj,
where
(2.3.15)
I(m+5,m+3,m+2'~+1,m,nJ.{-Io(m+5,n+1) I(m+3,m+2,m+1,m,n)
+2 I (2m+8,2n+l) I(m+2,m+l,m,n)
-2I(2m+7,2n+l) I(m+3,m+l,m;n)
+2I(2m+6,2n+l)I(m+3,m+2,m,n)
-2I(2m+5,2n+l) I(m+3,m+2,m+l,n)
+(m+5) I(m+4,m+3,m+2,m+l,m,n)1 /
,
/(m+n+6)
J
C?3 .16)
I (m+5,
~+4, m+2 ,m+1,m,n)' (-IOCm+5,n+1) I(m+4,m+2,m+1,m,n)
+2I(2m+9,2n+l) I(m+2,m+l,m,n)
-2I(2m+7,2n+l) I(m+4,m+l,m,n)
+2I(2m+6,2n+l) I(m+4,m+2,m,n)
-2I(2m+5,2n+l)
I(m+4,m+2,m+l,n~ /
,
1/(m+n+6)
(2.3.17) I(m+5,m+4,lfrl-3,m+1,m,n){_Io(m+5,n+1) I(m+h,m+3,m+1,m,n)
+2I(2m+9,2n+l) I(m+3,m+l,m,n)
-2I(2m+8,2n+l) I(m+4,m+l,m,n)
+2I(2m+6,2n+l) I(m+4,m+3,m,n)
-2I(2m+5,2n+l)
I(m+4,m+3,m+l,n~ /
,
J/(m+n+6)
(2.3.18)
I(m+5,m+4,m+3,m+2,m,n)~\-IO(m+5,n+1) I(m+4,m+3,m+2,m,n)
+2I(2m+9,2n+l) I(m+3,m+2,m,n)
-2I(2m+8,2n+l) I(m+4,m+2,m,n)
+2I(2m+7,2n+l) I(m+4,m+3,m,n)
-2I(2m+5,2n+l) I(m+4,m+3,m+2,n
J/
J/(m+n+6)
(2.3.19)
I(m+4,m+1,m,n)~f-Io(m+4,n+1)_t(m+).....n)
-2I(2m+5,2n+l) I(mJn)
-2I(2m+4,2ntl) I(m+l,n)
+(m+4) I(m+3,m+l,m,n)} /
/(m+n+5)
,
,
I(m+4,m+2,m,n)~ (-Io(m+4,n+l)
(2.3.20)
54
I(m+2,m,n)
+2I(2m+6,2n+l) I(m,n)
-2I(2~4,2n+l) I(m+2,n)
+(m+4) I(m+3,m+2,m,n)} /
/(m+n+5)
where
rf
=
[-Io(m+7,n+l) I(m+6,m+5,m+4,m+3,m+2,m+l,m,n)
+2I(2m+13,2n+l) I(m+5,m+4,m+3,m+2,m+l,m,n)
-2I(2m+12,2n+l) I(m+6,m+4,m-;'3,m+2,m+l,m,n)
+2I(2m+ll,2n+l) I(m+6,m+5,m+3,m+2,m+l,m,n)
-2r(2m+10,2n+l) I(m+6,m+5,m+4,m+2,m+l,m,n)
,
+2I(2m+9,2n+l) I(m+6,m+"m+4,m+3,m+l,m,n)
-2I(2m+8,2n+l) I(m+6,~"m+4,m+3,m+2,m,n)
+2I(2m+7,2n+l) I(m+6,m+5,m+4,m+3,m+2,m+l,n)}
,
~ere
(2.3.23) I(m+6,m+4,m+3,m+2,m+l,m,n)=
{ -Io(m+6,n+l) I(m+4,m+3,m+2,m+l,m,n)
+2I(2m+lO,2n+l) I(m+3,m+2,m+l,m,n)
-2I(2m+9,2n+l) I(m+4,m+2,m+l,m,n)
+2I(2m+8,2n+l) I(m+4,m+3,m+l,m,n)
-2 I (2m+7,2n+l) I(m+4,m+3,m+2,m,n)
+2I(2m+6,2n+l) I(m+4,m+3,m+2,m+l,n)
1
+(m+6) I(m+"m+4,m+ 3,m+2,m+l,m,n)JI /
/(m+n+7)
(2.3.24)
I(m+6,m+S,m+3,n~2,m+l,m,n)~
[-IO(m+6,2n+l) I(m+5,m+3,m+2,ffi+l,n,r.)
+2I(~n+ll,2n+l)
I(m+3,m+2,m+l,m,n)
,
56
-2I(2m+9,2n+l) I(m+5,m+2,m+l,m,n)
+2I(2m+8,2n+I) I(m+5,m+3,m+l,m,n)
-2I(~1+7,2n+l)
I(m+5,m+3,m+2,m,n)
j
j(m+n+7 )
+21(:1m+6, 2n+1) 1(lIl+5,m+3,m+2,m+1,nj
,
(2.3.25) I(m+6,m+5 ,m+h,m+2,m+l,m,n)=
( -10 (m+6,n+1) 1(m+5,m+4,m+2,m+1,m,n)
+2I(2m+II,2n+l) I(m+4,m+2,m+l,m,n)
-2I(2m+l0,2n+l) I(m+5,m+2,m+l,m,n)
+2I(2m+8,2n+l) I(m+5,m+4,m+l,m,n)
-2I(2m+7,2n+l) I(m+5,m+4,m+2,m,n)
+2I( 2m+6, 2n+1)
I(m+5,m+4,m+2,m+l,n~
J
(2.3.26) I(m+6,m+5,m+4,m+3,m+l,m,n)=
i
-1o(m+6,n+1) 1(m+5,m+4,m+3,m+1,m,n)
+2I(2m+ll,2n+I) I(m+4,m+3,m+l,m,n)
j
j(m+n+7)
,
r
57
-21(2m+10,2n+l) 1(m+5,m+3,m+l,m,n)
+21(2m+9,2n+l) 1(m+5,m+4,m+l,m,n)
-21(2m+7,2n+l) I(m+>,m+4,m+3,m,n)
+21(2m+6,2n+l) 1(m+5,m+4,m+3,m+l,n)\ /
/(m+n+7)
J
(2.3.27) I(m+6,m+> ,m+)t,m+3 ,m+2,m,n)=
1I -I() (m+6,n+l) 1(m+>,m+4,m+3,m+2,.'1,n)
+2I(2m+ll)2n+l) I(m+4,m+3,m+2,m,n)
-21(2m+10,2n+l) I(m+5,m+3,m+2,m,n)
+2 I (2m+9,2n+l) I(m+5,m+4,m+2,m,n)
-2I(2m+8,2n+l) I(m+>,m+4,m+3,m,n)
+2I(2m+6,2n+l) I(m+5,m+4,m+3,m+2,n)} /
/(m+n+7)
(2.3.?8) I(m+,,m+2,m+l,m,n)- [ -Io(m+,,n+l) I(m+2,r.l+l,m,n)
+2I(2m+7,2n+l) 1(m+l,m,n)
-2 1(2m+6,2n+l) I(m+2,m,n)
56
+2I( 2rn+5, 2n+l) I(m+2,m+l,n)
+(m+5) I(m+4,m+2,m+l,m,n)
J/
,
/(m+n+6)
(2.3.29) I(m+5,m+3,m+l,m,n>=[_IO(m+5,n+l) I(m+3,m+l,m,n)
+2I(2m+8,2n+l) I(m+l,m,n)
-2I(2m+6,2n+l) I(m+3,m,n)
+2I(2m+5,2n+l) I(m+3,m+l,n)
+(m+5) I(m+4,m+3,m+l,m,n)}
I
/(m+n+6)
,
(2.3.30) I(m+5,m+3,m+2,m,n)= {-Io(m+5,n+l) I(m+3,m+2,m,n)
+2I(2m+8,2n+l) I(m+2,m,n)
-2I(2m+7,2n+l) I(m+3,m,n)
+2I(2rn+5,2n+l) I(m+3,m+2,n)
+(m+3) I(m+4,m+3,m+2,m,n)1 /
)/(m+n+6)
(2.3. Jl) I( m+ 5,m+4,m+2,m,n)= { -10(m+5, n+l) I (m+4,m+2,m,n)
,
59
+2I(2m+9,2n+l) I(m+2,m,n)
-2I(2m+7,2n+l) I(m+4,m,n)
-2I( 2m+5 ,2n+l) I(m+4,m+2 ,n)
( ?, 3,32)
1/
,
j/(m+n+6)
I( m+5,m+4,m+l,m,n)= f-I O( ....5,n+l) I(m+4,m+ 1,m~ n)
+2I(2m+9,2n+l) I(m+l,m,n)
-2I(2m+6,2n+l) I(m+4,m,n)
+2I( 2m+5,2n+l) I(m+4,m+l,n)1 /
] /(m+n+6)
I
(2,3.33) I(m+5,m+4,m+3,m,n)= [-Io(m+5,n+l) I(m+4,m+3,m,n)
+2I(2m+9,2n+l) I(m+3,m,n)
-2I(2m+8,2n+l) I(m+4,m,n)
+2I(2m+5,2n+l) I(m+4,m+3, n)1 /
/(m+n+6)
J
and
(2.3.34) I(m+4,m,n)=i-Io(m+4,n+l) I(m,n)
+2I(2m+4,2n+l)
,
60
+(m+4) I(m+3,m,n)J j
j(m+n+5)
The probability
L~tegral
•
for s = 9 and 10 also have been obtained
but are not presented here because of the large number of terms
involved.
CHAPT-:I:R III
APPROXIMATION TO THE PROBABILITY
INTEGRAL OF THE LARGEST ROOT
3.1
Introduction
The expressions given in the previous chapter for the cumula-
tive distribution function of the largest root are useful for extensive tabulation.
