.,Trude M. :.>::(
ON THE DISTRIBUTION OF QUADRATIC FORMS
by
JAMES PACRARES
Special report to the Office of Naval ReBe~ch
of work at Chapel Hill under Project !ffi 042 031
for research in probability and ~toxistics.
Institute of StatisticB
Mimeograph Seriee No. 75
July 15, 1953
i1
AC ICNOHLEDGMENT
The writer wishes to express his
thanks to
Prcfe~sors
Harold Rotelling
end S. N. Roy fer their valuable suggestions and criticisms during the
preparation of this work.
Acknowledgment is also
Office of
ai.d.
N~val R~search
~ade
to the
for financial
1ii
TABLE OF CONTENrS
Page
11
INTRODUCTION
1v
Che,pter
I.
II.
III.
IV.
THE DISTFIBUTICN OF A DEFINITE QUADRATIC
FOR,\1 IN INDEPEIIDENT CENTRAL NOBMAL VARIATES
1
1.1 The Problem
1
1.2 The Solution
1
1.3 An Appllca,tion:
The Distribution of e.
Sum of Squares of Depend.ent Variates
7
1.4 Rotelling's Method of La.guerre Polynomials
10
THE DISTRIBUTION OF A DEFINITE QUADBATIC FO:RM
IN INDEPENDENT NON-CENTRAL NORMAL VARIATES
13
2.1 The Problem
13
2.2 The Solution
14
2.3
24
An Application:
The Power Function of
the Chi-Square Statistio
TEE DISTRIBUTION OF AN INDEFINITE QUADRATIC
FORM IN INDEPENDENt' CENTBAL NORMAL VARIATES
26
3.1 Special Ca.ee 1
26
3.2 Special Case 2
47
FUBTlIER BOUNDS ON THE C. D.F. OF A DEFINITE
QUADRATIC FORM IN INtEPENIlEN'l' IDENTICALLY
DISTRIBUTED VARIATES - EACR CENTRAL NOBMAL
OB MOBE GENEBAL
49
4.1 The Norma.l Case
49
4.2 The Genera.l Case
51
BIBLIOGRAPHY
54
INTRODUCTION
One could easily
j~stif.y
t.he study of the distribution of
quadratic for-ms from the standpoint that many of the tests in
statistics are based on the distributions of quantities which can
be thought of as opecial cases of either a q'ladratic form or functions of
~2~iratic
mention;
however,
(i)
forms.
~e
The applications are too
nume~ous
shell list a few as illustrations.
The distribution of a definite quadratic
form where the components have a multivariete normal distribution.
(ii)
The problem of finding the power function of the chi-square statistic, for
large samples,
C3n
be reduced to that of
finding the distribution of a positive
definite quadratic form in non-central
normal variates.
(iii)The distribution of a form of the serial
corre},a;l;ion coeff;.cient ca.n be expressed
in terms of the distribution of a ratio
of two q,uadratic forms.
See Anderson
["1_71
(iv) Of special importance is von Neumann's
J'tat:;stic, L.~.9_7,
1.
["20_7,
the ratio of
Numbers in square brackets refer to bibliography.
to
v
the mean square successive difference
to the
nsed to test weather
vari~nce,
obserYs,tions are independ.ent or whether
a trend exists.
(v)
Durbir- a~d Fatson
[6_7) use a similar
statistic to teet the error terms for
independence in least squares regression.
(vi) KoopmA.ns
£12_7,
says, "Assuming a normal
distr:bution for the
ran~om
math~~t1cal prere~uisite
disturbance, the
for an estimation
theory of stochastic processes is the study
of the joint
d1st~ibutions
forms in normal variables".
of certain quadratic
The problem
Koopmenscons1ders is that o' estimating the
serial correlation in a stationary stochastic
process.
(Vii) To
tE'!6G
hYTo'cheses concerning variance
co~por.ents
in the analysis of variance, we
require the distribution of an indefinite
quadratic form.
(Viii) Hotelling ~9-7, shows how the distribution
of the ratio of an indefin1te quadratic form
1n non-central normal variate. to a definite
quadratic form could be used in the theory ot
select1ng variates tor use in prediction.
vi
(ix) McCarthy ~1'_7, shows how the distribution
ot the ratio of two detinite quadratic torms
could be used to make an F test in the
analysis ot variance when the assumptions
ot equal variances and independence ot the
observations are not met.
It would make this report quite lengthy to discuss the
distributions ot all these tunctions ot quadratic torms. We
shall restrict ourselves to the study of a definite quadratic
torm in both central and non-central independent normal variates,
giving an important application tor each distribution.
Then we
shall discuss two special cases of an indefinite quadratic for.m.
Finally, we shall discuss a few inequalities.
We give below a
slightly more detailed chapter-wise breakdown.
In Chapter I we shall be concerned with the distribution of
a detinite quadratic torm in independent N(O,l) variates.
Robbins ~16J, has treated this problem but we have carried it a
bit turther.
Robbins and Pitman
L17_7,
have given an expression
for the distribution of a linear combination of chi-square variates.
We feel that we have improved on this torm.
We have derived an
expression which depends only on the value of the determinant of
the torm and on the moments of the inverse quadratic torm.
The
expression is an alternating series which converges absolutely
and is such that if we stop after any even power we have an upper
bound, and it we stop atter any odd power, a lower bound to the
vii
cumulative distribution function.
L-8_7,
Rotelling ['10_7 and Gurland.
have suggested the use of Laguerre polynomie,ls in finding
distributions of quadratic forms.
