Gertrude M. Cox
ON THE 2XACT DISTRIBUTIONS OF CERTAIN STATISTICS RELATED
TO THE "rTEilN
S(~{L~RE
SUCGE8SIVE DIFFERENCE
by
Seymour Geissor
Special
of Hork
Project
ability
"~
report to the Offico of Naval Research
at ChapGl H=-ll under Contract NR 042 031,
N7-Oi'lR-284(02), for rosearch in proband statis~ics
Institute of St2tistics
Mimeograph Series No. 127
February 25, 1955
ii
ACKN01rILEJGHENT
The l.yriter wishes to express bis thanks to
Profossor Harold Hotolling who directed this thesis,
for hLs valuable suggestions
am~
helpful guidance in
its preparation.
Acknowledgment is also made to the Office of
Naval Research for financial aid.
iii
TABIE OF CONTEIITS
PAG2:i:
PART
ACKN01'1LEDGMENT
INTRODUCTION
v
THE lvIEAN SQUARE SUCCESSIVE DIFl'EHEHCE AND ALLIED
STATISTICS
• • •
1
. . ....
·..
1.
Introduction
2.
The ¥.ean Square Successive Difference
3.
A Double Root Theorem
4.
5.
The Modified Hean Square Successive Difference
6.
Variance Ratio Analogues
12
2
7. Analogues of the Student t
15
Applications
8. l''Ioments
II.
ii
Remarks
10.
Tables
11.
Use of Tables
2
· ... . . .
·.....
..··
5
17
·••·
··•
..
•
24
•
34
·..····
44
THE DISTRInUTION OF THE RAiIOS OF CERTAIN QUADRATIC FOID1S
•••.
• • • •
·... .
1.
Introduction
2.
Preliminaries
3.
A Double Root Theorem for Ratios
4.
The Distribution of the Durbin and Watson Statistic in the Non-Null Qase
5. Approximations
3
11
..··
9.
1
·.
· . . . . ...
6. Homents of a Ratio
·.....
46
46
46
..
51
54
57
58
iv
7. The Distribution of the Modified Von
Neumann Ratio in a Non-Null Case and its Use
8.
Remarks
BIBLIOGRAPHY
......
....... ...
... ... .... . . . ..
60
66
69
v
lNTROllJ eTlON
The problem set forth in this paper is to find the exact distributions, wherever possible, of statistics used in time series analysis
and the pOimr functions of these statistics for specific alternatives.
Using the methods propounded by R. L. Anderson L-l_7, T. W. Anderson
£2_7,
J. Durbin and G. Watson
L-S_7,
l'lho r,10dified standard statistics
by assigning c:ouble roots to the quadratic forms and also to the model,
exact
distributions of these modified statistics and their power func-
tions can be found in many cases.
The justification for the use of
these statistics and models is their mathematical manageability and
the exhibition of the sort of properties Gxpected in simple time series.
The statistics used are the same as the standard ones, except for the
deletion of one or two terms.
They are very slightly less efficient,
and tend to the efficiency of the standarG statistics as the number of
observations increase.
The model used has several properties in common
with the stationary Markov process.
The advantage of using this new model and the related statistics
is the liwthematicDl simplicity of the distributions of these statistics.
This is not tho case with tho standard st3tistics in the stationary
Markov process.
I,
Nw~bors
in square brackets refer to bibliography.
PART I
THE 110DIFIED HEAN SQUARE SUCCES;:;IVE DIFFH'ltENCE
AND ALLIED STA.TISTICS
1.
-Introduction.
In ostimating the> variance of a norr,lal population one uses the
statistic
s
2
1 n
= (n-l)-
~ (x.-
. 1
~:z
~
2
i) because of its optimum properties.
In certain cases whero thero is Dn indeterminable trend in the data, it
has beon thought useful to estimate tho variance by another st3tistic,
namoly the mean
squ~r6
successive differenco, the moan of the squared
first differences, which eliminates a good deal of tho trend and under
same condit~ons is less biased than s2.
distribution of this
cult to obtain.
st~tistic
scems at
An explicit form of the exact
lc~st
for the present too diffi-
However, by applying thc device of Durbin and Watson
~S-l, that is, by dropping from the mean square successive difference
tho middle term for an Gvon number of
observ~tions
terms for the odd case, we find that the
quadr~tic
and the two middle
form has double roots,
thus enabling us to obtain exact distributions in terms of olementary
functions.
In Part I we givG a short and simple proof of the asymptotic normality of the moan square successive difference.
A theorem concerning dis-
tributions of double-root stntistics is derived and in consequence we
obtain. the exact distribution of the modified mecm square successive
difference and give a relatively simple Gxpression for tho moments of
this statistic.
2
In addition we define analogues of the student t 2 and Fisher Fusing
similarly modified st3tistics, and proceed to derive their exact distributions under the null case ('independence) and in a specific dependency
case.
2.
!he Mcnn Square Successive Difference.
Let
2
1
_In-l
2
5 : -2(n-l)
~ (x. 1- x.)
. 1
1=
1+
•
1
ThiS, except for the factor~, is the stntistic studied by Von Neumann, Kent, Bellinson and Hart
L-18-7.
They evaluated the first four
moments of 02, fitted a Poarson curve in order to find percentage points,
and concluded that for n >
mation.
This stntistic for
50 the normal curve
r~ndom
w~s
a satisfactory approxi-
samples from a normal population is
an unbiased consistent estimate of tho variQncc with asymptotic efficiency
2/3, and is
as~nptotically normal.
2
A proof that 5 is asymptotically normal will now be given using tho
central limit theorem for dependent rancom variables of Hooffding and
Robbins
.L8J .
2
Let Xi be N(O,cr ) and independent.
Now
and
By dofinition
3
I
./
0
if
j > 1
'.
<
204
if
j
(-
804
if
j =0
j
~
.,.-
(1.?2)
C.
~i~i+j =
Now the set
~1' ~2'
••• ,
~n-l
=1
•
is a I-dependent sequencG, i.G. the
is independent of the set (~ r+ 2' ••• , ~ n-1) for
r
==
1, 2, ••• , n-2.
Lot
p. ==
J.
Therefore
Pi
Q
r"';'
'-
r
2
""i+1
+ 2 1..; ~
1204 for all
r
'- "'i+1"'i
i.
for i
::a
1, 2
•
Since all the other conditions
of tho Hooffding-Robbins theorem are satisfied, we have for every real
a and b,
where
3.
F(x)
is tho cumu1ativG normal density function,
A Double Root Theorem
Let
(1.3.1)
XI' x 2, •.. , x 2m have density
f(X) ""
II!
2
2(2 2)-m -'2 x' \ X/o
\1 \1
no
G
J
A!
4
where
Xl
... , x 2m ) '
is tho row vGctor
/\
. ,
~
"f
positive definik matrix, and : ''. ~ is the d·.:;tcrminant of
is
:l
2m..rowod
t .
Let
(1.3.2)
~ Xf D X ;
q =
the following theorem will be provod:
If
1.
,.\ is positive definik with distinct double latent roots ).,1''>..2'' _.,
).,
2.
.
m'
D is of rank 2r
=
2(m-V) and is positive definite or semi definite
with double latent roots
3.
the determinant
II \-D
d
l
> d
2
m
= JT
_
mf
> ._. > d
('>...-d )
j-l J j
2
> 0
(f
e
0, _•. ,m-l);
;
then q has probability density
•
Proof:
Considor tho ch3.racteristic function ¢(t) of q.
mathod of M. D. McCarthy ~l~7 we got
Using tho
5
1
)1(t) =
(1.3.4)
I,
,
\
i"2
I
1
i to CJ2 D \ -'2
1\ _
Therofore
~(t)
(1.3.5)
=*
m-k'
2
rr
L(A..-itd.CJ )-1
j=l
J
J
J
•
Now by the inversion theorom
This integral may be
poles at
evalu~ted
by contour integration since it has simple
t::o A.(~id.)·l. Honce the integral
J
J
= -2ni
times the summation
of tho residues, since the integration is clockWiso; and the calculation
of the residues at t
A. (CJ2 idj )-1
::2
j
allows us to evaluato tho integral
and gives us tho desired result (1.3.3).