As the number of roots increases, the number of
incomplete beta functions involved in the expressions for the probability integral also increases considerably.
But if one proceeds
systematically from the case of two roots to higher values of s, through
consecutive values of the number of roots, the expressions of the previous chapter could be used with more advantage, since the nmaber of
incomplete beta functions to be considered at each stage will be less
in this case than if we skip certain values.
However, if one con-
siders only the problem of obtaining the upper percentage points
(S
9b
or less), the expressions given in the last chapter can be
avoided, and surprisingly simple expressions can be obtained, as given
in the following sections.
3.2 APproximate expressions for obtaining upper percentage points
(s
ro
or less) of the largest root
Let us first consider the case of two roots.
The probability
integral of the largest root given in (2.3 0 2) can be written in the
62
form
For integral values of m, integration by parts on I(2m+l,2n+l) and
I(m,n) will reduce the probability integral in (3.2.1) to the form
m (m) .r(n+l)x2m- i +1I._xt"+i+2
l:
l.
{ i=O
r(n+i+2)
2m+1 (2m+1)i r ( 2n+2 )x 2m- i +1 (1_x)2n+i+2
- 2
l:
i=O
r(2n+i+)
+ 2r(2m+2)r(2n+2)
r(2m+2n+4)
r(m+l)r(n+l)
r(m+n+2)
1J
xm+1 (1_xl'1+ 1
63
where (t)r = t(t-l) ••• (t-p+l)
In (3.2.2) if we neglect terms of order higher than xm, (l_x)n, we get
C(2,m,n) {2r(2m+2)r(2n+2)
(m+n+2)
r(2m+2n+4J
•
For small values of m, (3.2.3) will give the percentage points for the
upper 5
~
level or less.
(3.2.3) can be further simplified by in-
troducing the value of the constant C(2,m,n).
~ r (2m+2n+5)
)\
C(2,m,n)
2
Now
r (2m+2n+6)
2
:=
•
Using the identity
I)
22x-1r ( x ) r ( x+2
we have
~
",- r(2x)
I}{
,
64
r(2m+2n+5)
C(2,m,n)
-
=
22 r(2m+2) r(2n+2)
and thus (3.2.3) takes the form
r(m+l) r(n+l) r(2m+2n+4)
1 -
•
2 r(m+n+2) r(2m+2) r(2n+2)
If the probability in (3.2.7) has to be 0.95, then
r(m+l) r(n+l) r(2m+2n+4)
(3.2.8)
xm+1(1_X)n+l
= 0.05
•
2 r(m+n+2) r(2m+2) r(2n+2)
The expression in (3.2.8) is very simple for computation, especially when the values of m are small.
The error involved in using
this approximation has been computed and difference between the exact
and the approximate probabilities has been found to occur in the fifth
decimal place, and, on rounding off, there could be at the most a
difference of 1 in the fourth place.
65
Proceeding on the same lines as above i.e. as for the approximation in (3.2.7), we get the following approximations for s = ), 4 and 5,
(m+l)(2m+) )
- xm+2( I-x )n+l (2m+2n+5)
f(2m+2n+8) f(m+2) f(n+2) xm+)(l_X)n+l
().2.10) Pr(e ~ x)=l-
4
...
2f(m+n+) r(2m+4) f(2n+4)
r(m+l) r(n+2) r(2m+2n+8) xffi+2(1_x)n+l
4r(2m+2) f(2n+4) r(m+n+4)
- _..- - - - - - - - - - 4r(2m+2) r(2n+4) r(m+n+4)
]
66
r(m+n+4)
(3.2.11) Pr(e, < x)= - - - - 3r(m+l) r(n+3)
x
(2m+3(2m+,)
f5
o
8 (1-8,)nd8,
+ (2m+,) (2m+2n+7)(2m2+2mn+13m+12)
x
r e~l(l_e,)nde,
(2n+)
0
x
(2m+3)(2m+2n+7)(2m+2n+9Xm+n+4)
{
(2n+) )
o
2
5 (1-S,)nd8,
9 +
(m+3)( 2rn+,)( 2m+2n+,)( 2m+2n+7) xm+2(1_x)n+l
(2n+3)(m+l)
(2m+2n+7)(2m+2n+9) xm+3(1.X)n+l
+ 3
(m+l)
(2m+2n+7)( 2m+2n+9 )(m+n+4) xm+4(1_x)n+l
(m+l)(m+2)
•
The incomplete beta functions in (3.?9) and (3.2.11) can be
further integrated by parts and for small values of m the expressions
will become simpler and could be readily used for computation.
3.3. Percentage points for the largest root
An important problem in multivariate analysis is to test the
67
hypothesis that
f p-variate normal populations having the same var-
iance-covariance matrix, from each of which a sample is drawn, have
the same mean for each variate.
This, in fact, is the second one
among the three tests discussed in the introduction of chapter II.
For
this test it turns out that
m=
if-pI» .2. .
as given in (?.2.2).
and
p
For
:::>
n=
2 we have
N-p-K'-l
m=
2
(24 and the expressions
for the upper percentage points given in section 3.2 which are 'good for
small values of m become quite useful, because the number of samples,
I,
cannot be too large.
For example, the case of m :::> 4, i.e. (
relates to a problem with 12 samples.
5
%
and 1
0/0
= 12
Using (3.2.8) tables of upper
points have been computed for two roots for values of
m = 0, 1, 2, 3 and 4 and n
:::>
5,
10, 15, 20, 25, 30, 40, 60, 80, 100,
130, 160, 200, 300, 500 and 1000.
Significance levels for fractional
values of m and intermediate values of n can be obtained by interpolation.
The percentage points based on the approximate formula 3.2.8 are
given in Table 3c3.l.
In order to form an idea of the error of approximation, the probability based on the exact formulae is calculated at the percentage
points for the parameters, m = 2 and n ::: 10, 30 and 80 and m = 4, n
=5
and 100, picked out from Table 3.3.1 (based on the approximate formula
68
(3.2.8».
The difference between the exact and approximate prob-
abilities is exhibited in Table 3.3.2.
It is to be noticed that within
the range of values of the parameters consideredJthe approximation is
quite good.
The exact expressions for the probability integral for
m = 2 and m ::a 4 obtained from (3.2.1) are given below: For m = 2,
_ (l-x)2n+3x 4(2n+5)(2n+6)(2n+7)(n+4)
60
_ (1_x)2n+5x 2 (2n+6)(2n+7)
2
2n+6
)
- ( I-x)
x(2n+7
... (1_x)2n+ 7
For m =
4,
•
69
_ ~(?n+5)(2n+6)(2n+7)(2n+8)(2n+9)(2n+l0>(2n+ll)(n+6){1~x)2n+3x~
9J
_ ~ (2n+5)(2n+7)(2n+8)(2n+9)(2n+lO)(2n+ll)(n+6)(1-X)2n+4x 7
9J
96 (2n+7)(2n+9)(2n+lO)(?n+ll)(13n 2+115n+222)(1-x)2n+Sx6
-91
_ ~ (2n+7)(2n+9)(2n+ll)(SSn2+53Sn+1230)(1-X)2n+6xS
91
(2n+8)(2n+9)(2n+lO)(2n+ll)(1-x)2n+7x4
41
(2n+9)(2n+lO)(2n+ll) (1_x)2n+8x3
31
_ (2~+lO)~2n~11) (1_x)2n+9x2
2
•
71
Table 3.3.1 (continued)
;S:m
100
130
160
200
300
500
1000
·0
1
2
3
4
0.0515
0.0686
0.0835
0.0972
0.1100
0.0692
0.0878
0.1038
0.1184
0.1319
0.0400
0.0535
0.0652
0.0761
0.0864
0.0539
0.0685
0.0813
0.093 0
0.1039
0.0327
0.0437
0.0535
0.0626
0.0711
0. 0441
0.0562
0.0668
0.0765
0.0857
0.0263
0.0352
0.0432
0.0506
0.0576
0.0355
0.0453
0.0540
0.0619
0.0694
0.0176
0.0237
0.0291
0.9342
0.0390
0.0239
0.03 05
0.0365
0.0419
0.0471
0.0106
0.0143
0.0176
0.0207
0.0237
0.0144
0.0185
0.0221
0.0255
0.0287
0.00535
0.00719
0.00888
0.01045
0.01195
0.00725
0.00930
0.01114
0.01285
0.01448
72
Table 3.3.2
The error of approximation at the 5
70
level
(i.e. Difference in value between (3.2.1) and (3.2.7) for
s~2 at the 5 %
mgnificance level in Table 3.3.1)
~2
n
Difference"
(Approximate-Exact)
10
, O~00002693
30
0.00002697
80
0.00002757
Nanda
1
%
L-15_7 has
~4
Difference
(Approximate-Exact)
. °,00002574
100
.00003477
given significance levels (upper
for the largest root for s
n(m=0(~)2; n=~(~)lO),
n
5%
and
= 2 and very small values of m and
His table has been computed by using the Tables
of Incomplete Beta Functions
L-17_7
for some values of m and n and the
other values obtained by interpolation,
The significance levels are
given only up to two decimals and the range of m and n considered is
very small.
m, 0 -
4,
In Table 3.3,1, however, the range for n is 5 - 1000 and
The significance levels have been given correct to three sig-
nificant figures, in order to ensure sufficient accuracy for interpolation.
I
For testing the equality of variate-means from different popu-
lations, Table 3.3.1 should be enough up to m = 4, but if anything beyond this is needed one can obtain the significance level easily from
73
the expression in (3.2.7) for small values of m.
3.4
The distributions of the largest and smallest roots for s
=2
as
hypergeometric series.
The joint probability density function of 8 and 8 2 is given by
1
where C(2,m,n) is given in (J.?6).
Now consider the transformation
Then
where the range of variation of
~
as well as 62 is 0-1.