A brief account of Rotelling's
method will be given.
In Chapter II we have derived an expr.ession for the
distribution of a definite quadratic form in non-central
independent norma,l varia,tes which depends only on the value of
the determine,nt of the form and on the moments of the inverse
qUadra.tic form in normal variates with 1maginary mesns.
sta.tement will be IDS,de clearer later on.
This
This result enables us to
find the power function of the chi-square statistic.
In Chapter III we have discussed the distribution of
the difference between two independent chi-squares having different
numbers of degrees of freedom.
If the degrees of freedom are the
same, the distribution becomes the same as the distribution of
the sample covariance in sampling from a norma,l popula.tion.
We
have studied the properties of this distribution in some detail.
In Chapter IV we give some inequa.lit1es for the distribution
of a quadratic form in N(O, 1) varia.tes and also for the genera.l
case.
viii
NOTATION
All vectors are column vectors and primes indioa.te their
tra,nsposes.
"p.d.f." sta.nds for "proba.bl1ity density function";
"c.d.f. tI stands for "cUDlula.t1ve distribution function";
tlr. v. tI stands for "random vB.r1able";
"q.f." stands for "Que.d.rat1c form" ;
"N(J.1,a)" stands for "a r.v. ha.v1ng a. norma,l p,d,f. with
mean J.1 and standard deviation a".
CHAPTEB I
THE DISTRIBUTION OF A DEFINITE QUADBATIC FORM
IN INDEPENDENT CENTRAL NORMAL VARIATES
1.1 The problem.
Suppose we have a q.f.
~
1
=2
.
Y'A Y in Yl, ••• ,Yn , where
the Y are independent N(O, 1) variates, and where Y'
i
= (yl, ••• ,Yn ),
where the da.sh denotes the tranepoBo of the column vector Y.
Let Fn(t)
= Pr(~~
t).
Then the problem is to find Fn(t).
It
is well known tha.t we can make an orthogona.l transformation,
Y
=P
X,
sa~, where P P' = P'P = I, such that ~ Y'A Y = ~ X'P'APX =
elements a'l"" ,an in its main diagona.l, and where e·l , ••• ,an a,re
the latent roots of the matrix A.
Under such e.
Xl"'. ,Xn remain independent N(O, 1).
transforma~ion,
So the problem is now to
1.2 The solution.
Theorem 1.1.
independent N(O, 1), and where a i > 0, iel"'f,n.
*
Let ~=
1 n -1 2
E 81 Xi'
i=l
'2
Then,
E(~)k
00
1:
r(~+k+l)
k=O
* k 18 the k-th moment of
where E(Qn)
(b)
*
~.
The series is absolutely convergent and therefore it
is convergent.
(0)
2s-2
For
~
two positive integers r and s end every t> 0,
2r-l
E d > F (t) > E d , where
k=O k
n
k=O k
E(~)k
r(~+k+l)
Proof.
We shall make use of the Dirichlet integral;
f-l
x/
J J J=l
n
...
11
where
-00
< xi <
dXj
00,
1
Ill:
end
fj ,
P j ere all positive, and the
OJ'
expand the exponentie.l in the integrand, we get
(t)
n
F
= (2,,)
00
dx
t
k=O
We need to evaluate integrals of the following type:
d!.
It we expand the integrand according to the multinomial theorem,
we get
n
21 j
TI
x"
J=l ~
dx"
tJ
4
We now make use of the Dirichlet integral stated earlier, where
The problem now ie to eva,lua,te this last Qxpress1on.
Recalling
5
=
2-~J
t
11 +,··+1n=k 1
I
l'
l' · .. n'
n
=
- "2
1(
kJ
t
11+••• +1 =k 1 I
l'
n
1"·· n'
So that
and
E(Q:>k
r(~+k+l>
This proves part (a) of the theorem.
To show absolute convergence we note that if'
a 1 > a 2 > ••• > a > 0, then
_
- n
6
r(~+k)
This proves part (b) of the theorem.
The bounds we obtain are based on the fact that if rands a.re
two positive integers> 1, then for z > 0,
2r~1 (_z)k
~
k=O
2s-2
( -z )k
~
LI
k1
>e-z >
k=O
k1
This proves part (c) of the theorem.
In the case where some of the latent roots are zero, i.e.
when the
fo~
is positive
ee.mi-~~fin1te
of rank r, say, we need
only replace n by r in the theorem and in the proof.
Remarks.
(i)
* E(~)
* k , are easy to obtain from the
The moments of ~1
*
cumulants of~.
* 1s kr(~)=
* (r-l):
n a -r •
The r-th cUZllulant of ~
2
E
i
1
7
From Kendall L-ll_7, we have expressions for the first ten moments
1n terms of the cumulants.
(ii)
for Fn(t).
Let 5r be the sum of the first r+l terms of the series
Then 50' 52' 54' ••• 1s a sequence of upper bounds end
51' 8 , 55' ••• ie e. se~uence of lo,,"er bounds to ll'n(t).
3
If E2k+2
is the error committed by stopping with 82k+1 , then
r=O,l, ••• ,kj k=O,l, ••• ,
r=1,2, ••• ,k; k=1,2, •••
The values of l.u.b. 52r end g.l.b. S2r+l depend on the values of
the latent roots.
We note
tha~ E2k+2~52k-
S2k+l : the last
term included, e,nd E2k+1 :: S2k - 82k- l = the last term included.
Hence, ths error is lees then the last term included end it is
positive if we take an odd number of terms and
nega~ive
if we
take an even number of terms.