4.
Squ~re
The MOGified Moan
Successive Difference.
lTith the help of the theorem of the previous section we sh9.l1 derive
the distribution of the modifiod moan
°20 = 4-1 (m-l) -1 2m-I
(x.
. ...1
~
).
~+
I-X.)
2
~
squ~re
successive difference
under the null hypothesis of independence,
ifm
and also for an alternative hypothesis of a particular type.
We shall consider matrices
t, and
D of the preceding section capable
6
of nartitioning in the form
/\=
(1.4.1)
(~1 ~}
D
=
(:1
:J
uhere
1
-p
_Ip
0
0
1+
0
1\1
0
l+p -p
C
0
an m-rowed matrix.
Here the parameter
-p 1
p is not a correlation between
the vari3tes, but vanishes if and only if all the correlations among
them vanish.
(1.4.3)
Lot
D
1
-l( m-l )-1 A
=
2
1
where
1
-1
-1
2
2
-1
-1
1
7
Durbin and Untson
(1.4.4)
aj
=:
£5_7
4
find for Al tho latent roots
. 2 (m-j)n
2m
s~n
Now QQ 5 2=4-:1..(m-1 )-1 ~
0
I (
2 jn.
::0
x + -x
i 1
4 cos 2m' (J
)
i
2 whore
~I
(
)
::0
)
1, 2, •.. , m
2m-I
denotes
~
i=l
m
()
.
ir
It is cleo.r that
,
(1.4.5)
and by substitution in (1.).3) we got
.-2
Pm ( °0
(1.4.6)
whore {
=:
)=Ce
-2a-2 [) 2( m-l ) .
0
1 sincG the rank of
m-1
~
a m-
2
2
-1
3 -2[)O(m-1)(1-P)(a a )
k
•
G
k=1 k
~
is m-1 and
•
Now
let
8
(1.4.7)
and we will evaluate this exprasmo~By (1.4.4) and (1.4.1),
m-1
It
j=l
cos(k+j) ~
jfm-k
::: 4m- 2 lTtsin(k+j) ~
~
2m-l
1r
j=m+1
sin(k+j)
tmm
jf2m-k
We will further use a well known result (Schwatt
-/-15-7,
p. 238),
whi h states that
2m-1
(1.4.8)
\T
•
1\ SID
j=l
If now we lot k
m-1
=1
2m-2
4m- 2 lT sin(l+j) ~"IT
j=2
and
.:em j=mfo1
.
In
1 m
2m :: m4 -
m
f ::
•
l+j wo got
sin(l+j) ~
2m
m
I/,
2m-l
= 4Iil- 2 "IT sin ~ "IT
f=3.:em f=m+2
fL
sin ~
.:em
9
=z
2 2m-I
4m-
~lsin
.
¥m /
(sin(l+m) ~ sin ~ sin
7ni )
J=
::z
m/( 2sin
For k
2 ~) •
m
= 2 and (= 2+j we got
m-1
1t
sin(2+j)
. 1
J-
2m-I
n2m"IT
sin(2+j)
.
1
J=~
~
~m
:::I
2
-m / (2sin ~ )
m
jr2m- 2
jr2
In goncrnl we havo
'\7"1 (
J\
ak-a. )
J
. 2 kn
= ( -1 )k+1m / (2s1nm
)
Therofore
(1 • 4 • 10)·
2
pm(s:.u '• p)=2.4m- 3m- l Cc
o
•
\
-26 2a -2( m-1J~p
•
m-1
k+1. 2 kn
2m-6 kn
1: (-1)
Sln -cos
-
k=l
m
Under tho null hypothesis P
obscrvo.t.ions
0
1'10
get
m
= 0,
2(
2
- ( m-1 )( 1-P)oO 20" cos
2 kn)-l
2m
0
i. c. indepondonce of successive
10
(1.4.11)
m-l
.
6
m2
kn
2m- -kn
p (5 2) _
- 4 - ( m- 1)( I1U2)-1 LJ~ (l)k+l.
sJ.n2 -cos
Til
0
k==l
m
2m
kn)-l
-(m-l ) 502( 2(1 2cos 2 2m
e
The moments of 5~ when p::z 0 can cnsily bo computod since
t 5~r:
The
f
00
o
5~rpm(5~) d(5~)
c~~ul~tive
Hence
distribution function is
2
Now 502 is asymptotically normal, by the same proof as for 5,
and is also a consistent unbigscd ostimato of (12 with asymptotic
efficiency 2/3 •
If we let
D :;:
and
11
where
m-l
-1
-1
•
-1
Bl
( l.h.1S)
=m-1
-1
-1
or
q
1 , DX
= -2X
=
1( m-l)
-2
-lL- m.z ( x.-x-)2+.z2m
i=l
1
l
( x.-x
_)2_ 7 =
i=m+l 1 2
So2
,
2
we c,n find the distribution of sO' tho pooled v3riance, by substitution
in the Double Root Thoorom.
Thorofore
2
2 -2 (1- P +4.:p cos 2m
kn) ( m-l )
-sao
G
? -2
TNl1en p= 0, 2(m-1 ) sao
is distributoc1 liko chi-squ.1I'C with 2m-2
dogroos of fraodom.
5.
Applic'1tions.
As KnIlwt
£10_7
has pointed out, the ·,.lcan squaro succossivG diffor-
12
onco is likely to provide a bettor estimate of
of the
popul~tion
is undorgoing
Q
0
2 th~n s2 when tho mean
slowly moving shift.
2
cul'J.tod tho ',loan and varianco of 0
Karnot has cal-
and s2 and compared them for a slow2
ly moving shift.
The mean and variance of 0 aro for largo numbers of
0
2
observations pr3ctically tho same as for 0 , and 'J.ro given as follows
for the x. indopendont and
1.
(1.5.1)
~
~
02 ~
0
0
2
r 1+4-1 (m-l )-1
2m-I
Z
_
i=l
6~ 7
1. -
ifm
2m-l
(1.5.2)
Z
i=l
i,lm
6.
)
6.1. (6,1. -6 1.'+1)
3
Variance Ratio Analogues.
If we to-ke two indcponc13nt drawings from tho samo type population
(1.3.1), i.G. ono drawing will be from a population (1.3.1) with
2~
observations and the othor from 2m observations, and compute the statis2
tics ql from the 2m
l
observations and q2 from the 2m observations whore
2
ql and q2 each s2tisfy tho conditions of the Doublo Root Theorom, and consider their ratil' wo hnvo an analogue of Fisher t sF.
Lot
"'Y. ""
ql/q2 •
A subscript 1 or 2 in the following donates that it belongs respectivoly
to tho first or second s:lmplo or popul::'.tion.
prorluct of the latent roots of
"1 when
Thus
1\ is
1t
A' is tho
j=l J l
~-rowcd.
13
Now if ql is independont of q2 the prob2bility density of
'Y
is given by
oe
h(~)
(1.6.1)
=
Jo
Q2Pml('YQ2)I1n2(Q2)dq2
Since tho usu::ll convolution formula
pc;rk~ins
to the ')dcJition of
vnriatos we sholl colI this tho quotient convolutirn formuln.