Now inte-
grating out ~ from p(~,62)'
i
1
P(S2) = C(2,m,n)sr'2(1-8 2 )n
or
o
(l-M/\m(l-A)dA
74
p(8 ) = q(2,m,n)_
(m+1)(m+2)
2
2m+2(
82
( 3.4.6) where F( m+l,-n,m+3,8 2 )= 1.
)n (
)
1-82 F m+1,-n,m+3,8 2
'
n(m+l)
n(n-1)(m+l)(m+2) 2
3 82 +
(L){ 4)
8 2 - •••
m+
21 m+3 m+
Again starting with (3.4.1) and effecting the following transformation
we get
It may be
~bserved
gives (3.4.5).
that a change from 1-61 to 82 and m to n in (3.4.9)
Hence
75
=1
- Pr(Ql < 1 - X; n,m)
•
Hence lower percentage points for the smallest root can be obtained
from the upper percentage points of the largest root when the parameters m and n are interchanged.
- -7
The result (3.4.10) is true for any value of s as Nanda /-12
has shown starting from distribution (2.2.1).
CHAPTER IV
MOrmNTS OF THE SUM OF THE CHARACTERISTIC
ROOTS OF A MATRIX
4.1
Introduction
Among the various test criteria proposed for problems in multi-
variate analysis there are at least three which are important.
These
criteria are based respectively on:
(1) the largest and smallest characteristic roots of the matrix for
(1) under section 2.1 and just the largest characteristic root of the
matrix for (ii) or for (iii) of section 2.1,
(2) the sum of the characteristic roots of the matrix for (i), (1i)
or (iii) of section (2.1) and
(3) the product of
(1-9.)'5,
where the e.fs
are the characteristic
1..
1.
_
roots of the matrix for-(i), (ii) or (iii) of section 2.1.
On the corresponding null hypothesis we have formally just one
distribution problem common to (i), (ii) and (iii) under (1), just
another distribution problem common to (i), (ii) and (iii) under (2)
and another one common to (i), (ii) and (iii) under (3).
The
-7
test criterion (3) was suggested by S. S. Wilks /-21
-
and the statistic turns out to be the ratio of two determinants which
we can easily pick out from (i), (ii) and (iii) of section 2.1.
The
(common) distribution on either of the three null hypotheses was ob-
77
1 and
tained in an exact form for s ::
the moments.
2 by ';nJilks
£21_7 by
- -7
For larger values of sJM.S.Bartlett /-2, 3
observing
has sug-
gested a chi-square Gpproximation which appears to be good for
practical purposes and C. R. Rao
L- 18_7 has
obtained a series of
chi-squares of which Bartlett's form is a first approximation.
A
beta function approximation and a corresponding series also have been
The distribution of the largest root, the smallest root or of
-
any root has been studied by Roy /-20
,..
7 and
- -7 and further
Nanda ;-12
studies have been presented in chapters II and III.
As shown there,
it is much easier to compute the values of the largest roots at the
customary significance levels (say 5
%
and 1 %) by using the
formulae given in chapter III for approximation to the probability
than using the exact formulae stated in chapter II.
In fact, for
relatively large s, these approximations alone will make the computation feasible.
In chapter III the actual expressions for computations
- - 5.
have been given for 2 < s <
handle the case of s >
The same method would enable us to
5.
The sum of the roots suggested by H. Hotelling
1:7, 8, 9_7 is
another useful statistic for testing certain hypotheses in multivariate situations.
He has given some interesting ballistic examples
where this criterion is the most appropriate one.
For example if
78
th
where the coordinates on impact of the B bomb are x lB and x 2B and
the matrix ij ) is the inverse of the covariance matrix from an old
(K
sample with n degrees of freedom, T~ is applied to measure the
accuracy of the Bth bomb. If the new and old populations are identical, T~ has the T distribution ~5_7. For m new bombs, a combined
2
measure of accuracy, or rather inaccuracy, is defined as
2
T
m
2
E T
o = B=l
B
= m Z.
E fJ.' '
J
where s~. is a new estimate of covariance with
J.J
ill
degrees of freedom,
Calling the old and new covariance matrices Sand S* respectively
(these should not be confused with the notations in previous chapters) we have as a measure of multivariate dispersion
•
Since the trace of a matrix is the sum of its latent roots,
2
To/ m is the sum of the characteristic roots of the matrix S-lS*.
In
chapter VI we shall be discussing the moments and in chapter VII, an
approximate distribution of the sum of the charaoteristic roots of the
n
-l*"
matrix ( m) S-1S.
This sum lies between zero and infinity.
(Again, m
79
and n of this section should not be confused with the same symbols in
the previous chapters).
In the rest of this chapter as a Iso
be considering the statistic V(s)
= i e.,
. 1 J.
J.=
istic roots defined in section 2.2.
in the next chapter we srian
where 9. are the characterJ.
These srs as already stated, lie
between zero and unity.
4.2 The moments of v(S), in general
Consider the recurrence relation (1.7.12) putting r
qi
=n
= m+i-l, k = s and x = e. We get
m+s-l,n
(4.2.1)
(m+s+n)
•
U 1;
•
•
• m,n
•
•
•
•
m+s-l,n
• m,n
j
f
{m+S_2,n•• m+i,n m+i-2,n••m,n)
(-1)s-i-lI (1;2m+s+i-2,2n+l;2t)U 1·
;t,
J.1
. m+J."2 ,n •• m,n
,
m+s- 2,n••m+1,n
=2~~
1
;t
+tull;
m+s-l,n+1 • • • • m,n
•
•
•-
•
•
m+s-1,n+1
· ••
· m,n
··
•
;t
80
(It is tCl be noted that for brevity in notation the upper limit unity
will not be given explicitly hereafter).
Multiplying both sides of (4.2.1) by C(s,m,n) and using properties of Vandermondets determinant given in 1.2, it can be easily
seen that we have a recurrence relation for the moments of the sum of
the roots, v(s).
In fact, we get
(~n+s-t) E(e tv
(s)
) + tE(V
() tv (s)
S e
)
2C( m ) 6-1 s-i-l
I(2m+s+i-2,2n+l;2t)E
=·C( s2 ;n j Z (-1)
s- ,m,n i=1
~.
(Zel •• e
. l)e
tV( s-2) }
S-1-
where zel •• es_i_1denotes the (s_i_l)th elementary symmetric function
in 61 • • • e s _2 '
It may be noted that for obtaining (4.2.2) we have
made use of the relations
m+s,:,l,n+l ,
C(s,m,n)U
.
.m,n
• •
• ••
•
•••
•
•
•
m+s-l,n+l ••• m,l).
81
and
m+s-2,n • • m+i,n m+i-2,n • • m,n
• •
(4.2.4) C(s-2,m,n)U
•
•
• •
..
••
•
;t
m+s-2,n • • m+i,n m+i-2,n • • m,n
The relation (4.2.2) gives the moments of the sum of the roots for
s roots in terms of the expected value of the elementary symmetric
functions, their powers and products for s-2 roots. The relation
(4.2.2) therefore is a fundamental one for our work.
The moments of v(2)
When s
= 2,
the relation (4.2.2) reduces to
= 2C(?,m,n) I(2m+l,2n+l;2t)
•
But it may be observed that
1
r(2m+l,2n+l;2t) =
.
{ 6 2m+\1_e)2n+le 2t6ds
o
82
;::: r(2m+2)r(2n+2)
r(2m+2n+4)
F(2m+2;2m+2n+4;2t)
,
where F(a;p;z) is tho confluent hypergeometric function defined by
F(ao8'z)
'. ,
= I + ~ z + a(a+l)z1
~
+ • • •
~ ( ~ +- I )21
Substituting from (4.3.2) in (4.3.1) and equating like powers of t on
both sides of (4.3.1) we get recurrence relations between the moments
of the sum of two roots.
tv(2)
<
Expanding e
.
ts
c~en
0f
as a power series and equating the coeffi-
th e ~.th power of t on both sides of (4.3.1) we get
( 4. 3. 4) (m+n+ i +2 )l-Li'( 2t:4L <i' (12).. _2~-.;·(....2_m+_2.)~. (~2_m_+ _i+_1..:...)(~m_+_n+_2~)
(2m+2n+4) ••• (2m+2n+i+3)
(i
where l-L!
~
(2)
denotes the i
th
1,2,3 •• )
raw moment for 2 roots.
(m+n+2)l-L,(2);:::(m+n+2)
o
:=
or l-L ,(2)
o
=0
For i ;::: 0
1, as it should be
•
83
(4.3.4) gives the i th moment in terms of i_I th and hence by successive
substitutions of the lower order moments given by the respective recurrence relations, we get
From (4.3.5) putting i
=0
I we get
(2m+3).. •
m+n+3
(4.3.6)
Similarly, putting i = 2 and simplifying, we get
1(2)
~
_2
=0
2
2(2m+3)(2m +2mn+IOm+4n+ll)
(m+n+3)(m+n+4)(2m+2n+5)
-
.
For i ::: 3
(4.3.8)
_.
~ i (2) ::: 2( 2m+3) (l~.m3+4nP!!+ 34m~~8mn+86m+20n+65)
3
(m+n+3) (m+n+4) (m-'-n+5)(2m+2n+5)
..
84
(4.3.9)
For i
=4
(2) A!42
1-.1.'
4 -- -B'42
where
22222
+30m n +406m n+1212m +74mn +720mn
and
B1
42
= (m+n+3)(m+n+4)(m+n+5)(m+n+6)(2m+2n+5)(2m+2n+7)
(4.3.10) For
i
=5
where
22
+ 17720m+8m4n2+88m3n2+358m2
n +638mn
and B',2 = (m+n+3)(m+n+4)(m+n+5)(m+n+6)(m+n+7)(2m+2n+5)(?m+2n+7) •
From the raw moments given in (4.3.6) - 0-1-.3.9) we can obtain the
central moments as given below:
85
IJ.