(11i)
The above theorem seemB to be in severa.l
improvement over the method given in Robbins
1. 3 An e,ppl1ca.tion:
Tr._~ di~~t'ibution
WB,yS
an
£16_7.
of a. sum of squares in
deper-dent variates.
Suppose that Xl"",Xn have-a joint multivariate normal
distribution with zero means and cova.rience ma,trix equa.l to A- l •
8
Then the problem is, what is the distribution of
Make en orthogonal
X = L Y,
tranefo~at1on,
the la.tent roots of the ma.trix A.
l
L.-i5
'-
e~
1 n
2
2
t Xi?
1
where L L'
Now
= L'L
~
I.
Now make the transformation
J... J
-1 2
.. -
Exp
7
- J:
2
ZtZ
dz
-
E 8'1 Zi< t,
= PrL- ~ ~ ~.7,
end we cen make use of
theorem 1.1.
Remark.
Combining the results of theorem 1.1 end the above
a~plicat1on
it 1s easy to show that we could find the distribution of a definite
q.f., X'A X, 1n Xl' ••• ,
xn where Xl'
••• , In have a. multivariate
normal distribution with covariance matrix B- 1 , end this distribution
involves as parameters the chara,cter1atic roota of AB" l •
9
We shall now state, without proof, an obvious corollary
to theorem 1.1,
o~tained
by letting the first
~
latent roots be
Corolla.rz 1.1
Let Sr
=-:L-:; ~ e 1
t:.
" + ... + e,
X ~.~
r
-'1
X;r)'
Wflere
the
~i
are
independent rove 'e having e. oentral chi-sg,uB,:r.e dietribution with
mi
d~groes
of freedom.
IJst 8* = -;.',1(-1
e,.J..
J..~
X 2 +, , ,+
~
.J..
-1 x2
)
.....
6,
r
='
r
r
M = I:1 mi , A._~ > 0, Gr (t) = Pr(Sr_
< t), then
00
G (t)
I:
k=0
r
* k is the k-th moment of Sr'
*
where E(Sr)
The aeries 1s absolutely
convergent, and for any two positive integers e and 3 end every
t> 0,
2e-2
23-1
dk > Gr(t) > I: dk , where
k=O
k=O
I:
10
Remarks.
The moments of S* aore ea.sy to obteoin from the
r
cumula.nts •
We can find the first ten moments in terms of the cumu1ants
from Kendall
(ii)
1'11_7.
The above corollary gives a method which seams to
be an improvement over the one given in Robbins and Pitman
1.4
L-17_7.
Rote 11 ins I a method of Laguerre polyncmiaols.
In this section we ahall give a very brief a,ccount of a
where the Xi Bore independent N(O, 1) va.rie,tes and where e'i > 0,
i=l, ••• ,n.
Let g{q) be the p.d.f. of Qn and let
f(x)
=e
-x m-l
r(~)
,
X>O
,
n
where m = '2'
Then the suggested expansion for g( q) is the La,guerre aeries
11
where L(m-l)( q ) , raO, 1, ••• , is the sequence of Laguerre polynomia.1s
r
satisfying the relation
= 0,1, ••• ,
i,J
The Laguerre polynomi8.1
L~m-l)(q)
for m > O.
hs.s the following explicit
represents,tion:
8
L (m-1) (q) .. 1:
b
s
8=0,1, •••
v-O
s
See Sgego
(s+m-1) ~
s-v
vI
'
£18_7.
= s~{~~~
It follows from the orthogonality condition the.t
Jo
00
g(q)
L~m-l) (q)
dq,
end so b 8 is a linear funotion of the moments of
~.
The series
12
converges uniformly over the whole real axis and eo we can integre.te
term by term.
Remarks.
(i)
The main drawback in using the above expansion ie
that no convenient bound is known for the error committed by
stopping after a certain number of terms.
(11)
Rotelling suggests that the above series be used
to find the p.d.f. of the ratio of a definite que.dratic form to
a sum of squares and the p.d.f. of an indefinite form by
convolutionl since an indefinite form is the difference of two
definite forms.
CHAPTER II
THE DISTRIBUTION OF A DEFINITE QUADRATIC FORM IN INDEPENDENT NON-
CENTRAL NORMAL VARIATES
2.1 The prob1en4
Suppose we have a q. f. Qn· ... %ytAy in Yl' ... , Yn where the
Yi are independent N(~i' 1) variates, and where y!
= (Yl'
••• , Yn ).
Let
Gn (t;~l' ••• , ~ n ) = Gn
(t; .~) ... P.r(Qn-< t)
then the problem is to find Gn(t; ~).
Let us make an orthogonal
transformation Y a rx, say, where rtr = rr t = I, such that
Y'AY
D
Xtr'ArX
a
XtD X =
a
where Da is a diagonal matrix having the elements aI' ••• , an in the
main diagonal, where ~, ••• , an are the latent roots of the matrix A.
~t
D
(~l' ""
~n)'
Hence, under such a transformation Xl' ••• , Xn
remain independent with the same variances as Yl' ... , Yn ' but the
14
•••
+ Yn~, ~ n )" where (Y'l"
.~
••• " y n~.)
are the elements of the i-th rOT/v of r I •
So the problem is now to find the
Pr
n
1 ~a ,Xi2 < t 7v"where
-2
L l~ - -
the Xi are independent N(~i' 1).