Henco by (1.3.3) anJ (1.6.1)
If
.'10
lot
(1.6.3)
then the density of lyJiS
•
14
~-1
"IT
whero
j:=1
A. =(l-~ )-1
Jl
m -1
i~ 'i2
• (l_pj-l
1;\ 11
(1.6.6)
then
"lvO
lot
~1
i:l
I;\11 when A1 is !!l2-
For p =0 this ro]ucGs to
If
when
ffi
1
-rultlGd and
rcMcd •
15
7.
i\n~logues
~ve
2
of the student t •
will nOH clorivc the
'listributL~ns
of st:1tistics
(1.7.1)
~2
.
h
were
q 1S 0
Lc..t
q
0
=:
2
or So
°02 •
'1nd
t
x.1
= I-l. •
In orecr for the quC'ticnt convolution formula of
~2
(1.6.1) to hele', the numer"ltor ane denonil1::ltor of
This is oquivalent tel the independence of
x2
-2
The matrix of the quo.'Jratic form of x
and
must be independent.
og
is propcrtL·nal to
J,. • • • • 1
E.
1 . . . . 1
For independence of quadratic forms Hith matrices E nnd A, Matern
["13_7
ancl others have shown that it is nccoss[,ry :md sufficient that
1\-1
E"
A=O.
Now
Thorefore
/\=-
1 + peA-I)
E 1\-1A "'"
Ell /\-1
so that
=::
0 •
ll.l\=::I\:'., h::.nce
A-lit =
I\-lll..
16
iNow the distribution of u
= 2m(X-i-L)2/<,.2 can be found by in-
specting its characteristic function l
_1
¢(t)
(1.7.)
or (l-p)u
=
(1_21t(1_1~-1) 2
is distributed like chi-square with one degree of freedom
and so the probati1ity density of I.l.
is
•
The distribution of
o;/a 2
has already been given in (1.4.10).
Hence by the quotient convolution formula
WG
get the probability density
of ~2
1
1
2
2(5- 2m ) m-l m-l
k 1
(1.7.4) v 2(~ ; p)=(m-l)m- 4m- (l-p')
1T A. Z (-1) •
6
j=l J k=l
2
o
For
( 1.7.5 ) v
p= 0
,,10
get
m1
2k
2
2 -~
kn
( l; 2) =(m-l )4m- 2m_lkE- ( -1 )k+l cos 2m-6~sin
r?L~ +(m-l)sec ~7 .
l
o2
1
o
If we let
q
2
= sO'
the product of matrices whose vanishing is nee-
17
Gssary for independonce is
(1.7.6)
and since (1.7.9)
dOGS
hold
W3
may again usa thG quotient convolution
formula.
Honce
1
v 2(~ ; ,P)=(l-P ) 2
s
p
?
-m(m_l)m-
1 m-l
~2
2(m-l)
m-l
2
kn
A. £.~ ( - l)m+k+l.
Slnm
j=l J k2 1
"IT
o
and for P = 0,
8.
1
2 -2
2
•
is distributed liko tho 3 tudent
Moments.
!flo shall now Gvaluat G the moments of
approximations for tho moans of
Now by (1.4.12)
where
(1.8.1)
S
::a
°
2
0
and
So2
6~ in the null case and find
in the non-null case.
18
=4
m-l
~ (-1)
k-l
2(m+r-l)kn
2m -4
cos
k=l
81=
m-l
~ (-1)
2(m+r)kn
2m
cos
k=l
m-l
~ (-1)
k-l
2(m+r-l\n
~;
cos
k=l
Now accorJing to Schwatt,
8 2=-
m-l
~ (-1)
k-l
2(m+r)kn
cos
-2'
k=l
L-16, p.
m
222_7
,
m+r-l(2m-2r-2J 8 =41 -m-r ~
(
) / 1 ..
1
a"",l '
m+r-l-a
(1.8.2)
k-l
()m.-2
-1
cos
(2m-l)an
.2
m
7
~cos 2m
2111+2r-2)
(
and similarly 8
Nm'l' for
8
1
Til
[
2
is the sarno as
'
=41 -m-r
with m+r
r
I
~os(2m-l~~
1+ __'--an-_,m
m+r-l;"a "
t
~
+ ~
m+r-l-a
a=O
_7 '*
cos 2m
m+r - 1 ( 2m+2r-2) m+r-l( 2m+2r-2 )
a-I
in place of m+r-l
oven
=41 - m- r ~r-l(2m+2r-2
a=l
8
1
m+r-l
m+r-l-a
2m+ 2r - 2\ (
C
m+r-l/ )
L
cos(2m-l) ~
cos
an
~
•
19
~
4l-m-r( 811
where 8
+ 8 )
12
is equal to the first sum above and
11
8
12
is equal to the
second sum o.bove.
Now
8
11
= m;r-l (2m+2r-2) = ~ .l"22m+2r-2 _ (2m+2r-2)
a=l
m+r-l-a
_7.
m+r-l
Further
cos( 2"1,1-1 >¥m
:=
cos
an
(_l)a
for all integral valuos of
a
except
a=(2y-l)m,
2m
Howover
y=l, 2, •••
cos( ?m-l~2n
lim
m
x .-;:.( 2y-l)m
life shall· evaluntc
cos
8
12
m+r-l
812
=
~
a=O
m+r-l
... 1 - 2m
"2iii.
:s ;3m-l
2m+2r-2 )
-2m
(
( 2m+2r-2)
\
2m+?r-2)
(_l)a _ 2m
( m+r-l-a
1 (2m+2r-2)
= ~
for r
xn
r-l
r-l
20
Thorefore
S
1
= 41 -m-r
f
22m+2r-3 - 2m
(
2m+2r-?J
r~
2m+2r-2)
(
r-l
c.md s imi12r ly
S --
m42 -(lil+r)(2m2~m-r )
( 2m+?r-2)!
rH 2m+r)!
•
Substituting in (1.8.1) we got
(1.8.3)
for r
=5
~
S.clffiG
NOH
(2iii+r) !
3m-l •
SL~lQrly
tho
(2m+?r-2 )1
2r 2r(
-r 1-r
2
)
=cr
m-1) 2
(2m -m-r •
0
~ 6
for m odd the same type Corivation is carried out with
rosult (1.8.3).
we "'rill evaluate the menn of
6
2
0
and
2
So
in tho non-null case.
By finc1ing the moment generating functicns of these st,1tistics from
21
(1.3.5) we get exprossions for the moans by differentiating onC0.
Therefore
(1.8.5)
NOH
for large
by (l'!proxim'1ting the sums in (1.8.4) and (1.8.5) by integrals,
m
we
got
[ L
n/2
~
-1
l + P (4cos2x-l)}
dx-(l-p )-1 )
o
°
I!' 2
( 1. 8 .7 ) c..
0
r
n/2
2(
-1
~ 4am-I)
mn -1
L-1+ p (4cos 2x-I )_ 7-1dx
cos 2x
o
ThCSG integrals arc Gvaluatcd by Dicrcns De llaan ~4-7 p.16; and
we obtain
(1.8.8)
C. s~
rJ
1
1
a (m_l)-1(1_p)-lL-m(1_ p )2(1+3 p )-2 - 1
2
_7
or
•
(1. R.10)
~
r.
°
I I I
---1
2
2(
2
~2a 1+3 p)
0
L- (1+3 p) 2
+
(
1- p)
2
_7
22
Fran (1.8.9) and (1.8,10), the :lsymptotic cxpectnticns of
s~
and
6;, wo computo tho
~symptotic biases
of
s;
and
1
-13
6~
Thus
and we compute the binsos to be
)2
(1. 8.11)
,
(1. 8.12)
A short to.ble is givon for the COT:lpo.rison of tho biases for
different valuos of
p
p
Bins of
e6;
(i
(i
r;i
-.3
3.338
.830
-.1
.267
o
o
.1
-.160
.2
-.268 a 2
2
1.770 cr
.443 cr2
.139 cr2
o
r;i
2
2
-.028 cr
2
-.1l6
-.113
-.130
(]
2
.3
-.345 a
.4
.5
-.40 3
c/
-.447
a
.6
-.482
r;/
-.054 (]
2
.037 (]
.7
.8
.9
s~
t
Bias of
-.508 a
2
2
-.527 a
2
-.536 cr
cr
2
cr
2
-.106 cr
2
2
.212 cr2
2
.644 (]
23
If we differenti3te the
and s; twice ancJ set t
::e
mom~nt g~ncrnting functions
of
5~
s(;c'~md
5~
and
0, we get the
moments of
rhcref0rG we get tho variances to bo
v( 6~ )==4cr4 (m-l) -2
m-l
(1.8.13)
(1.8.14)
V(so)=cr (m-l)-
4
r
2
4.