(2)
(2m+3)(2n+3)(m+n+2)
=--;r.2
(m+n+3) (m+n+4)(2m+2n+5)
1J.(2) = 2(.n...m)(2m+3)(2n+3)(m+n+l)
,
3
(m+n+3)3(m+n+4)(m+n+5)(2m+2n+5)
(4.3.13)
and 1J.4(2)
~ AB·~2
where
42
~ 3(2m+3)(2n+3)
A42
[[(2m+2n+6)2+(2m+3)(2n+3)(m+n)} (m+n+2)(m+n+5J
-16(m.n)2(m+n+3~
and B = (m+n+3)4(m+n+4)(m+n+5)(m+n+6)(2m+2n+5)(2m+2n+7).
42
It may be observed that the central moments become surprisingly !imple compared to the raw moments. . The raw moments given in
.
-
(4.3.6) - (4.3.10) will have to be used to get the moments for s
= 4.
86
(4.4.1) can be fully written as
(m+n+3_t)E(e tV
(3)
)
(3)
) + tE(V(3 e tV )
r(2m+2n+6)r(m+n+4)
= 2r(2m+2)r(2n+2)r(m+2)r(n+2)
¢
Again,
I
(4.4.3)
I
o
2te
e2~n+3(1-e2)2n+le
2dB2= r(~m+4)r(62~)1+2) F(2m+4;2m+2n+6;2t),
r 2m+2n+
87
and similarly, writing the other three integrals in terms of confluent
hypergeometric series,we observe that the right hand side of
(4.4.2)
takes the form
{ (2m+3)(m+n+2) F( 2m+4; 2m+2n+6;2t )F(m+1;m+n+2; t)
-
(m+l)(2m+2n+5)F(2m+3;2m+2n+5;2t)F(m+2;m+n+3;t~
Replacing the right hand side of
(4.4.2)
by
(4.4.4),
expanding
.
(3)
tV
E(e
) as a power series and equating the coefficients of like
powers of t on both sides of the relation
(4.4.2),
currence relation between the successive moments.
(4.4.5)
=
we can get a reThus
r( i+l). (m+n+J)
(n+l'
cr
where
r:J=(2m+J)(m+n+2)
i j
ij=O 2(i-j)Jjl
- 2m+4), .. (2m+i-j+J) m+l) ••• (m+j)
2m+2n+6) ••• (2m+2n+i-j+5) m+n+2) ••• (m+n+j+l)
)
(i=O,1,2,J, "~. )
•
88
where it must be remembered that the first term in the hypergeometric
series is unity.
putting i=O on both sides of (4.4.')}after necessary
simplifica~i~n, we
get
~b(3)=1 as
we ShoUld: For i=l, the right hand
side of (4.4.,) is given by
(4 ••
4 6)
l
(m+n+3) "2m+3)(m+n+2) {2(2m+4 ) + m+l
(n+l)
(2m+ 2n+6) m+n+2
-(m+l)(2m+2n+,) {2(2m+3)
(2m+2n+,)
+
J
m+2
m+n+3 ~
J11..
'
.
which, after simplification, gives 3m+,; and hence from (4.4.,) we have
~I(J) = 3(m+2)
1
m+n+4
•
In a similar manner putting i=2 in (4.4.5) we can show, after necessary
simplification, that
~2 0)
=
2
3(m+2)(6m +6mn+14n+39m+,,)
(m+n+4)(m+n+,)(2m+2n+7)
(4.4.9) Similarly we have
~I 0)
3
AI
=--11
B'33
where
•
89
A'33 = 3(m+2)(18m4+243m3+1152m2+36m3n+333m2n+959mn+2287m+18m2n2
2
+90mn +112n2+872n+1620)
and B'33= (m+n+3)(m+n+4)(m+n+5)(m+n+6)(2m+2n+7)
(4.4.10)
and
AI
lotIO) =~
4
BI
43
where
A'43 = 9(m+2)(36m6+108m5n+828m?+108m4n2+1944m4n+7587m4+36m3n3
+1404m3n2+133l5m3n+35405m3+288m2n3+6492m2n2+43420m2n
+88709m2+764mn3+l2796mn2+67617mn+113247m+672n3
2
+9120n +40356n+57708)
and Bt
43
= (m+n+3)(m+n+4)(m+n+5)(m+n+6)(m+n+7)(2m+2n+7)(2m+2n+9)
•
Even though the raw moments seem to be of an irregular nature,
the central moments take somewhat simple and neat forms.
moments for s
(4.4.11)
=>
The central
3 are given by
= 3(m+2)(n+2)(2m+2n+5)
...
,'2
,
(m+n+h) (m+n+5)(2m+2n+7)
(4.4.12)
lot(3) = 12(n-m)(m+2)(n+2)(m+n+l)(2m+2n+5)
3
(m+n+4)3(m+n+5)(m+n+6)(2m+2n+6)(2m+2n+7)'
90
A
(4.4.13)
A43
43
-1343
lJ. (J) -
and
4
where
~ 9(...2)(n+2) ~+n+3)(m+n+6) {2(m+n+4)2
.
,f
2 (!i.+n l1-88(m+n)+
1~(m+2)~+21
+(n-m)2ta(...n~+3B(m+n)2~24(m+n)-i34}~.
and
B
43
= (m+n+3)(m+n+4)4(m+n+5)(m+n+6)(m+n+7)(2m+2n+7)(2m+2n+9) •
4.5 The Mo~~nts of V(4)
From (4.2.2) putting s
(m+n+4-t)E(e tV
(4.5.1)
=
20(4,m,n)
C(2,m,n)
f
\
(4)
= 4 we
get
)+tE(V(4)e tv
)
(2)
iI( 2m+S,2n+l;2t)E(etv m,"')
-I( 2m+4,2n+l;2t.)E(V
+
(4)
(2)
1
m,n e
tV(2)
m,n)
(2)
C(2,m,n I(2m+3,2n+l;2t)E(etVm+l,n)
C(2,m+l,n)
}
where V(s)
is an explicit symbol for the sum of the roots when
m,n
the parameters are m, nand s.
After necessary simplification we have
91
C(4,m,n) = (2m+2n+5)(m+n+3)f(2m+2n+9)
C(2,m,n)
22 f (2m+4)f(2n+4)
and
C(4,m,n)
C(2,m+l,n)
=
r(2m+2n+9)
•
Working on similar lines as before and equating the coefficients of
like powers of t on both sides of
(4.5.1), we get
(4.5.4)
where
.
J.
2i - j (2 m+.
5) •• (2 m+J.-J+
. . 4)
..
~ ,(2)
.
-(2m+2n+5)(2m+2n+7)(m+2) Z
J+l,m,n
j=O jl(i-j)J(2m+2n+7) ••• (2m+2n+i-j+61
.
\
i 2i - j (2m+4) ••• (2m+i-j+3) ~,(2)
+(2m+2n+7)(2m+3)(m+l) Z
.
j,m+l,n
j=O jl(i-j)!(2m+2n+6) ••• (2m+2n+i-j+5)
For i
= 0,
we have form
(4.5.4)
92
(h. 5.5) (m+n+h)=t (m+n+3)'( m+n+h) \ (2m+2n+5)( 2m+5)( m+2) .
1
(n+l)(2n+3)
-(2m+2n+5)(2m+2n+7)(m+2)~1,(2)
,m,n
+( 2m+2n+7)( 2m+3) (m+l)
. that
Now, remembermg
should, ~~ (h)=l.
1(2) '" 2m+3
,m,n m+n+3
~l
}
, (4.5.5 ) gives us, as it
To obtain the moments of v(h), the following results
have to be substituted in (4.5.4):
,(2)
(h.5.6)
~l,m+l,n
=t
(4 .5.8 ) ,,'(2)
~3 ,m+ 1 ,n
(h.'.9)
and
•
2m+5
m+n+h
""
2(2m+5)(2m2+2mn+14m+6n+23)
(m+n+4)(m+n+5)(2m+2n+7)
,
= 2(2m+5 4m3+46m2+l66m+4m2n+26mn+42n+189)
(m+n+4 m+n+5 ) (m+n+ 6 2m+2n+7 )
A' 42,m+l
1(2)
~4 ,m+ 1 ,n
'"
B
42,m+l
where
93
2
A'42,m+1 = 8(2m+5)(4m5+86m4+716m3+2884m +5616m+4221+8m4n
+128m3n+742m2n+1852mn+1680n+4m3n2+42m2n2
+146mn2+168n 2)
and B f 42,m+1
:I
(m+n+4)(m+n+5)(m+n+6) (m+n+7 )(2m+2n+7)(2m+2n+9)
Making use of (4.5.6) - (4.5.9) and (4.3.6) - (4.3.10) in
(4.5.4), the first four moments can be obtained as follows:
1J.1'(4) = 2( 2m+5)
m+n+5
(4.5.10)
2
1J.1(4) = 2(2m+5)(8m +8mn+64m+22n+111)
2
(m+n+5 5(m+n+6)(2m+2n+9)
(4.5.11)
,
where
A'34
22
· 2+2452mn
= 2(2m+5)(32m4+64m3n+528m3
+32m
n +712m2n+3076m2
+184mn
+7512m+264n2+2676n+6552)
and B'34 = (m+n+4)(m+n+5)(m+n+6)(m+n+7)(2m+2n+9)
(4.5.13)
and
where
,
94
A'44
=
8(2m+5)(128m7+3968m6+51008m5+352448m4+14l44l2m3+33002l2m2
+4l52207m+2l77532+5l2m6n+13056m5n+133824m4n
+706084m3n+2025148m2n+2999798mn+1797351n+768m5n2
+15360m4n2+1l8072~n2+437088m2n2+782l56mn2+543282n2
+512m4n3+7424m3n3+38704m2n3+86884mn3+71292n3
+128m3n4+l152m2n4+344Bmn4+3432n4)
and B'44
=
(m+n+4)(m+n+5)(m+n+6)(m+n+7)(m+n+8)(2m+2n+7)(2m+2n+9)
( 2m+2n+l1)
The
central moments, however, are
by
far aimp1er and are given
below:
(4)
1J.2
IJ.