2.2 The solution.
Theorem 2.1
"-~~
Q.,r~
Let
ln
11)
y'-, where the
2 k=l n
k
... _
n
~
a-
Yk are independent N(~k' 1), i ... ;:i. Then
e-A t n/ 2
where
(b)
00
n (t; 1:) ... _..:.~.;..;--~)""'1~72~
(al,··an
(a)
~
G
Os
= E( 0"""niHl-)S
,
1 n
an d '\~""- ~
2'1
~...
s·O
2
~.
~
r(~ + s + 1)
•
The series is absolutely convergent and therefore it is convergent.
(c)
,
For any two positive integers r and k and every t > 0,
15
2r
Z d > G (t;~) >
8-0
n
8
2k-l
Z d, where
8",0
B
r(~ +
Proof.
~
-2i
8
+ 1)
We know that if the X. are independent N(!J.., 1) and if
~
1
1 n 2
!J.i ' and Y
e
_~
=2 i
1 n
n
~
e- Y Y
2
Xi ' then the p.d.f. of Y is
-1
00
E
m=O
f .. J
n
j\
j=l
-----(-~~~­
•
mJ r(m + ~)
1
-Y
-G j Y 2
j
00
Z
1.=0
J
dy.
J
16
::0
e
-'A.
00
Z
k=O
where
I( t;2;;n)
::0
j... j
-ra7-
Exp
n
- Zy.
1 ~
n
1.
y.J
j=l J
rr
- -2
1
dy.
•
J
Expanding the exponential and making use of the Dirichlet integral
stated in (1 0 2), we obtain:
i1
Xl
i
).,
n
••• n_.
1) .
r( i +
n 2
00
Z
r=o
-
r(i1+j~{~)
••• r(i +j ~)
~ 2
n n 2
•
•• t
17
This can be rewritten as
00
(_t)S
Os
2:
saO
°S
n
61 r(2
+
S
' where
+ 1)
S
d,
Z
=t
k=O
where d ...
The problem now is to evaluate
Let
*'II-
~
Cst
1 n -1__2
(
)
r;-
= 2 Z ak-Yk ' where the Yk are independent N ~k' 1 , i= ,-I,
k=l
then
00
Ey2r = (2n)-1/2
J
-00
18
and
s1
j
• 2""2
Z
jl+· .. ·+ j n=S
t •• ,J'
1·
n1
28 - k
jl . . jn
jl J ···jn~al ••• an
81
8
=
£:
k=O
d
jl
£:
i =0
1
., .
jn
£:
i =0
n
h ,
19
where d
=
But this is just c.
s
Hence,
n/2
00
(t·~)
n'-
G
=e
-"A.
(a
.t
1Q ,·an
)1/2
Z
s=O
.
(_t)S
sl
r(~ + s + 1)
This proves (a) of the theorem,
If kr(~) is the r-th cumu1ant of Qn , it is not difficult to
l3how that
n
Z
•
j=l
·Y.-J~
Hence, to find the r-th cumulant of Qn we must replace
a by a -1 , getting,
j
j
k
r
(Q**)
n
fore, c s is the s-th moment of a
(r-1H
Q
-
-
2
r.v~
n
Z a'-r (l-I'lJ. j2)
j=l J
~j
and
by
~j
•
There-
whose s-th cumulant is
20
1 n a -1 Xi'
2 where the Xi are independent N(O, 1),
If o~* = 2
i
i
r(~ + s)
• Consequently,
aSr(~)
n
e-), t n / 2
. -
00
-
E
1/2
(a1 •• • aJn )
8=0
'-
r(~ -tr s)
tS
.
sJ
'-
ans r (n
"2 + s + 1 )
e.
e
-A.
t
n/'2
e
This proves (b) of the theorem.
t/an
~ 00
for finite t •
The bounds stated in theorem 2.1
are based on the fact that if rand s are any two positive integers
~
1, then for z > 0,
28
Z
k=O
(_z)k> e- z >
kJ
This 1'roves (c) of the theorem.
2r-l
Z
k=0
(.·z.Lk
ka
21
Theorem. 2.2.
Under the conditions of theorem. 2.1,
,
where G (tj 0) equals the Fn(t) of theorem 1.1.
n
-
Proof.
Jj-R_7...J 1:1
:(
n
G (t il:!-)
=
n
-~i
e
e
-'3' 1
-1/2
'3'1
00
!:
.1 =0
1
("j.Y:!)
Ji
~'i!r(Ji+ ~)
Now,
2
wherex2"m+l is s. r.v. ha,v1ng e, centra.l chi-eq"as.re distribution
with 2jm+1 degrees of freedom.
Hence,
'
J1 ...
..
1
LI~ ~
'.I
oJ
d'3'1·
22
and so"
G (t; ~)
n
-
>
-
e- A G (t; 0)
n-
•
lATe note that equality is attained for any given t" by putting A = 0
(in which case ~
Theorem 2.3.
= Q) •
Under the conditions of theorem 2.1,
G (t· II.) < G (to 0)
n ,~ - n ' -
•
Proof.
Usinr a special case of a more general unpublished result due to
SigeityMoriguti, we find that
We can generalize the above inequality to the case n = 2 as follows:
=
Genera.lizing the method used for the cese n=2, it is not
difficult to show that the above inequality could be established
for n=3,4, •••
This canpletes the proof.
Combining the
results of theorems 2.2 and 2.3 we have the very useful inequality:
The proof of theorem 2.3 was suggested to the writer by
Professor S. N. Roy.
Remarks.
(1)
is merely
The introduction of an imB.g1nary mean in theorem 2.1
8.
mathematical conveniencej we could he.ve omitted this
fo~ulation e~together
and merely stated
tha~
Os is the s-th
24
** ).
moment of a. r. v. whose s-th cumulant is k a (Qn
(11)
The moments Os aol'e ea.sy to oompute when we know the
cumulants.