Z cos
j=l
¥m
m
2.
{l+ p(4 cos
~ -2
¥m -1) )
1-2
2 m-l
2•
.Z tl+ p(4 cos ~ -1»)
J=l
We C2n ap~)r-'ximak, the sums in (1.3.13) and (1.8.14) by integrals
of the form
I
n/2
n/2
o
2
(1- p+4p cos x)2 dx
r
and
fo
cos 4x(l-p +4 p cos 2x)dx.
n/2
Tho intO[;ral
2
(l-p +4p cos x)2c1x
is eV21uatcd by Biercns De
o
Haan ['4_7 p. 78; and WG got
1
(1.8.15)
_1
V(s;) '" o4(m_l)-2(1_ p )-2fm (1+ p )(l-p )2(1+3p ) 2_1 _ 7
•
H0wover, the evaluati'n c.f the second integral lC::lds to a hypergeometric series which is not in a
sli~plor
form than tho OX::lct value
of the variance (1.8.13).
If wo
11S0
the Euler-Maclaurin Sum Formula, on (1.8.4) and (1.8.5)
we got the maximum errors in using (1.G.8) and (1.8.10), which arc
respectively;
24
(1. 8.16)
I cr
2
(2m-2)
-1
(1- p +4
p
~n
cos ~ - 1)
Both (1.8.16) and (1.8.17) tonu to mora ns
(4
2n
-1
co s ~ -1)
< p <
9.
1
-1
gn ,
cos --rm
•
m incrcnsos for
•
Remarks.
Tho rcnson for tho usc of tho po.rticll.1nr alternativG
A of
2
(1.4.2) whore (1-4 cos ~)-l <p <
matricos of the quadratic forms
1
is that it commuted with the
Comnutativity is a
necessary conditim for the assumptir)ns of the doublo root theorem of
section 3 to hold.
In other words, wc arc choosing matrices associat-
cd with tho modol ani the
with roughly the sort of
when
p
cated
st~tistic
pr(~erties
is not teo large.
that are mnthomatically manngeable,
expected in simple time series,
They arc, of course, not uniquely indi-
by any law of nnture.
In ordor to discuss some of tho properties of this model we must
find
A-1
(1.9.1)
Now
A -1
whore
25
-1
-1
2
•
A1 =
(1.9.2)
NOll
1
2
-1
-1
1
the same orthogonal matrix which reduces A to diagonal form
1
likewise reduces
1\ l'
1
2- 2
1
2 2
Such an ortho::,onal matrix is
rmn
. · · . . . cos
(m-l)n
2m
cos 3n
. · · . . . cos
3 (m-l),t
2m
cos
2m
.'
•
1
2
2
cos
(2m-l)n
2m
• • • • cos
( 2m-l)(m-l)n
2m
Therefore
(1.9.1-1-)
and
pr~p =
A-
m-I
= !im+l_j
26
1\
(1.9.5)
-1
1
=
p D-1
pI
Am+ 1 -J.
It follows from these forms (1.9.4) and (1.9.5) that the element
.
f ik ln
the 1. th row and k th
1\ -1
column of ,1
is, for i
-1 1
'f",
k..J
r 1
m
(1.9.6) K'k~2m-l'~lcoS(k-1)(2j-l~COS(i-l)(2j-l)--2n ~
J=>
1
'2m
m
while
1
for
m
f ll = m- J=
'~1;-1
-
(1.9.7)
2
+ p(4cos (m+1-j)~ - 1)
~m-
-1
7
i =- k .... 1 ,
1
-2 -1 m
( n ..
2
n
)
(1.9.8) f lk:=2 m .~ cos(k-l) 2j-lTmLl + p(4cos (m+l-j)~ - 1
_7
-1
J=l
for
i
= 1 and k .; 1 •
If we let P ik be the corr031ation betwGen Xi and x k then for i .; k
P
Now when
p::: 0, P ik
ik :::
=:
0 for all m.
This is obvious by consider-
27
ing
AI.
(1.9.10)
He vlill now show that
-1 m .
i'I
=2m
lC
lim
~
-1 m
Trrl+
P (aa
-l)J, Z
-
a=l
1
P ik
1.
Q
Now
m
cosf(k,j)cosf(i,j)
J=l
a=l
a"j
where
(1.9.11)
cos f(i,j)=coS(i-l)(2j-l)~
and
(1.9.12)
cos f(k,j)~cos(k-l)(2j-l)~
and
(1.9.13)
Hence
m
Z cos f(k,j)cos f(i,j)
j
Jr /-
_7
a=l -
a-:Jj
When
p
=
rrrl+
P (aa -1J]
-
1 ,
••• , m
28
:: 1
(1. 9.16)
A similar
<r
gumcnt also shows that
lim
.,
p ~1;.;,4cos
m Ip ik'
2 kn
He will now proceed to show that
1
=
p
•
ik diminishes monotonically
for k :: 1, 2, •••
and fixed i by considering the clements of
-1 .
/\ 1 ' 1. e. f ik , in another manner.
It can be shown by a Laplace expansion that, for
(1.9.18)
where b
j
•
p
k-i b
k
i-Ib m-k
is the determinant of the j-rowed matrix
1
-p
l+P
..
0
0
. .
\
B
j
==
0
0
•
-p
0
0
-p
l ... P
~
i
29
This follows from the fact that the cofactor
b
ik
cf tho G10mont
in the i th row and k th column of
1
-p
-p
l+p
•
1+ P
-p
-p
1
is the determinant
B _
i 1
l\ik
0::
(_l)i+k
0
f
R
--II s--
0
i-I
R is the
matrL~
l\ik::Z
i-I
R
k-i
0
k-i
I
m-k
B
m-k
m-k
-P in the
consisting of zeros oxcapt for
lower left hand corner and
in the main diClgonal and
I
0
I- - -
-I-
whero
f
I
I
S is the triangular matrix with
0
is the null lililtrix.
ThereforG
( -1 ) i+k( - P )k-i b. Ib k'" pk-L
o .1- Ibm- k
1- m-
•
-p
30
Holding
i
fixed we will show that for all k
and
0 < P <
1
In order for (1.9.21) to hold we must first show that
b. > 0
(1.9.22)
J
for all
j
= 2, 3 ,
for all
(1.9.23)
Now we define
I L j +l '
j
= 2,
... ,
0< p -<
1
3, .•• , 0 < P < 1
to be the determinant of the j+l rowed
square matrix
(1.9.24)
.
f~ 1'+1
J
b
1
... p
-p
l+P
o•
...
o
•
o
o
o
o
l+p
... p
-p
1
Hence
(1.9.25)
since 1. 1
J+
= "b.
J
2
p b, 1 >
J-
0,
for 0 < P <
1
is positive definito.
It is clear that (1.9.24) implies that b , b 2 , •.• , b s
1
are < 0
•
31
an~
b s + ' b s +2 ' •.. , aro
1
(1.9.25).
0, sinco any other case would contradict
>
Now b 2 = 1+ p -
p
2
for
> 0
o<
p <
1, henco
b. > 0
J
for all j.
Further b
and
3
> b
for 1 > p >
2
rovo that this implies that
D
O.
v1e now aSSU111C that
b j+2 > b j+l.
b. 1 > b.