(4.5.16)
(4)
3
=
. 2(2m+5)(2n+5)(m+n+3)
2
.
(m+n+5) (m+n+6)( 2m+2n+9)
=..
8(n-m)( ?m+5)(2n+5)(m+n+l)(m+n+J)
(m+n+5)3(m+n+6)(m+n+7)(2m+2n+8)(2m+2n+9)
and
where
9,
A
44 •
24( 2m+5)( 2n+5Jf (m+n'4)(m+n+7J t2(m+n+5J2.(m+nJ3+12(m+DJ2
L.
+44(m.nJ+h9)(2m>5J<2m+5 J] +
(n-mJ2(2(m+nJh+17(m+n)3+20(m+nJ2-111(m+DJ-224}~
B = (m+n+4)(m+n+,)4(m+n+6)(m+n+7)(m+n+8)(2m+2n+7)(2m+2n+9)
44
(2m+2n+ll)
4.6 The moments of V(s)
From the results of the previous sections it is possible to
generalize the lower order moments for any value of s.
By a compari-
son of the results on central moments in the preceding sections one
can easily see that for s roots the first three central moments are as
follows:
(s)
(4.6.1)
!J.l
I.t~s)
:::
8+1,)
s ( m~2
=
m+n+8+1
8(mIS2l)(n~S;1)(2m+2n+s+2)
(m+n+s+l)2(m+n+s+2)(2m+2n+2S+1)
s+l)
(4.6.:3>.
,
5+1
2s(n~m )( m~ (n~)(2m+2n+2
and
) (2m+2n+s+2)
=
(m+n+8+1)3(m+n+s+2 )(m+n+s+3 )(2m+2n+2s) (2m+2n+2s+l)·
96
~4
however does not seem to occur in a form which w0uld suggest a suit-
able generalization.
From (4.6.1), (4.6,,2) and (4.6.3). one can retrace the steps
to get the general form for the raw' morr.ent.
=
Thus
'VT8
get, for example,
s(2m+s+l)(4m2s+4msn+6ms2+8ms+2s2n+2s3+5s2+2sn+3s+4n+2)
4(m+n+s+l) (m+n+s+2) (2m+2n+2s+l)
The moments obtained in this chapter will be used to study the distribution of V(s) in the next chapter:
CHAPTER V
ASYMPTOTIC AND APPROXIMATE DISTRIBUTIONS OF V( s )
5.1.
Introduction
An asymptotic form for the distribution of the sum of the roots
-7
has been given by T. W. Anderson r l
Hsu
-
L-ll_7.
following the method of p. L.
Anderson .has shown that V' (stnv(S) ;,s 'Ssymptoticallydistri-
buted as a chi-square.
An alternative derivation of the asymptotic
form as a garrnna function is given below which is believed to be simpler
than his derivation.
5.2
The asymptotic distribution of v(s) as a garrnna function.
Starti~ ~rom
the simultaneous distribution of the roots 9
as given in (2.2.1) it has been shown £13
.. ~ -
Wl'
2, ••• , s), the distribution of
7 that,
if '¥ i=ne. (i
J.
1
• •
= 1,
••• , '¥ s can be obtained in the
form
S
...
(5.2.1)
- 1:
S
p(
W-
l
, ... ,
W-
s
)
::0
const.
"IT
i=l
W.m e i-I
-
'¥ J..
T(
i>j
J.
('¥. - W-.)
J.
J
(0 < '¥1 ~ W- 2 :; • • :; Ws < 00).
t
Then the moment generating function of V
(s)
s
::0
1: W.
. 1 J.
J.=
is given by
98
f crr
w
.
¢( t)=const.
(W S
00
J d1jr
o
d 1jr
S
S
2
0
1·.
s-
0
.. Z (l..t)V.
s m i=1
1
'If. )e
T{' ('If. -W.)d ""I"
'=1 1
• >j
1
J
1
1
NoW consider the transformation
y.= 1V.(I-t)
1
1
(i
= 1,
2, ... .,s) •
¢(t), mder the transformation (S.?3), takes the form
(5.?4)
¢(t)
00
s+1 {
(l_t)s(m~)
= ·Const.
1o
dy
s
Ys
(Y2 s .. .
{o dys-10. 0' .C1f1y. )me
s
E y.
1
1=
1
1=
,
and the distribution of V'(s) is given by
1
)('
'
1
'
J
(y. -y . )dYl
1
J
•
99
•
The a~ymptotic distribution (5.?6) can be obtained from the approximate distribution for v(s) suggested below, by making n very large.
5.3.
An approximation to the distribution of V(S)
The nature of the central moments given in sections
3-5
of the
last chapter indicates that the distribution of the sum of the roots can
be approximated by a beta function.
The following seems to be a suit.
able approximation.
. 8+1)
s(n"7 -1
s+l
s(m+T)-l
(v(s»
(l_V(s)/s)
=
S+l)
s
(
s~
•
100
It is well known that the first four central moments of u,
following the distribution
p(u)
ua-l(l_u!r)b-l
=:
~(a,b)ra
,
are given by
\.Ll
a
=:
r a+b
2
r ab
lk4
Hence
~l
and
~2
=>
3r4ab 2(a+b)2+ab(a+b-6)
(a+b) (a+b+l)(a+b+2)(a+b+3)
are given respectively by
•
101
(5.3.8)
and
If we set a
8+1)
= s ( m'7
and b =
8+1
s(n~)
and r
= s,
. the
we obtam
first four central moments of the right hand side of (5.3.1) which is
an approximation to the distribution
ofV(s)~
These central moments
might be denoted by ~i(A) (i=1,2,3,4) and the corresponding ~1'~2 by
~l (A) '~2(A)
1'1here A st~nds for approximation to what we would get from
the exact distribution. We give below the explicit forms of
~l(A)
and
~2(A) (since they will be repeatedly needed).
4(n_m)2 ~ s(m+n+s+l)+l}
and
It may be observed that the first moment is the same for the exact
and approximate distributions and the other exact lower order moments
102
tend to the respective approximate ones when m+n is large and s
For the exact distribution,
~l
sma]~.
is given by
where
A22=3 (m+n+4)( 2m+2n+5)
and B
22
=
f{
~
J(...n+2)(...n+5)
(2...2n+6)2+(2m+3) (2n+3) (m+n)
-16(m-n)2(m+n+~
(2m+3)(2n+3)(m+n+2)2(m+n+5)(m+n+6)(2m+2n+7).
where
A23=(2m+2n+7)(...
n+5)~n+3)(m+n+6)f2(m+n~4)2
.2
+(m+2)(n-..2)(l2(m+n)
.
and B
23
=
+88(m+n)+12~}
+(n~m)2{8(m+n)3+38(m+n)2-24(m+n)-134J=r
(m+2)(n+2)(2m~+2n+5)2(2m+2n+9)(m+n+3)(m+n+6)(m+n+7)
103
where
and
A
24
~ 6( m+n+6)( 2m+ 2n+9) t-+n+4 )(m+n+7)
t
2
2(m+n+5) +((m+n)3
+12(m+n)2+44(m+n)+4~(
2m+5)( 2n+5)]
' 2
+(n-m) 2{ 2(m+n) 4+17(m+n) 3
+20(m+n)
-UICm+n)-224D
and B
24
= (2m+5)(2n+5)(m+n+3)2(2m+2n+7)(2m+2n+ll)(m+n+4)(m+n+7)(m+n+8)
For m = n, it is interesting to observe the
as well as the approximate distributions"
seen from (5.3:11) that
that fO:: m = n,
butions.
~2
~l
~is)= 0
~2
from the exact
In this case it is easily
(5.3~9)
that
~l(A) = 0,
so
is the sa.me for the approximate and exact distri-
from the approximate distribution is given by
~2(A)
and
and from
~ts
= 3 {s(2m+s~1)+.~1
s( 2m+s+l)+.3
from the exact distribution, is given by the following:
3(m+2)2(4m+5)
(m+l) (m+3 )(4m+7)
,
104
8(3)
·2
e
2
(48m +l76m+133)(4m+7)(2m+5)
(4m+S)2(4m+8)(2m+7)
= 3(8m3+48m2+8Bm+5l)(2m+6) (4m+9)
(m+4)(2m+3)2(4m+7)(4m+ll)
It may be observed that
~2
and
•
simplifies considerably in the case of equal
values for m and n.
S.4
Comparison of the
~IS
from the exact and approximate distributions
From the moments as well as the Pi~s (i
= 1,2)
it is easy to·see
that the closest approximation to the exact distribution of V(s) is
obtained when m = n, in which case, as noticed already in the previous
section, PleA)
= 0 = ~is), and P2(A) and p~s) are also very close.
This latter result is checked by comparing the approximate P (A)I S in
2
(S.3~lO) with the exact and general p~s)(~ = 2,3,4) given by (S.3.12)(5.3.14) and ~h~ particu:ar P2(A) in (~.3.15) with the particular
p~s) gl:vmby(S.3.l6 )-(S.3. 1B) for m = n. It may be observed that
general
when
p~s)
m~n,
there is a term with (m_n)2 as a factor
in the
Wh~ch vanishes
thus leading to a considerable simplification.
On the other hand, for a given m+n, the largest difference between the approximate
~i(A)
and
~is) (1.
1,2;
s = 2,3,4) is found
105
to occur when the difference between m and n is the largest.
If, for
a given m+n either m or n= -~, then we get the maximum difference
between the exact and approximate distributions as judged by the
~IS
alone.