Kenda.ll
£ 11_7,
gives the first ten moments 1n terms
of the cumulants.
(111)
It oan be shown tha.t Gn(tij!) 1s 1nva.ria.nt under
an orthogona.l transformation
and
r'r
:II
YarX,
say, such that
r 'A r
= D
a
r r' = I.
2.3 An applica.t1on:
The power function of the chi-squaol'e stat1etic.
Suppose tha.t the observations from a random experiment
can fall into any one of k cells and that the expected number
of observations in the i-th cell under
EO
and
Hi
is mo
and mi ,
i
respectively.
Let
That 1s:
,
n
i
l-~
where
1s the observed number of observations in the 1-th oell.
1s the power of the test, then
If
25
approxtmBtely, for sufficiently large n1 • Putting xi
=
ni -m1
..[m
i
Renee 1 the a:r;pl1ca,tion of theorem 2 ~ 1 will give us the power
function of the chi-square statistic.
1
CHAPTER III
THE DISTRIBUTION OF AN INDEFINITE QUADRATIC
FORM IN INDEPENDENT CENTRAL NORMAL VARIATES
3.1 Special case 1.
As a preliminary step to the study of the distribution of
an indefinite q.f., this section is concerned with the
distribu~ion
of the difference between two independent chi-squares having
different numbers of degrees of freedom.
If the degrees of
freedom are the same, the distribution becomes the same as
the.t of the semple cova,rience in sampling from e, nOrID8,l population.
We sha,ll study some of the properties of this distribution in
order to ant1cipa.to the behavior of the distribution of e, more
cona1e..ered this problem from a slightly different standpoint.
The main results of this section e,re:
(i)
(11)
(i11)
(iv)
Recurrence rela,tiona 3.1. 4, 3.1.5, 3.1.6, 3.1. 7, 3.1.20
Inequa,l1ties
3.1.9, 3.1.21
Further properties 3.1.12, 3.1.13, 3.1.15, 3.1.16
An application of a. result due to Berry
If' T
n,m
= Xn - Ym,
L-3_7,
3.1.19
where X and Y ere independently
n
m
distributed with p.d.f. hn(x) a,nd ~(x) respectively, where
n
-x ~c; - 1
e x , x >0,
•
27
then if the p.d.f. of Tn, m is gn, m(t),
Gt.,m(t)
__
~
fOO
hn(x + t)~ (x) dx,
t> 0,
o
t <
o.
We see that given the p. d. f. for t> 0, to get it for t < 0, replace
t by -t end interchange n and m.
Therefore, we she,ll consider only
the case t>O, for definiteness.
Hence,
The moment generating function of Tn,m is M(e)
n
= Ee te =
m
(l-e)· 2 (1+e)- 2. From M(e) we see that
(1)
If n,m --+
00
n
so the.t in
---t
1, then Tn ,m is
n-m rn+m)
asymptotice.lly normal (2 ' "'2'" ·
(ii)
If n
~
00,
but m remains finite, then Tn ,m is
28
asymptotioal1y normal (n~m), ~n).
(11i)
If m --.
00,
but n remains finite, then Tn, m is
n-m , ~m
r) •
asymptotioa1ly norma.1 (~
In 3.1.1 let x/t
Let n/2
= P,
m/2
= q,
= 1,
getting
and let
00
3.1.3
I(p,q)
=~
e- 2yt (1+y)P yq dy.
o
Integra.te 3.1.3 by parts three eepa,ra,te times a,s follows:
(i)
U
= (l+y)P
yq,
(i1)
u
= (l+y)P
e- 2yt ,
(iii)
u
= (l+y)P
yq-1,
dv
= e -2yt
dv
= e -2"'+'
""y
dy,
dy, getting
29
(i1)'
I(p,q)
2t
I(p-l,q+l) + (q+l) I(p,q+l),
on
= -(q:r)
m>-2, q+l> 0,
(iii)'
I(p,q)
= ( 12)
4t
L-(q-l) I(p,q-2) + p I(p-l,q-l)
+ 2t(q-l) I(p,q-l) + 2tp I(p-l,q)_7,
m> 2, q-l> 0, respectively.
The restrictions placed on m and q are to prevent the
pe.rt from becaning infinite.
(iv)
I(p+l,q)
= (~)
integra~ed
Equate (1)' to (iii)' getting
I(p+l,q-l) + (~;l) I(p,q).
In (i)' repla.ce p by p+l and q by q+l B,nd then substitute from
(i1)1
e~ (iV)
(v)
into (i)' getting
2
4t I(p+l,q+l)
= 2(p+l)(q+l)
I(p,q) + q(q+l) I(p+l,q-l)
+ p(p+l) I(p-l,q+l).
Now q(q+l) I(p+l,q-l) + p(p+l) I(p-l,q+l)
= q(q+l)
I(p-l,q-l)
+ L-p(p+l)+q(q+l)_7 I(p-l,q+l) + 2q(q+l) I(p~l,q), and I(p-l,q+l)
+I(p-l,q)
= I(p,q).
Substitute into (v) getting
30
so tha.t fine.lly,
(t)
~+4,m+4
3 1 4
••
=f
7
n (n+2) -Rn(m+2 +2 (n+2) (m+2)
(t)
4(n+2) m+2}
- ~+2, m+2
t
The same rela.tion holds for t <0 if' we repla.ce t by -t and
interchange n and m.
and the
rele~ion
Note tha.t if n=m, the la.st term vanishes
reduces to
From the first integration by parts we get the simple rela.tion
We now make use of' the p. d. f. 'a to obte.1n the c. d. f. 's.