J+
J
It is clear that
(1.9.26)
Henco
1'1C
must show thA.t
2
(1+ p)b. I- P b. > b. 1
J+
J
J+
,
or
(1.9.28)
Since
o
< P < 1,
which is surely true if
(1.9.28) is equivalont to showing that
b j +1 > b •
j
This comp1etcs the proof.
Thorofare in this model dist1nt observations have lower corre1ntions
th~n
~ve
ncar onos.
will now show thnt
of p for
Now
a<
p < 1 .
2
P ik is an monotonic incroasing function
32
,
(1.9.31)
Now
b
. 2
b
. I
.( m-1- )( m-1- )
bm-1. 1 bm-1.
Thus
p
k.-i
b
m-k
0---
can be split up into k-1
factors each of
m-i
the typo
•
Similarly
b
Now
t-1
P
p~>
k-1· b.1-I
b _
can bo split up in tho same manner.
k 1
0
for
0 < P
<
1
•
If we can show that
we
hnve proVGd that P
t
bt _1
P~
is an incrc8sing function of p
t
an incroasing function of
for
2
ik
is
O<p<l.
Now
(1.9.34)
p
(1+ p )-p
2 bt 2
(r-)
t-1
•
33
t "" 3
For
(1.9.35)
which is an incrcosing functir:oll of p since
> 0 fnr
O<p<l.
b _
vie then o.s sumG
th at this implies
t l
is an
p ----
b
th~t
incre~sing
function of
l+P -
I( p )
and provo
is an increasing function of p •
p
p
whero
p
t
I ,( p ) > 0
and
p
f or
Now
,
I( p )
0 <
~.
< 1
Furthor
dp
==
1+ p2 I I(p )
(1+ P _ pIC P »2
This completes tho proof.
Thus
2
Pik
> 0
•
is an incroasing function of p
Hence the model under consideration has tho following properties:
1)
lim
p~
.,
p == 0,
lvhon
1
,.
,I
•
34
3)
for
0 < P < 1, P ik
4)
for
0 < p
k
=t
1, 2,
~
1,
...
P
is a monotonicDll;y incrG::~sing function of
p ;
is a monotonicQlly decreasing function of p
ik
for every fixed
i < k.
These properties tend to shr)w thQt this mr;del behaves somewh3.t
like tho
stati~)nary }1Olrkov
process.
It is also possible to rclClX the necessnry condition of cornmutativityand still get a dnuhle root theorem
L-lS_7.
2S
was done 1)y J. Pacheros
Howevor, this involvos tho chnr2ctcristic roots of
which in general is difficult to find for the usual medels.
AA- l
It would
also vitiate the independence of tho numerator and denominator in the
Student t
2
analogues.
Tho compClris('n mode in the previous socticn of the binses of
and
soems to indicate that
is tho preforable statistic
sinco it is loss hiased for the smallor values of p
approxima.ti::m for
tho variances of
Since no good
could bo found we are not able to compare
Howeve~ since both V(O~) and V(s;)
arc continuous functions of p for
- "31
( 2 '.
2)" .
8 )< V(OO
for p=o,
0
< P < 1 and V
it follows a.t least fer sufficiently small values of p that
10.
Tahlcs.
One of the pleasant features of the statistics discussed here is
thllt their cUlUlllntivc distributi"n functi-,ns ca.n all be expressed in
3.5
olement~ry
terms.
Sov"rnl cf those st,tistics h::lve ')ccn tabulated.
Inted exnctly cnly up to
n = 20
Thoy 1r0 klbu-
since it beccmes 1,'1bori(us
t(;
com-
pute further with a desk calculator.
Tables
h~ve
coon c31culntod for
several vnlues of
5~
6~/a2
p and for tho student
when p = 0,
t-analoguG when
p
=0
anel
is used as an estimate of the variance.
Tho density
~f
~
in this case is
'1
m-l
2m-6
2
2 --1m
kn/2
(
kn 72
(1.10.1) v (~ )...4m-2( m-l ) m-1 ~ ()k-+-l
-1
cos
~sin ~ -+- m-l)sGc -2
k=l
~m
mm-
whoro
-00
<
~
<
00 •
It c:::n 1-,0 shrwn that ~
~
('if
2
~
)2/
= 2m(x-~
/
°02 •
- > N(O,l)
by c'nsidcring
If we divide the nWl1crnte>r:md thu dun0min::lt c·r
2 by a,
2 the denrminator ccnverges in probal ility t" 1 as
m
incronses and the numerator is a chi-square variablo with 1 degroe of
freedom for nll m.
~
Honce
~2
is symmetric it tends to
converges to a
3
Xi
N(O,l) variable.
It als0 CQn be shown that tho density cf
ncrmcl1 :lensity for large
variable and since
tends to tho
m.
The characteristic functi<n d'
is givon as follows
36
m-l
¢(t)::z JT J.. .•
j=l J
(1.10.2)
m-l
JT
j==l
l
"' -1
A.-it(m-l)-l)
J
2.
where
Aj == 1- I' -10
4P cos
Jfl
2m •
m-l
Now
m
A. == (1- p)-l lT A.
j=l J
j=l J
JT
and
m
1
Z.
J..j=m log(l-p)-Io Z (l-104p (1- p ) - cos ~) •
j=l
j=l
m
log
(1.10.3)
For large
JT
m we approximate the sum of (1.10.3)
by an integral
thus
m
1
2.
Z log(1+4p(1- ~- cos ~)
(1.10.4)
rv
n/2
1/
1
2
2mnlog(1+4p(1- ~- cos x)dx
j==l
0
This integral is evalunted by Bierens De Haan
(1.10.5)
f
fl~
109(1+4p(1-p)-\os2x)dx
~
L-4_7
page 441
and
1
n
lO~(l+(1+4p(1_p)-1)2
o
Hence
(1.10 .6)
Again approximating a sum by an intogr21 for large m ;
) •
Honce
1
•
-2m
L-l+(1+4p(1- p _it(m_l)-l) -1)2_7
Now consider
y -2m
(1.10.9)
= L-1+ (1+ 4
1
P (1- p -it(m-1) -1)-1)2_
7
-2m
Let
(1.10.11)
b
:2
-1
1+ 4 p(1- p-it ( m-I ) -1 ) -1 =1+4p(1-p )-1 (I-it ( 1~) -1( m-1 )-1 )
Further
1.
1.
(1.10.12) b 2=( 1+4p( 1- p) -1) 2L-1+4pit( 1+4p( 1- p) -1)
-1
(l-p) -2(m_1) -1+ 0 (m- 2
.1
17:
Expanding the second expression in (1.10.12) we get
1
(L 10.13) b2c (I+4p(1-
Therefore
1
~-1)2L-l+2pit(l+(I-p) -14 P)( 1- p) -2(m_1)-1+0(m-2 )_7 •
38
(1.10.14) y -2m
Hence if we let
_1
_.2
(1.10.15)
w( p)=2p(1-p) 2 (1+31)
1
_1_1
2 (1+(1+3p) 2
) 2 )
(l-p
,
or
(1.10.16)
¢(t),OJ (l-it(l-p)
-1
(m-1)
-1 1-m
-1 -1 -m
)
(1+2mitw( p)(m-l) m )
•
Now
(1.10.17) l.l. =
1
t
1
S~/(l"2 rYL m(l- p)2( 1+3 p)-2-1_7( 1-p)-\m-l) -l=(l_p) -1- 2m(m_l) -lw( p)
Consider the variable
(1.10.18 )
•
which has the characteristic function
3('
- ;1
.