Thus for a given m+n, when m=n we have the smallest difference
~l's
(no difference at all) between the exact and approximate
very small difference between the exact and the approximate
~Js
= 2,
in this case are shown in Table 5.4.1 for s
10, 20, 30, 40, 50 and 60.
and
~~s).
~l(A)ls
and
~2(A)ts and ~~B) are compar;d
and
~~s)
are compared
S~4~3 tor
It may be ob-
~l(A) and ~iB) or be-
served that so far as the difference between
~2(A)
~~s)
in Table
s = 2, 3, 4 and m+n=lO, 20, 30, 40, 50, 60 and 100.
tween
3, 4 and n+m =
~l (A) and ~i s ), and also between
For the case mFO, the
in Table 5.4.2 and
The
~2Js.
Again, given m+n, when m or n= - ~ we
ha i.e the lar ge~t difference between
~2(A)
and a
is concerned, it is more interesting.and signi-
ficant to study how these differences behave under variation of n-m
for a given n+m, rather than under
~riation
of n+m for a given n-m.
This is because, apart from a factor, n+m is practically of the order
of the total number of observations (this may be easily checked, for
example, in the multivariate analysis of variance s.tt.uation) and it is
rela tively of less interest to verify as we
Cal
do from the tables that
in both the cases m=n or m=O, t he agreement between
between 13 2(A) and
~l (A)
and
~~s)
or
~~ s) gets closer as ·.it should, with increase of n+m.
106
Table 5.4.1
Values of approximate and exact
~2
when m=n and 8=2,3,4
p2
s~
s~
n+m or
just 2n ~~A) ~~2) AlE
P2(A) ~~3)
AlE P2(A) ~~4) AlE
or 2m
10 2.7931 2.8357 .9850 2.8667 2.9088 .9855 2.9048 2.9424 .9872
20
2.8776 2.8924 .9949 2.9200 2.9354 .9947 2.9418 2.9563 .9951
30
2.9130 2.9206 .9974 2.9429 2.9508 .9973 2.9580 2.9657 .9974
40
2.9326 2.9371 .9985 2.9556 2.9604 .9984 2.9672 2.9719 .9984
50
2.9450 2.9480 .9990 2.9636 2.9669 .9989 2.9731 2.9763 .9989
60
2.9535 2.9556 .9993 2.9692 2.9716 .9992 2.9772 2.9795 .9992
Table 5.4.1 shows
tha~
@2isnearlythe sane in the two cases i. e.
approximate and exact.
A
comparison of the values of
~1
from the approximate and
exact distributions in the nearly most unfavorable
situation,i.~
for a given m+n, m=O, can be made from Table 5.4.2 and a similar
comparison for
~2
from Table 5.4.3.
when,
107
Table 5.4.2
Values of approximate and exact
n+m
or
just n
6=2
B1 (A)
f3
1
when m=O and 5=2,3 and
6=3
~i2)
A/E B1 (A)
~i3)
4
6=4
AlE ~1 (A) ~i4)
AlE
10
.7986 .6315 1.265 .3702 .2517 1.471 .2031 .1220 1.665
20
1.0121 .8877 1.140 .4848 .3903 1.242 .2761 .2068 1.335
30
1.1040 1.0078 1.095 .5357 .4606 1.163 .3098 .2532 1.224
40
1.1550 1.0771 1.072 .5643 .5026 1.223 .3291 .2818 1.168
So
1.1875 1.1221 1.058 .5827 .5304 1.099 .3417 .3011 1.135
60
1.2100 1.1537 1.049 .5955 .5502 1.082 .3504 .3151 1.112
100
1.2570 1.2209 1.030 .6225 .5932 1.049 .3690 .3457 1.067
.
-
-
.
Table 5.4.3
Value s of (approximate and exact ) ~ 2 when rrll"O and 6=2,3 and 4
s=2
8 (A)
2
~~2)
6=3
AlE
~2(A) ~~3)
8=4
AlE
~2(A) ~~4)
AlE
10
3.9497 3.6706 1.076 3.4096 3.2206 1.059 3.2046 3.0826 1.040
20
4.3647 4.1372 1.055 3.6374 3.4674 1.049 3.3519 3.2296 1.038
30
4.5450 4.3639 1.042 3.7387 3.5986 1.039 3.4195 3.3150 1.032
40
4.6457 4.4967 1.033 3.7958 3.6785 1.032 3.4582 3.3690 1.027
SO
4.7099
4.5837
1.028 3.8324 3.7320 1.027 3.4833
3.4060
1.023
.
...
...
.
.
60
4.7544 4.6452 1.024 3.8579 3.7704 1. 023 3.5008 3.4328 1.020
100
4.8478 4.7770 1.015 3.9117 3.8540 1.01503.5380 3.4925 1.013
108
It may be observed from Tables 5.4.2 and 5.4.3 that, unlike
the case of
~l's,
approximate
~2's
5,,5.
for
~21S
the difference between the exact and the
does not increase with increase of s.
Upper bound to the error of approximation
-
<----
As stated in the previous section,for a given n+m,
t~e
mnximum
error of approximation occurs when m or n=~. For 8=2 and 3, and m=O
I
(which are really very special cases), Nanda L-14_7 has obtained the
probability integral for v(s) and the expressions are given below:
and
(5.5.2)
-11.2
+
n+3 3-z 3n+6 (.
)
(
}
(2")
lI2/(3_ Z )(2n+4,n+3 -I(2-z)/O-z) 2n+4,n+ 3
)J
~(2n+3) 11_(1_z/2)2n+5l
(2n+5)
l
j
109
(0 ~ z ~ 1)
,
- 2n+3(3T Z )3n+6 A { 1-12_3 (2n+4,n+3) ]
3-z
+ t~:'W
(1 -
(l_Zj2)2n+,)
+-l...
n+3
----
and
(5.5.$)
:=
C
(1 ~ z ~ 2)
,
110
where A=~(2n+4,n+3)
and C= (n+2}(n+3)(2n+5)
•
Now from the approximate dJi.stribution (5.3.1), the probability integral can be obtained as an incomplete beta function given by
I z/ s
lf s (8+1)
~ ,
S+l
2 )J
s(n
l)
•
Hence for s=2 the upper bound (under variations of n-m for a
given n+m) to the error of approximation (ignoring the case~m=- -~)
is .between
•
I z / s (3, 2n+3)
and the respective expressions given in (5.5.1) and (5.5.2) for
different ranges of values of Z; while for 6=3 the upper bound (ignoring tho'oase ~-~) is" given by the difference between
and the respective expressions given in (5.5.3) for 0 :: z :: 1, in
111
-
- ...
(5.5.4) for 1 < z < 2 and in (5.5.5), 2 ...< z < 3.
It is interesting to compare, for m=O, s=2,3 and different
values of n" the c.d.t's of the exact and approximate distributions
over different z t S and also the 5 % points of the same distributions.
The following tables supply part of this information.
Table 5.5.1
Values of the exact and the approximate c.d.f.ts and of
their upper 5
0/0 points for m=0 and s=2
Upper 5 0/0
significance
_.4
,I
.2
.3
1eye1
App.
Exaot App .. Exact App• Exact A.PP. Exact ApE' Exac.t
10 0.1271 .1192 .4629 .4534 •7463 .7471 .9018 .9084 .462 .451
n
z
-
15 .2542
.2466 .6937 .6924 .9130 .9174 .9810 .9840 .338 .332
20 .3923
.3865
.8410 .8432
.9735 .9762
.9968 .9976 .268 .264
25 .5232
.5195 .9226 .9253
.9925 .9937
.9995 .9997
30 .6370
.6353 .9640 .9661 .9980 .9984 .9999
.9999
.220 .218
.187 .186
112
Table 5.5.2
Values of the exact and approximate c.d.f.ls for m=O, s=)
n=10
n=20
.6
.2
z
.3
Appr.
.856
.993
.726
Exact
.868
.996
.731
z
Since the exact c.d.f. for
~-o
is not available for values of s
above 3,a method of comparison of the exact and approximate c.d.f.ls
based on the table of percentage points of Pearsonian distributions for
different values of
~1
and
~2
given by E. S. Pearson and Maxine
Merrington ~16_7, seems to be of great advantage.
The validity of
this method can be jUdged to some extent by a comparison of the upper
5
%
point of the exact distribution with that of Pearsonian dis-
tribution having the same
~1
and
~2'
It may be observed that the
approximate distribution itself is Pea:' sonian so that this comparison
does not have much significance.
113
Table 5.5.3
Upper 5
0;0 points (for m=O) from the exact and approximate
c.d.f.'s and of the Pearsonian distributions with
the same
~1
and
~2
(as the exact and approximate distributions)
n=10,s=2
na lO,s"'3
Distribution
~'s
s=3,n=:40 6=4,n=,O
Distribution
~IS
~IS
~ts
Approximate
.463
.
.463
.718
.718
.236
.282
Exact
•451
.451
.697
.697
.234
.280
Table 5.5.3 suggests that the second method of comparison might
be reasonably reliable.
From the last column of Table 5.5.1 it is evi-
dent that, for 6=2, the error becomes negligible when n=3 0, and from
the last two col'UllUls of Table 5.5.3 that, for s=3 the error becomes
negligible when n reaches 40, and, for 6=4, when n reaches ,0.
This,
in fact, is the stage where the approximate and exact standard denations almost agree in the second significant figure, as will be ob..
served from the following table.