For x>O,
Prl-Tnm>x_7
,
=
J
x
00
gnm(t)
dt
,
31
Integrate by parts where dv
= e-tdt,
00
and
u=~
o
B-1 '!-l
e- 2Y (y+t)2 y2 dy,
getting
Hence, if n is even,
PrL-T m>x 7 -=
n,
-
(n-2)/2
E
gn-2j,m{x), where
3=0
If n is odd
From 3.1.6 and 3.1.7 it is seen that if we have a table of -n,m
~
(t)
and if we know prL-Tl > x 7 for all m, we can find prL-T > x 7
,m n,m -
32
for all n and m.
Our first objective, then, is to be able to find
S:n ,met)
for any n and m.
If we consider a rectangular ta:ble
of 8n,m(t) having n colU!lUlS and m rows, we will find that
given
~
met)
~,
for n,m=1,2,3,4,5 and
~
,met)
form=1,2, ••• , we
can complete the table using 3.1.4 and 3.1.5.
First of all we ca.n fill in the second colmm using the
fe.ct that
-t
82,m(t) = 2:/2
g4,m(t)
=
m=1,2, ••• ,
'
-t
;m72 (t +~)
,
m=1,2,3,4,5
end
•
If we let n=m, we obtain, a.s we she.ll see la.ter, tha.t
1
t
Letting n=1,3,5 we find that gl,l(t) =; KO,(t), g2,3(t)=;Kl (t),
t
2
and g5,5(t)=3nK2(t), where ~(t) is the modified Bessel function
of the second kind of order n.
We.tson L- 21_7 •
See McLa.chlan £14_7, and
33
We shall now find the p.d.f. for ca.ses where n=m+2.
we complete the square in ,.1.1 B,nd let n=m+2, we get
Using the fact that
... 1/2(~)n
" - :2 ~(t) = -
r(n~)
J
nJ:
00
e-tY(y2 -1)
2 dy,
end
1
If we now put m=1,3 in the above expression we find that
If
34
We can now use 3.1.5 to find g5,1(t), S3,1(t), gl,5(t), end
g3,5(t).
Ther·3
8J:."Q
still six va.lues remining to be found,
which can be expressed in terms of the inoomplete gamma. function.
In fe.ct we find the,t
t
00
'!!-l
y 2
e
e- y
dy,
g 1"\ (t) =
n,
n /1"\
e
2 er(~) 2t
J
end
'
Letting n=3,5 we get g3,2(t), g5,2(t), g3,4(t), and g5,4(t).
Using 3.1.5 we obte.1n gl,2(t) and gl,4(t).
Now we have all
~
-n,m (t) for n,m=1,2,3,4,5, and together with 81"\e,m(t), 3.1.4
and
3.1.5 we ce,n S'3t 8.11 the rema.ini~ gn,m(t).
OUr second objeotive, then, is to find PrL-T
all m.
l
,m> x_7
for
We shall first giva e. method of eva,luat1:lg this when m
•
35
is even, and then a method for any m.
Evaluation of prL-T1 ,m> %_7, when m is even
00
prL-T1 m>
,
Ret)
%_7 =
=
J
11 m
J e-t R(t) dt, where
r('2)r('2) x
00
e-
2y
m
1
--1
- y2 (y+t) 2
dYe
o
Expa,nd a( t) in a. Mac1a,urin series about t=O.
1
00
a(k)(t) • (_l)k~:2r(k~)
I
Now
!!!-l
-(k.J)
2 dy,
e -2yy2 (y+t)
o
and
We may wr1 te
Ret)
=
r
k
t ~
k=O k.
)
a(k (0)
+ R (t),
r
where
36
R (t)
r
=1,
r.
J
t
(t_y)r H{r+l) (y) dy.
o
<
Therefore,
Hence, the error committed by stopping
a~ter
in magnitude than the first term neglected.
7
Evaluation of pr/-T ,m>x_ for a,DY m.
l
1
Let n{t)
= {1+t)-2
,
then
any term is lese
37
1
R(t) = Jf "2
R (t)
r
=~
r.
r
~ ~j
-I
J~'
f
1 + R (t), where
r( j + ,,)
~
r
t
(t_y)r R(r+l)(y) dy, and where
o
Therefore
I
r(r~)r(~+r+l)
m
m
'2+r +1
Jfr('2) (r+l):2
I
~
r""2
1 (x)
J~
(x), where
f
~
I
l(x)
J"*2
=
J
x
(1)
00
e
-t - J~
t
dt,
JaO,l, ••.
Hence, the error committed is less in magnitude then the first
term neglected.
We find that
-( j1)
-x
I
1 (x)
= --e_x~l--
j~
j=O,l, •••
j-'2
By repeating the above recurrence relation several times, we
find the.t a.ll I
1 (x) can be me.de t,o fa.ll back on
J""2
f
00
e-ttl / 2 dt,
x
which we can get from a table of the incomplete gamma. function.
Briefly summarizing, then, we have shown how to evaluate
to find a.ny c.d.f.
Inequal1ties.
In whe.t follows, under thie hee.ding, we she.ll discuss
certain inequalities
rele~ed
to the distribution of the difference
between two independent chi-squares considered in the preceding
sub-section, 3.1, of section 3.