)-1
(I-it ( I-p ) -1 (m-l)
I-m
( 1+2mitw(p) (
- 1m-1)
m-l)
-m
or
-it( 1- p)-1
(1.10.19)
0'(t)r'"'e
-1
(l-it(l-p)
(i-I)
2itm(m-l)-lw(p)
e
(l+2mitw(p) (m-l)
-1 I-m
)
-1 -1 -m
m·)
Hence it can be shown that
-t 2( I-p)-2( m-1 )-1
¢(t)
tV
e
•
e
or
2
--'12 t ~r
,L 2(1- p) -2 (m-1 )-1+8m(m-1 )-2/_ w( p )_ 7 _ 7
(1.10.20)
¢(t)
/V
e
If
v 2 = 2(m-l) -1 J.r( 1-p )-2 + 4m( m-1 )-1( w( p) )2_7
,
then
(1.10. 21)
v -1
L-
2
0
~-
s
rj
7
IJ._
---------:>
N(O, 1)
40
TABLE I
Cumulative
o;/a2
Distribution
n/Pm
.975
.025
4
3.689
3.071
2.694
2.458
2.294
2.172
2.078
2.001
1.938
1.866
1.810
1.761
1.718
1.677
1.645
1.612
1.583
1.558
1.534
1.511
1.491
1.472
1.454
1.437
1.000
.026
.106
.172
.225
.269
.3 06
.338
.366
.391
.405
.423
.442
.463
.483
.502
.520
.538
.554
.570
.585
.599
.612
.624
.637
·1.000
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
00
In this table
in tho tabln for
n=2m
n < 20
is the number of observations.
The values
have been co:n')uted directly from the cumu-
lqtive distribution function P given above. For
m
n > 20
it was con
jectured that for a given percentage point
(1.10.22 )
P
m
The method used is to assl.L'llC that
(1.10.23)
Since we have exact values for
P
m
fer
n < 20
we choose three
of those valuos anJ get threG simultaneous equations in aI' a , and a3"
2
Ho then use tho values for
table for
n > 20.
l~ilo
~,a2
and a
3
in (1.10.23) to Gxtend tho
exact bounds on the error cannot be determined,
sevoral of values were chocked and thoy
sc~m
fairly close to the true
values especially for the upper percentage points.
42
TABIE II
Cumulative
2
So
Pm(2; )=2m-1
0
p
n/Pm
4
6
8
10
12
14
16
18
20
00
m-l
i-m "IT
A
2 2 Distribution
sO/o
m-l
2
Z (_l)m+k-l sin kn
h-l h k=1
m
= -.25
.033
4.49
4.37
4.12
4.07
3.74
3.59
3.49
3.42
3.35
1.74
p ""
A~I(l_C
p
.025
.975
s2
- ~Ak(m-l)
~172
.305
.420
.462
.500
.535
.566
.593
1. 74
n/Pm
4
6
8
10
12
14
16
18
20
00
.25
)
0
=0
.025
.975
,025
.141
.207
.273
.325
.367
.402
.431
.451
1.00
3.69
2.78
2.40
2.19
2. 05
1.94
1.86
1.80
1.75
1.eo
p
:z
.5
n/Tlm
.975
.025
n/Pm
.975
.025
4
6
8
10
12
14
16
18
20
2.95
2.36
2.09
2.oh
1.96
1.87
1.80
1. 75
1. 70
.872
.020
.099
.160
.216
,246
.265
.283
.290
.315
.812
4
6
8
10
12
14
16
18
2.46
2.19
2.05
1.95
1.88
1.80
1. 74
1. 70
1.66
.894
.017
.086
.156
.209
.240
.260
.278
.295
.310
.894
00
20
00
TClblG II (continued)
p
nipm
::0
.75
.975
.025
4
6
2.11
2.10
.014
.077
8
10
12
2.16
.134
.186
14
16
2.19
2.06
.320
18
20
2.00
.370
.380
.901
2.19
2.20
1.95
.901
00
43
.248
.340
Table III
Cumulative t- Distribution
1
2 1 m-l
k+l
2m-4
2
2
-'2
m
V (l; )=h - m- k~l( -1)
cos
~ sin kn 't;' .r't;'2+(nr-l)Roc kn 7.
m
m
m s;L.s
m p
5
%
level of significanco for various n
n
5
4
6
4.30
2.82
30
32
8
2.5S
10
12
2.39
2.34
2.31
2.29
2.27
34
36
38
14
16
18
20
22
24
26
28
0/0
2.25
2.22
2.20
2.18
2.16
n
40
42
44
46
48
SO
00
5 0/0
2.14
2.13
2.12
2.11
2.10
2.09
2.09
2.08
2.08
2.07
2.06
1.96
44
Tho values in the table for
manner
as
in Table I, i. e.
m
n > 20
o:aV
Here cornputGd in the same
assuming for a given percentage point
00
v
rfuilo for
n > 20
00
l:
+
j=l
a .m- j
•
J
the values given in Tables I and III are
approximate, they may be useful until the oxact values can be calculated by hic;h spced computors.
11.
Usc of Tables.
If we use
to estimate
(J
2
when p = 0, we may set up a con-
fidence interval ldth confidence coefficiont
a and b from the tablos for a givGn
(1.11.1)
Prob.
L-
6
a <
n
.95
by finding two numbers
such that
2
-22
< b -
7
= • 9;)c
(J
or
S~nilarly
So2
for difforent values of
and get for corresponding numbers
(1.11.3)
Proh.
L-
c
p
~ivcn
c
and
in Table II we may use
d
for a givGn
.95
n
45
or
2
(1.11.4)
Probe
So
L
d
s
=
2
<
2
(J
<
So
c _7
:=
.95
Fur thor
III
(X"1!)
IR 0
so that the confidcncG intcrv'11 for th,-.-; Hl.can when p = 0
and this
student t-nna1ocuc is usod is
1
Probe
where
.L x~ .05 n- 2
~.05
50
<
~ < x+
1
s .05 n-2
is taken for the corresponding
50
_7 = .95
n
of Tnb10 III.
PART II
THE DISTRIBUTION OF THE RATIOS OF CERTAIN QUADRATIC FORMS
1.
Introduction.
In testing the hypothesis that successive members of a series of
observations are serially correlated a number of statistics have been
proposed.
Most of these are discussed by T. W. Anderson
G. S. Watson
L-19_7.
R. L. Anderson
L-l_7 gave
L-2_7
and
the first exact dis-
tr1bution of a serial correlation coefficient and Durbin and Watson
L-5~ gave the exact distribution of several other statistics using
double root methods.
In this Part we shall extend the work of Durbin and Watson for
a non-null case of their statistics.
We shall also find a simple
expression for the moments of another of their statistics.
By using the non-null alternative of Part I we shall also
derive the distribution of a modified form of the ratio considered by
Von Neumann
L-11j
and show that this ratio prOVides a tlbest tl test
for the parameter involved.
2.
Preliminaries.
... , x
n
have Joint density
1
(2.2.1)
where
f(X)
= 1/\1 2
tP f < 1, 't Xi
= 0
and
47
(\=
(2.2.2)
• 1+ p 2
-p
-p
1
This type model 1s referred to a.s the stationary Markov process
and can also be written as
(2.2.,)
1
n
-2
1 -2 f 2
2 n-l 2 2
n-l
S
lx+(l+P) 1:
x~+xn-2p 1: x + x ) •
1
1:=2'"
1=1 i 1 1
- - ---0
= (1_ p 2) (21t02 ) 2 e 2
One now wants an estimate of p.
The estimate usually given is
or
(2.2.4)
where
A
=
0
1
1
0
1
2
•
1
1
0
R
= X'AX'
/ XIX
48
Unfortunately the distribution of R seems difficult or impossible
to obtain in elementary terms.