111.~
Table 5.5.4
Standard deviations of the approxima te and exact distItibution;;;
for m=O
n=30
5==3
n=40
s==4
n=50
s=5
n=60
s=6
n=70
5=7
n=80
s==lO s=10
n=lOO n=110
Approximate
.0509
.0542
.0561
.0572
.0581
.0587
.0651
.0597
Exact
.05°1
.0532
.0549
.0560
.0568
.0574
.0635
.0585
s=2
At the stage where the maximum error becomes negligible, the ratio of
-r,he approximate to the exact f:3 fa become
l.~, correct to the second
1
decimal as may be observed from Table 5.4.2. Also the agreement be-
tween
t~e
creases.
corresponding
~2rs
does not become less close as s in-
Hence the approximate distribution will give sufficiently
accurate values for upper percentage points when the standard deviations almost agree in the second significant figure and the ratio AjE
of the
~l's
is 1.1. Since we'have.the exact expression for
~1
.for
any value of s, it is possible to find out at what stage the approxi.fnate distribution is likely to give good results, as judged by the
standard deviations and
~l'~
alone, assuming the role of
~2
to be
negligible for this purpose.
The follovJing table gives the compariRon of
mate and the exact distribution.
~1
from
t·bf approxi~·
115
Table 5.5.5
~1
Numerical values of
from the approximate and exact distributions
for m=O
s=2
n=30
8=3
n=40
s=4
n=50
8=5
n=60
s=6
n=70
s=7
n=80
s=lO
8=10
n=lOO n=110
Approximate
1.1040
.5643
.3417
.2288
.1638
.1231
.0619
.0628
Exact
1.0078
.5026
.3011
.2006
.1432
.1073
.0531
.0546
Ratio (A/E)
1.0954 1.1229 1.1346 1.1407 1.1443 1.1466 1.1655 1.1502
From the above table it may be seen that the approximate distribution
can be used to obtain the upper 5
0/0
points for 8=2, when m+n
~
30
(or even> 20 for two decimal accuracy); for s=3, when m+n > 40, (or
-
~ 30 for two decimal accuracy); for s=
4, when
-
m+n .:: 50, for 8=5, when
m+n ~ 60; for 8=6,wmen m-n ~ 70; for s=7, when m+n > 80 and for 8=10,
when m+n > 100.
Consider now a situation not covered by the foregoing
paragrcp~
for example, s=4, m+n=30 (say), and m differs rather considerably from
n.
For this situation it is better not to use the approximate distri-
bution but to calculate the
then use the Table of E.
the percentage point.
s.
~l
and
~2
from the exact distribution and
Pearson and Maxine Merrington to obtain
This method should take care of all cases under
8=2, 3, 4, which have not been covered by the previous paragraph.
CHAPTER VI
THE MOMENTS OF THE DISTRIBUTION OF U(S)
6.1 Introduction
In Chapter IV towards the end of the introduction it is stated
that the statistic suggested by Hotel1ing, viz,
(6.1.1)
2
s
(s)
To = n . E
1. 1
= nU '
-.
i=l
where li's are the characteristic roots of ~ (S~lS*) defined in
section 6.1, will be studied in this chapter.
The moments of this
statistic can be obtained by methods stmilar to those developed in
Chapter IV using (1.8.12).
This chapter will be mainly devoted to the
problem of finding the moments of
T~ or rather of U(s).
6.2 The Distribution of lIs
From the distribution of e's given in (2.2.1), the simultaneous
distribution of the lIS can be obtained by the follOWing transformation.
(6.2.1)
(i
= 1,
2, ••• s)
•
117
Thus the distribution of the X's is given by
}\ (X.-X.)
s
"IT
i=1
(I+X.)n
t
i >j
~
J
~
If
(6.2.3) where K(s,m,n') - }\s/2
f(2n'-s+i)
1=1
2
it r(2m+i+1) r(2n'-~-2s+i-1)
i=l
and n' = m+n+s+1.
2
2
For simplicity of notation n J may again be referred
to as n, since this would not make any confusion here.
6.3
The
mom~nts
of u(s), in general
Using the result given in (1.8.12) to evaluate the expected
value of e
-tu( s)
with the help of the distribution (6.2.2), we get
(c.f. section 4.2)
(6.3.1)
(n-m-s+t)E e[
{
_U(s)
tu(6) \
J+tE U~:~ e',' .m,n }
118
2K(
) 8-1
. 1
tU(s-2)
• ( s,m,n) . z~lf-~- I(2m+s+i-2,2n-l,-2t)E (ZA1 •• As-~. l)eK 8 - 2,m,n ~=l
where ~Al •• \s-i~l sta~ds for the (s_i_l)th elementary symmetric
function in
~ ••A8_2
•
6.4 The moments o~ u(2)
Putting s=2 in (6.3.1) and working on the same lines as in
section 4.3, we get the following central moments:
1J.(2) _ 2m+J
1 - n-m-3
1J.~2)
(6.4.2)
(6.4.3)
(6.4.4)
(2)
1J.3
=
=
,
(2m+3)(2n-J)(n-m-2)
(n-m-3)2(n-m-4)(2n-2m-5)
2(2m+3)(2n-3)(n+m)(n-m-l)
J
(n-m-3) (n-m-4)(n-m-5)(2n-2m-5)
where
,
and
,
119
C42 = J(2m+J)(2n-J) [{(2n-2m-6)2+(2m+ J )(2n-J)(n-m)} (n-m-2)(n-m-,)
+16(n+m)2(n-m-J~
and D42
= (n-m-3)4(n-m-4)(n-m-5)(n-m-6)(2n-2m-5)(2n-2m-7)
6.5 The moments of U(3)
In a similar manner the central moments for u(J) are obtained
as follows:
!J. (3)
(6.5.1)
1
!J. ( 3 )
(6.5.2)
2
(3)
!J.
3
and
=
=
=
3(m+2)
n-m-4
3(~2)(n-2)(2n-2m-5)
(n-m-4)2(n-m-5)(2n-2m-7)
12(n+m)(m+2)(n-2)(n-m-l)(2n-2m-5)
(n-m-4)3(n-m-5)(n-m-6)(2n-2m-6)(2n-2m-7)
where
(6.5.4)
C
43
= 9(m+2)(n-2) f(n-m- 3)(n-m-6)l-2(n-m- 4)2.
t:
'\.
+(m+2)( n-2) (12( n_m)2-88 (n-m)+12,)
.1
120
+(n+m)2
{8(n-m)J-J8(n-m)2-24(n-m)+lJ~
and D = (n-m-3)(n-m-4)4(n-m-5)(n-m-6)(n-m-7)(2n-2m-7)(2n-2m-9)
43
6.6 The moments of U(4)
The central moments of U(4) are obtained in a similar manner
and are given below:
,
,
(6.6.2)
(6.6.)
and
(6.6.4)
(4)
IJ.)
8(n+m)(2m+5)(2n-5)(n-m-l)(n-m-)
=
3
(n-m-5) (n-m-6)(n-m-7)(2n-2m-8)(2n-2m-9)
1J.(4)
4
_ C44
-
l544
where
l
44-24(2m+5) (2n-5l (n-m-4)(n-m-7)[2(n-m-5)2
C
+(2m+5)(2n-5)(Cn•m»)-12cn-m)2
+44(n-m)-49)]
+(n+m)2{2(n-m)4-17(n-mlJ+20(n-m)2+111(n-m}-2~
and D =(n-m-4)(n-m-5)4(n-m-6)(n-m-7)(n-m-8)(2n-2m-7)(2n-2m-9)
44
(2n-2m-ll)
"-
121
6.7 The moments of U(s)
The nat ure of the first three moments given in the previous
sections suggests the following general formst
s(m..S;l)
()
S
lJ.1
=
n-m-s-l
(6.7.1)
s+l
lJ.~s)
=
,
s+l)
s(m'7)~"2 (2n-2m-s-2)
(n-m-s-l)2(n-m-s-2)(2n-2m-2s-1)
and
lJ. ( s) ...
3
~ne
.
s+l (n s+l)
2s ( n+m )( m+-)
- - (2n-2m-2)(2n-2m-s-2)
2
2
(n-m-s-l)3(n-m-s-2)(n-m-s-3)(2n-2m-2s)(2n-2m-2s-~
can retrace the steps to get the raw moments.
For example, the
second raw moment is given by
I(S)
lJ.2
~
C'28
-,-D 29
where
(6.7.4)
e'2SoS(2m+s+1) (2s2n+2sn+4rnsn+4n-2s3.6s2m-4sm2·5s2-8ms-3s-2)
and D'2s= 4(n-m-s-l)(n-m-s-2)(2n-?m-2s-1)
•.
122
It may be observed that the raw moments of u(s) can be obtained
from those of V(s) in a simple way.
In the numerator of any raw
moment of V(s) if one keeps the coefficient of the highest power of
n positive and put alternatively positive and negative signs for
successive powers of n one obtains the numerator of the corresponding
raw moment of U(s).
The denominator of any raw moment of U(s) also
can be obtained in the same way from that of the oorresponding moment
of v(s), so that the linear factors in the denominator have positive
coefficients for terms involving n and negative coefficients for the
other terms.
The same is true for the central moments also.
CHAPTER VII
AN APPROXIMATION TO THE DISTRIBUTION OF U( s)
7.1
Introduction
The moments of U(8) obtained in the last chapter suggests the
possibility of an approximation to the distribution of U(s).This
chapter is concerned with the approximate distribution and a discussion
of the error of approximation.
Approximation to the distribution of U(s)
While the approximation suggested for V(s) is a beta distribution, the one for U(s) seoms to be an F distribution of the form
. ( 7.2.1), ~f
. we put s ( m"7
s+l) -l=a and s ( n-T
s+l) +1=b, we get
In
124
The first four central moments of U(s) from (7.2.2) are given below:
IJ.1
•
s(a+1)
b-a-2
,
,
(7.2.5)
and
Hence
B
.1
=
4(a+b)2(b-a-3)
(a+1)(b-l)(b-a~4)2
3(b-a-3) {2(b-a-2)2+(a+l)(b-ll(b-a+4)}
(7.2.8) and
~2
=
(a+1)(b-l)(b-a-4)(b-a-5)
125
In the expressions for the lJ.' s given above as vrell as
for the
~I
~IS
for the distribution (7.2.1) which is suggest-
ed as an approximation to the exact distribution of U(s).
moments and
in
~he
those
. we replace a by S( ~
S+l) -1 and b by s(n-T
s+l) +1, we
s, ~f
get the moments and
lJ.i(A)(i
in
= i,
~IS
These
of the approximate distribution will be denoted by
2, 3, 4) and ~i(A)(i
= 1,2).