39
If n
= 1,
1
=
f
-t '2
e t
r(~)r(~) o
.:L
and since e
2t
e -27
00
(l<il ~
.;t
1
d:r,
1
Y
< (l-tt)
-"2
~
1, it follows the.t
and
m
3.1.9
2 f
2
m
l - Hl (x)_7=? prL-T1 ,m>x..7?:..
~(x)
where
=-
fo
m
2""2(1~)-2L-l-Hl(X)_7,
x
hn (t) dt,
X>O.
If n,m;: 2, then sinoe
(y+t)
.!!.:!!!-2
2
n
~ (y+t)
"2- 1
m
2-1
Y
n+m
~-2
'?:.. y
, it follows that
40
f
~-2
00
!1.:t!-1
2 2
r{E)r(~) 2t
2 2
-y
2
e
y
d
y~
Sn,
(t)
m
~
e-tr(!!:!:! - 1)
n-tm
2
--1
2 2
r(~)r(~)
!!-l
(n-2)y
Again, since l~ (1~)2 S e 2t ,it follows from ,.1.1 that,
n-2)
for t ~ (T '
m
,.1.11 2"2 hn(t) So
Now letting t
~,m(t)
~
m
S 2'2 hn(t) L-l -
00
(4t )_7
2
,.l.~].
in ,.1.8 and
m
2
we have
m
2
,.1.12
Sn,m(t)N 2
hn(t), so that
So,m{t) has the same order of contact at
2
X
+00
with n degrees of freedom.
order of contact at
-00
as the p.d.f. of
Similarly, -n,m
ll.
(t) has the seme
2
as the p.d.f. of X with m degrees of
freedom.
In '.1.10 letting t--> 0, we he.ve
41
r(E.:!:!
- 1)
2
8n , m(O) =------.!!±!!-l
2 2
, and so
r(~)r(~)
the frequency curve does not meet the origin.
The case n=m.
under this heading we shall consider certa.in properties
of the p.d.f. and c.d.f. of Tn, n'
In this case the p.d.f. of
Tn,n is the same a.s tha.t of the sum of products of pa.irs of
independent N(O, 1) variates.
On the p.d.f. of Tn1n •
The p.d.f. is symmetric and so we shall consider 11.-rl,n(t)
for t > 0.
If we put n=m in 3.1.13, 23 get
We can show that 11.-uJn (0) is a decreasing function of n.
Differentiating 3.1.14 we have
_ r l (¥)
- r(n;l)
42
From Cramer
CL- 5_7,
p. 131), we ha.ve tha,t
r' (n) /r (n)
ia an
increa.e1ng function a,nd so g n(O) 1a a. decreasing function of n.
n,
That is,
F1na.lly, it is ea.sy to show tha.t
d
.g
3.1.16
dt
consequ6:ltl;;r the,to
e~l
.I.•.•'
11
n,n (t)
B.t t=O, and
= 0
(t) ha.s its maximum a,t the origin•
If in 3.:.1 we ~ut n~, and let 2x
= t(y-l),
we get
(t)
n , n'
g
If n=l, gl,l (t)
=~
Ko(t).
It is known tha.t KO(t) 1a a,eymptotic
to both axes ano. ha.s s. 10ga,rittJD.ic singularity B,t the origin.
Using a welJ.-lmon expa.n.sion for Ko(t),
£14_7,
a,s t
we find tha.t
--+
O.
The moment generating function of Tn,n beoomes
n
M(e)
2 -
= (1-9)
ET28
n,n
'2
,and ao
= (28): r(~+8)
s:
r(E)
2
2s
It is worth noting tha,t if we expand Tn ,n and use the known
moments of X2, we get as a by-produot ths.t
where 8=0,1, •••
and
n=1,2, •••
On the o.d.f. of Tn,n.
For large vB,lues of n we may wish to use the normal
approx1ma.tion, since Tn ,n is as;ymptotica,lly normal (0, .[n).
here appropria.te to use a result due to Berry
where the xi a,re independent r. v. 's.
£3_7.
It is
n
Let x= t xi'
1=1
2
Let EX= a , Var Xea , the
44
J
x
t
2
- '2
e d t , then
-00
r F(x) -
Sup
-00<
x <00
G(x-~ )1~ c ~ ,
where
1
(2n)
- '2
'5 C ~ 1.88, a.ocording to Berry.
1 2 _~
n
If we let Ui .. '2(X1 - Y1), then we me.y write T
= E U ,
n,n 1 i
where Xi and Yi are independent N(O, 1) varia.tes.
and 2x
= t(sec e -
Putting n=Dl.=l,
1) in ,.1.1, we find tha.t E ,uil
' .. 8/1f.
If Fn(x) 1s the c.d.f. of Tn ,n then
'
'.1.19
Sup
x<
-00<
" n (x .[n) - G(x) , ~
00
00
Consider next!
x
t
n
~(t)
dt.
Integre,te by parts where
U I : t ~ (t), dV
I:
n
t - l dt, and use the rela,tion t rrn(t)
I:
~(t) - t ~+l(t), getting
x
x
Replacing n by (n-l)/2., we have
=Fn (x)
- !
2
(x)
n ""'n,n
-x
Hence, mowing tha.t g2.,2.(X)
= e2.
,84,4(x)
=T-x (l+x),
-x
F2.(x) = 1 - e 2.
,we can obta.1n all F2n+2 (X).
Similarly, having
a table of the Bessel functions KO(x) , Kl(x), ••• we can get all
F2n+l (X), it we know Fl(x).
Evaluation of Fl(X).
We have
1(£1 - Fl (xL7
I:
f
x
00
KO(t) dt.