L-6_7
L-16_7
Von Neumann
and later Gurland
have shown that it involves elliptic and hyperelliptic integrals
= O. There have been
Hsu L-9_7, Koopmans L-ll..1 and
In 1942 R. L. Anderson L-l_7
even when p
approximations to this distribution
by
Von Neumann
L-17_7.
changed the matrix
A
slightly
enabling him to get the exact distribution of a related statistic
(2.2.6)
where xl
n
2
xi+1X i / 1:. x.
i=l
i=l ~
n
R = 1:.
= xn+l
R
or
= X'AX
and
0
1
0
0
1
1
0
0
1
A=2
0
1
0
1
and
(2.2.8)
=
0
0
1
0
/ X'X
49
This was done by changing the four corner elements of
1+p2
-p
A so
that
a
0
-p
o
a
o
- p
-p
R. L. Anderson obtained the exact distributi(;n of the circular
serial correlation coefficient
In 191-1-5 !'ladow
L-l'2_7
R of
(2.2.6) when
1\ =
I.
found the distribution of this R when /\ is
given as (2.2.9).
In 19S1 Durbin and Watson
L-S_7
gave exact distributions for a non-
circular statistic which involved a slight loss of information.
considered the case for
They
n == 2m (reduced the ('dd case to the even by
dropping out the middle observation) and then deleted the middle cross
product.
(? 2.10)
where
They found the exact distributiun of
R :: X'AX/X'X
or
and
o 1
I
1
1
0
This was possible because of the duplication of the roots in the
ratio of quadratic forms (2.2.10).
Now if we consider the model for
which the Durbin and Watson statistic is "best" we roadily see it to
have /\ of the form
0
1\=
(2.2.13)
,
1\
0
where
l+p
2
-p
(2.2.14)
1\1
=
0
0
l+p
2
0
-p
-p
-p
0
-p
1+p2
Now it is clear that this is also essentially the Markov process
model with four elom'3nts changed.
The ciistributicn density is
and tho autoregressivG formulation leading to this model may bo found
in lrlatson £"20_7.
He also shows how gooJ an approximation to the ivlorkov
process this model really is.
We shall derive the oxact distribution
of the R of (2. 2.10) for the /\ of (2. 2.13) •
3.
A Double Root Theorem for Ratios.
Lot
Xl' x 2 ' •.. , x
n
have tho joint density (2.2.1) where
positivG definito; let A be real symmetric of rank
positive definito or semi-definite of rank
G( z) = P
L
XI tUC
XIX
r
< z
2
~
r
n.
l
~
n and
J\
B be
Let
_7
Lemma 1.
If
A
:mel
B
commute then
1
00
2G(z)
= l-n
-1
j
u-2it(a.-b.z)
J
2 dt
J
-00
whore
D=:P'/\ P,
plp=I,
P'AP=D
a.
J
,
pI BP=D,
D.
J
f
00
D
aj
and
Db.
arc diagonal matrices and
J
Principal Vahle of the intGgra1.
-00
is
is tho Cauchy
Freof,
-7
vTc usc t:'l0 folL\wing th30rom from rciJt:dc11gchrD (H. Hey1 /-21
-
p. 28):
If
a'1C'
only if twn HormitLm rwtricos commute can thore be
simulta'10011S reduction of those mE!triccs to diagonal form.
and
B
arc rc~l and symmetric, an orthogonal matrix
P
Since
8.
A
will satisfy
the theorem.
Lot
a.
J
A, b. of Band A.
be the latent roots of
J
J
Now make the orthogonal transformntion
joint density of
X'=y'P'.
of "
•
Therefore tho
Y is
1:
w(Y)=' D I
2
_:g
2
(2na)
2
_1:a - 2
2
0
y'DY
where
y'
1.
(2.3.4)
n
n
)
and
,
I
-I D-2iT I 2
,
¢(T)
n)
is the characteristic function of
2
l: t.y.
i=l
I DI 2
¢(T) =
... , t
... , Y
(y1' Y2'
:::
J. J.
•
- - 7 which
NOH by app1icct ion of a theorom by Gur1and /- 7
J
states
00
(?3.5)
2G(z) = l-(ni)-l
-00
¢(t(al-blz), ... ,t(an-bnz) )t-1dt
53
the lemma is proved.
If further
1\
commutos with
B it is clear that
I
00
f
A ane
1
tI\J2t-l-rr ,[A.-2it(a.-b.z)
j=l J
J J -
-00
7-2 dt
which loads to the main theorem.
If
1\,
B
commute pairwise;
1.
A,
2.
tho latent roots of
1\
and thoso of
A and those of B consist
of pairs with two roots in o3ch pair equal,
3.
B
4.
tho distinct roots of
is of rnnk
tho
2r which is
A
distinct roots of
tho d.istinct roots of
B
/\
>
are
are
TI'e
n
= 2m ;
Y=
rmk of A and
aI' a2 ,
m - r;
... , am
bl' b 2 , ••• , bm
AI' A2 ,
... , Am
-~
m-f
• • >
m...
.
arc the positivG non-zero roots of B and all tho non-zero roots of A
lie among
then
.
>a
-
1/;
m-x
54
aL+l
where
b L+l
<
Z
-
<
-
'.1
L
bi.
for
L
1, ••• , m-l
=
Proof.
2G(z)=1-(ni)
-1 m
Jr
A.
j=l J
=1-( ni)
-1
m-f
JT A.
j=l J
II
00
!f
t
-1 m-x -
Jt LA.-2it(a.-b.z)_7
j=l
-00
J
J
J
-1
dt
Not lot
a
>
':;.'his
inkgr~~l C1n
Whittokor and Ucttson
-2ni
L-
z
1.+1
> '5i:l
for
1 ::: L < m'"
f
be Gv!"tlu'"1tod by contour integration (Cf. o. g.
L-22_7,
p. 117 ) in the form
sum of the rosidues at t= - %iAj(Oj-bjZ)-l +
%
tharesiduo at t=~
Evaluation of the rosidues gives tho result (2.3.7).
4.
The Distributicn of the Durbin
.mel ~T[ltson
St'1tistic in the
Non-i~ull
-
Case.
Using the result (2.3.7) nnd the model (2.2.13) with tho quadratic
form (2. 2.10), we may find tho distrHmt.ion of tho Durbin nnd Watson
statistic.
55
Now
m
JT
2
jn
a j = cos m+l ' Aj =l+p -2 p cos
jn
m+l , b.=l, thcrafcrc
J
2
.
(l+p -2p cos 1..!!...-) •
. 1
~
J=
m
A.=
. 1 J
J=
1t
FUrthor
rrhen
2m+2)( 1-p2)-1
(
( ) =l-l-p
GZ
L
l: ( a.-z) m-1Al-1
k=l
K
C
TrLI) 7 -1
J\_
A. ( ak-z)-Ak(a.-z_
J
j
J
NOH
A -A.
k J
= 2p
(a.-a)
J k
Henco
2m+2)(
G(z ) =l-(l-p
2)-1(
1-p
2
l+p -2pz
)l-m
L
(
m-L -1 \'71 (
-1
l: :lk- z ) ""k J\ ak-a.) ,
k=l
J
Fllrthcr
\7"' (.,
)_2,,1-1
J\
u, -n. ~{J
~
J\
j=l
jr'm
.
Sln
(k+ j )n
2(
1)
- m+
2m+1
.
~
~\
J=m+2
Sln
j7'2(m+1)-k
(k +J") n
2(m+l)
,
-
•
56
and by (1. 4. 9)
2
~'(
kn
1\
a. -a. ) = (m+1 )( -1 )k+1 2 -m cse--..,..