It may be observed that, as
case of v(s), the first moment is the same for the exact and
the approximate distribution of U(s) and the other exact moments tend
to the respective approximate ones when n-m is large and s small.
In explicit forms, we have
~l(A)
= .
4(n+m)2 [s(n-m-S-l)-l}
.
s+1
s+1)
(m~)(n-~
fts(n-m-S-l )-2 )
J
2
and
It is interesting to compare the approximate
ones given below:
SiS
with the exact
126
(s)
(7.2.11) e1 D
2
)2
('.
4(n+m) (2n-2m-2 (n-m-s-2) 2n-2m-2s-1)
8+1 ( 8+1)
()2
2 '
s(m~) n-~ (2n-2m-s-2) n-m-s-3 (2n-2m-28)
(7.2.12)
where
C22a)(n-m-4)(2n-~5)~-m-2)(n-m-5){(2n-2m-6)2+(2m+)(2n-)(n-m)}
+16(n+m)2(n-m-~
and D22 = (2m+3)(2n-3)(n-m-2)2(n-m-5)(n-m-6)(2n-2m-7)
C2)a(n-m-5) (2n-2m-7) t-m-)( n-m-6){-2( n-m-4) 2
+(m+2)(n-2) (12(n-m)2-88(n-m)+125)J
+(n+m)2(8(n-m)~)8(n-m)2-24(n-m)+1)4J1
and D23 = (m+2)(n-2)(2n-2m-5)2(2n-2m-9)(n-m-J)(n-m-6)(n-m-7)
and
where
127
I
24=6( n-m-6)( 2n-2m-9) (n-m-4 )(n-m-7)
C
f n-m-5 t
.
2(
+(2m+5)(2n-5)(n-m)3-12(n-m)2
.
+44(n-m)-49)]
+(n+m)2l2(n-m)4-17(n-m)J+20(n-m)2+111(n-m)-22~
and D24=(2m+5)(2n-5)(n-m~3)
2
(2m-2m-7)(2n-2m-ll)(n-m-4)(n-m-7)(n-m-B)
7.3 Comparis on of the aEproxima te and exact cUImllative distribution
functions for s=2
Hotelling ~9_7 has obtained the exact cumulative distribution
function of U(2) in terms of incomplete beta functions.
The exact ex-
pression is given by
1
(7.3.1)
n-m-2
(~)
P=I (2m+2,2n-2m-3)- (2n-2)1
.
w
2(2m+1)1 (2n-2m-4)J
where
B.
2(m+1,n-m-1)
w
•
The <¥,proximate cumulative distribution function (e.f. (7.2.1) ) is
given by
Iw(2m+3,2n-2m-5)
•
,
128
The accuracy of the approximate
cumul~ive
distribution function can be
judged from Table 7.3.1.
Table 7.3.1
Values of approximate and exact probabilities for u(2)
for different values of w, m and n
w=.20
m= 4
n= 30
w=.10
m"" 0
n= 25
w=.20
m= 0
n= 10
w=.50
m= 5
n= 15
Approximate
0.6051
.8617
.6904
.6494
Exact
0.6069
.8600
.6970
.6551
Probability
From Table 7.3.1 it can be seen that when n-m is 25, the error
is the approximate probability is below two units in the third decimal
place.
Consequently, for practical purposes the approximate distri-
bution can be used for n-m above 30.
It is interesting to compare the
respective 8's for the approximate and exact distributions.
7.3.2 gives the values of
~l
Table
from the approximate and exact distribution.
129
Table 7.3.2
Values of the approximate and exact
~1
for m=O and 5 and for
different values of n
m=O
n
A
m=5
~1 (A)
8(2)
·1
AlE
~l(A)
-I
,/2)
A/E
15
2.1120
2.7259
.7748
1.6461
2.5962
.6340
20
1.8581
2.2207
.8367
.9880
1.2751
.7748
30
1.6506
1.8454
.8944
.6448
.7396
.8718
40
1.5606
1.6927
.9220
.5305
.5828
.9101
60
1.4783
1.5583
.9487
.4400
.4662
.9439
100
1.4174
1.4619
.9695
.3805
.3932
.9679
comparison of
~2
•s
for s=2 can be made from Table 7.3.3.
130
Table 7.3.3
Values of
~2
(approximate and exact) for different values of m and n
m=O
n
~2(A)
~~s)
m=15
m=5
AlE
~2(A)
~~s)
15 6.6046 8.0365 .8218 6.2391 8.9238
AlE
~2(A)
~~s)
A/E
.6991
20 6.0706 6.8777
.8827 4.8382 5.5541 .8711
30 5.6421 6.0588
.9312 4.1371 4.3568
.9496
4.5720 4.7034
.9721
40 5.4584 5.7359
.9516 3.9011 4.0244 .9709
3.8598 3.8892
.9924
60 5.2915 5.4566 .9697 3,7259 3.7827
.9850
3.5110 ).5179
.9980
100 5.1686 5.2594 .9827 3.6071 3.6337
.9927
3.3424 3.3441
.9995
It will be seen that the error of approximation, i.e., (E-A)/E
changes with change of m or n, and that, as m or n increases, the
for
~2'
is
in
a more favorable direction than for
approximation depends more on
exact
PI
PI
PI'
chang~
The error of
and the ratio of the approximate to the
can be shown to be a function of n-mfor fixed value of s,
we set n-m=h, we have
If
131
which is a function of h for constant s, and hence, for this ratio to
be near unity,
h=n~m
has to be large.
But if m is moderately large,
n has to be comparatively larger which seems to indicate that for the
goodness of approximation m ~is tbe most favorable case, for a givan n.
D
Also, for a given n, the
~2
:"
from the approximate"distribution stays
close to that from the exact distribution even with increase in m.
most unfavorable situation occurs when m is closest to n.
The
Since n is
a simple function of the total number of observations, it is more interesting to observe what happens as m varies (for a fixed n) than
what happens as n varies (for a fixed m).
The closer agreement that
we get by increasing n(for a given m) is a result which Was only to
be expected and would thus seem to be relatively trivial.
From Tables 7.3.1 and 7.3.2 one can expect that, when the ratio
~l(A)/~is) is correct to the first decimal place, we can use the
approximate distribution without any great fear of error.
desirable to compare the f3 IS for s=3 and
Hence it is
4. A comparison, for the
case m=O is sufficient to show the accuracy, because the ratio depends
on just n-m (except for s).
for 8=3 and
4.
Table 7.3.4 gives the values of the ~lfs
132
Table 7.3 .. 4
Values of ~l (approximate and exact) for m=O and s=3 and
4 and for different values of n
s=3
n
~l(A)
~i3)
20
.9872
30
s=4
AlE
~l (A)
~i4)
AlE
1.3353
.7393
.6414
.9738
.6587
.8570
1.0339
.8289
.5398
.6978
.7735
40
.8019
.9191
.8726
.4983
.6000
.8305
50
.7716
.8588
.8985
.4758
.5503
.8645
60
.7523
.8217
.9156
.4616
.5203
.8872
100
.7161
.7541
.9496
.4353
&4669
.9324
Another way of comparison is in terms of the standard deviation and it
appears that the approximate distribution can be used when the standard
deviations of the approximate qnd the exact distribution more or less
agree in the second significant figure.
Table 7.3.5 gives the standard
deviations for different values of s, which would indicate the smallest
value of n-m, or just n (if m=O), for which the approximate distribution can be used with reasonable safety.
133
Table 7.3.5
Values of the approximate and exact standard deviations for m=O and for different values of s
deviations
s=:6
13=5
s=7
n=60
n=80
n=70
Sta~dar.d
s=2
n=30
n=40
8=4
n=50
Approximate
.06653
.07023
.07240
.07382
.07483
.07558
.07700
Exact
.06777
.07186
.07421
.07573
.07680
.07759
.°7908
8";3
s=l~
n=110
For each value of s, the approximate and exact standard deviations
show a difference of 2 in the second significant figure.
Hence for
an increase of unity in s, n-m has to increase by 10 so as to guarantee
sufficient accuracy.
For a value of s as large as 10, the approximate
distribution can be used for values of n-m slightly above 100.
7.4
General behavior of V(s) and U(s)
Frem Chapters IV-VI and the previous sections of this chapter
it appears that V(s) and u(s) behave somewhat like duals.
The moments
of the one can be derived from those of the other, and the approximate
distributions alse show a certain duality.
To bring out a kind of
duality between the two, the same letters m and n have been used to
denote parameters in both cases, even when n has different meanings
134
in the two cases.
For the exact distribution to be close to the approxi-
mate ,in the case of V( s )" m+n has to be large, and in the case of U( s)
n-m has to be large.
The error of approximation for V(s) is the least when m=n and
that for U(s) is least when n-m is the largest (under the appropriate
defini tion of m and n in the two cases).
If now we remember that in
the case of the hypothesis of equality of two population dispersion
matrices, thaOElSe.af'm=n (for v(S»
is a common occurrence and in the
analysis of variance situation a large value of n-m (tor u(S»
is a
corrnnon occurrence, then it would appear that, from the point of view
of approximations,
V<4) should be more suitable for the first situa-
tion while u(s) should be better for the second situation.
This is
with regard to the goodness of approximations for the c.d.f.fs of the
two statistics.
For a comparison of the performance of the two
statistios (in terms of exact c.d.f.'s) in different situations we
have, of course, to compare their powers against appropriate alternatives in the different situations.
This is under investigation, but
no results on this has been offered here.
1.35
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