Using ta.bles of
46
Ka(t) we oould eva.lua.te F (x) by numerica.l qua.dra,ture; however,
l
we give an alterna.tive method here.
U
= lit,
dV
= tKo(t)
Integrate by parts where
dt, end use the relation
J t~n_l (t) dt
=
x
x
If we ca.rry out this integration by parts repeatedly, we find by
induction tha,t we get an a,lterna.ting series, in whioh the error
oommitted by stopping with a. given term is less in magnitude
than the first term neglected.
If dr is the r-th term of the
series, then
r=1,2, •••
Furthermore, if sand k are
~
two positive integers,
47
,.2 Special case 2.
In this section we shall give an expression for the p.d.f.
of an indefinite q.f. when the latent roots of the
me~rix
of the
q.f. are eque.l in pairs.
Theorem
,.2.
Let
where the a i > 0, and the Yi are independent r. v. 's ee.ch ha.ving
a chi-square distribution with two degrees of freedom.
is the p.d.f. of Qn , then
f(q)
q>O
=
q<O
Proo~.:.
The moment generating function of
•
~
ie
•
If f(q)
f( q)
J
= ~n
00
e-itq M( it) dt.
-00
Let us ta.ke a.B
We ca.n evalua,te f(q) using contour integration.
our oontour in the oomplex plane, the real line from -R to +R
and then the semi-circle of radius R,. 1nthe lower half· plane if
q> 0, and in the upper ha.lf-plene when q < O.
In both oa.ses the
value of the integral around the Bemi-circle tends to zero as
R
--+
00,
if n ~ 2.
Rence
t(q)=2ni L-Su~ of residues of the integrand at
t=_1-,
3'
j
J=1, ••• ,~7
,
q>o, and
=2~i
L-sum of residues
i
of the integrand at t_ a. , J=k+l, ••• ,n-l,
j
Eva.lus.t:t..ns the rasicluee, we get the form eta,ted in the theorem.
CHAPTER IV
FURTHER BOUNDS ON TEE C.D.F. OF A DEFINITE QUADRATIC FORM
IN INDEPENDENT IDENTICALLY DISTRIBUTED VARIATES - EACH
CENTRAL NORMAL OR MORE GENERAL
4.1 The centre.l normal case.
If we make use of the ];l.d.f. of Q
2m
when the latent roots
of the matrix of Q2m ere equal in pairs I we Clm obtain some
convenient bounds on the p.d.f. of Q2n when the latent roots
are not necessarily equa.l
1n
pairs.
Let
function of Q2m is
of Q2m is h(q) =12Jt
-
f
00 •
-Hq M(tt) dt.
-00
Using the calculus of residues we readily find that
50
m
h(q)
4.1.1
=:
t e
j=l
Suppose now that we want the p.d.f. of
We form the two
eypr~e9~onR
We can find the J? d.f. for ~ B,nd ~ by ueing 4.1 eo that we have
bounds on the c.d.f. of Q2n'
p.d.f. fa of Qr'" Q" •
'J
Con
f
o
t
fU(q) dq
~
Pr
Let fu(q), f(q), fL(q) be the
~
.... respectively.
L- Q2n ~
t_7
~
f
0
Then
t
fL(q) dq, where
51
-
=
n
t
e
q
~2J-l
.1=1
n
n-l TI
a 2.1 - l
k=l
k1J
Remarks.
(i)
The above inequality we.s suggested by Professor
Rotelling.
(11)
The above method could be extended to cover the
case of an indefin1te q. f.
4.2 The general case.
In this section we shall discuss briefly a system of
ineque.lities for the distribution of a definite q.f.
Pr
£Y~ ~
c_7
= p,
pr(~<c, ... ,~<c)
1=1, •.• ,n.
= (l_p)n.
Let
Then
It follows that
52
Let a 1
~
s'2
~
•••
~
a'n ~ 0, getting the following system of
inequal1ties:
Pr
Pr
L-Qn < '2c( a'n_l+ an)_ 7~
c
L-Qn< '2(a'n-2+
e'n_l+ an )_
7~
(l-p )n + (1n) (l-p) n-1p,
2
(l-p )n+(n)
1 (l-p )n-lp+ (n)
2 ( l-p )n-2p,
The a,bove system wa.s suggested to the wr1ter by Professor
Harold Hotelling.
The following improvements are due to
Professor S. N. Roy_
2
Suppose tha,t the distribution of Y is known and that
i
53
pr(~ ~
a
n-2
~n-l +a.)
= P3 '
n
Pr (y2 >
20
1 - e'l +••• +an
Then it follows
= Pn •
th8~:
Henoe we have one lower bound B,nd a. whole system of upper
bounds.
pra.ctioe.
Obviously, we would we,nt the lea,st upper bound, in
In01denta.lly, Pl So P2 So ••• So Pn •
BIBLIOGRAPHY
L-~7
Anderson, R. L.,
"Distribution of the Serial
Correlation Coefficient",
Anna.ls of Me.thematica.l Sta.tistics,
XIII (1942), 1-1;.
"The Proba:bil1ty Function of the
Product of two Norma.lly Distr'ibuted
Va.ria.bles", Annale of Me.thematic al
Sta.tistics, XVIII (1947), 265-:271.
"The Accuracy of the Gaussian
Approximation to the Sum of
Independent Varia.tee", Trenee.ctione
of the American Mathematical
Society, XLIX (1941), 122-136.
"On the Frequency Function of XY",
Anna.ls of Mathematical Sta.tistice,
VII (19;6), 1-15.
I
Cramer, R.,
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