K
J
m+.L
(2.4.1)
Thereforo
( 2.4.2 )
2m+2)~(
G ( Z; P ) =1- ( 1-p
c m+1 )-1 (l-p 2)-1( l+p2 -2pz )l-m
~
<.J
(1)k+1(
k=l
-
kn )-1
kn
)m-1. 2;:n (1+p2-2 pens m+i
cO&---:-l -z
S1.11 .. +1
- m+
m+
,11
c:md
•
L
2
~ (_1)k+1 (
kn
kn
cos -;"";
-z )m-2.
sm 'm+!
k=l
m+~
<.J
where
cos
For
(L+l)n
m+l
<
Ln
z :; cos m+I
P '" 0
,
and
57
m
1
G I( z ) =g ( 7, ) = 2 ( m- 1) ( m+ 1) -
L
1- 1
k
2
21
'\' (_1)
n -Z )m-.
- ".+ ( cos--y
Sln ,m
k=l
m+
m;r •
~
It thun follows that
2 -l( l+p2-2pz) -mg(z).
G r (z,p ) =g(z,p )::a(l-p2m+2) (l-P)
(2.4.6)
5.
Approxim~tiGns.
In n pLpur by T. W. Andersen ':Ind H. I... Andorsen
L-3_7
in which thu
circul::l.r serial corroL:ltion coof.f'iciont is ,Jiscusscd for fitted trigcn: 'metric series for tho menn, thoy hClvG fitLd tho trigonomotric sorios for
semi-mmu~l
d[1t" to correct fur v.'}riati.,n of period tWG and get
Q
quClcl-
ratic form
(2.5.1)
f';'r
n
q
= 2m
::2
X'CX/XIBX
•
Thoy reduco q to tho f0rm
2m-2
2
L C .y.
j=l J J
wh0ro the c.
J
~c
2m-2
2
1: y.
j=l J
identicCll with tho
0..
J
,
of the previ('us section.
Thorofcro the distributi'\n in this particulnr
v.'1tic·ns is eX.1ctly tho samo
Durbin :md vvatson for
2m-2
.'13
c:~se
fer
2m
obsur-
that for the non-circulnr c ':Ise nf
obsorv.':ltinns ,,,hen p =
They nlsn Ed VG tho apprexim'1to distributic\n,
a.
cf their circular
58
statistic as a beta distribution, and if we put
2m-2
in plo.ce of
2m we got tho o.pproximate distribution density of R of (2.2.10)
for
2m-2
obsorvations when p = 0
to bo
6. Moments of a Ratio.
The previous work was based on the assumption that
known.
r~tio
However, if
is unknown,
l-I.
~no
e. x.
~
=
l-I.
11l':lS
of tho sto.tistics used is tho
of the moan square successive differonce to the variance;
(2.6.1)
The distribution of
plicit ova uo.tic'n.
~
is for the present too difficult for ex-
The momonts of ~ h(~vc '!Joen found by Williams /-23 7
- -
and much light shod on the distribution by Von Neumann ["18_7.
However,
the exprossion for tho r th moment given by Williams is in terms of tho
r th derivntion of a function.
Durbin and Watson
-
for
-/-5-7 suggested a modified statistic in this case
n = 2m •
Let
(2.6.2)
whore
or
59
with
with
l~tcnt
roots
.
8j
=
4sin
2
1
-1
-1
2
(m~.21
2
-1
-1
1
::: 4 cos
2
jJt
~
and
2m-I
-1
• .
-1
-1
-1
-1
-1
2m-1
,,
Tho distribution of R is givon to bo
'"\T' (
(2.6.3)
By
(2.6.4)
J\
simplific~ti'-n
it can DG
roc~ucod
to
;)k-n j ) -1
60
The moments of this ratio can be c~sily found for
r < 3m-I,
since we already have evaluated the moments of the numerator in Part
I and the l11C'ments of the denominator are well known.
For independence
of successive observations the moments of this ratio are the ratio of
the moments.
Therefore
(?6.5)
c..t::
r l
2
2 + ( 2m+2r-2) 1(2m -m-r)
::
Rr~
r
_
L-(2m+r)!(~-I)(2~n+l) ... (2m+2r-3)_7
The Distribution of the Modified Von Neumann Ratio in a Non-jull
?•
If we consider the ratio
2m-I
~ (x i + l - Xi)
i=::'
2
irm
i.
G.
twice the ratio of the modified moan square: successive difference
to the pooled variance, we arc able to
find the distrihution of
USG
tr.::
~()lJ.tlG ~oc+, '::"l.'orc~; r~
'1')0 in tho non-null caso given by (1.3.1) ami
(1.4.2).
By tlw Double Root Theorom
G(
T1o; p)
i
-1
:z
1 - 2m
( 1- PI- P
ro 2-m
J
m-1
T7'
1\
X.
j=l J
61
,
and for p::: 0
,
GI( 11~ ) =? ( m-2)m-1
(2.7.5)
'U
for
2
kn
a k = [I cos -2m
L
2
z.;( -1 )k+1( a -n_) m-J sin -kn
l{: U
k=l
m
and
T
Thorofore ue got
(?7.6)
To justify tho usc of
that
ro
is a monotonic function of the: asymptotic maximum likolihood
estimate of P
The
(2.7.7)
T10 in tests concerning p we will now show
•
model usod hero for unknown lucan
~
is
62
lvhora
-p
1
0
(1
-p l+P
0
1\1=
0
0
I+p
-p
-p
I
.
~l
and
= (~ ,
...
,
~)
,
.
HcncG
1
-~
2cr~
2m
L- z
I
(x~-~)
~
2
2m-1
2m-1
i=2
ifm,
G
2
+P Z (xl.-p.) -2 p Z (x.
1-~)(x.4J.)
i=I].+].
-
7
irm
m+l
2.
whGr.:;
A ;; 1- p + 4p cos
j
¥m .
Now
2,,1-1
2
•Z2
(X.-II)
-2
].=
]. l"'"
ifm,
m+l
2m-I
2m-1
2 2m
'Zl (x.].+14J.)(x.-I-l.)=
,Zl (x.].+ I-X.)
-Z1 (x.-~)
].=
].
].=
].
].
ifm
ifm
2
•
63
Hence
2 m
1
2m
..2
2
log f(x , •.• x 2 )=-m 10g2na +~log A.- ---2 L-(l-p)Z (x.~) +pZ'(x. I-X.) 7
1
ffi
1
J 2a
1
~
~+
~whore
Z' (
)
=
2,,1-1
Z (
)
i=l
ir
ffi
NOH
SO that the
ffi~~imum
liklihood ostimate of
is x.
~
Further
So that
A2
a
1
= -2m
2m
2
I
/-(1- p) Z (X.-IL) + pZ (x. I-X.)
1
~
~+
~
-
2
-
7
2
2
Therefore if we use x for ~, m-l for m and So for s
.
1'2
2
is also
in (2.7.10) i.e. a = (l-P)S~ + 2p o~ we can show that a
unbiased for any £,ivon P •
2.
Now for
a.
J
= 4 cos ~
c:m
64
dlog f == ~~
,-l(a._l)
~ 1\
dP
j=l m J
1 /- Z I( x. I-x.
-2
-
20' -
d log
Ci
f
p
0
=
Xi'
m
2
Z(x.-X )
1 1. 1
+
2m
Z (x.-X )
m+l 1. 2
1
m
2
whore
niX1 = Zx.
J.
ZI(X·
o
1.
2
2m
m+l -
==
1
1.
Z (x. -IJ.) , and 10 t
in placo of
'Y)
1.+ 1
m
z(x.-i1 )
1
2
-::;)2
~.
2m
~
+
m+l
1.
(x.-x 2 )
2
l
(2.7.11)
Furthor frOl;l tho idontity
m-l
(2.7.12)
-1
m-l
A. + P ~
j=l J
j=l
Z
a
m
2
Z (x. I-X.) - Z (x.-IJ.)
2m
= Z
-
2
2
t
l+
TTlX 2
1.
2
Z (x.-IJ.)
1
1.
ill
If we ll'rite
1
,
2m
1
Z A~ (a.-l)=m /j=l J
J
-
)2 - Z2m(x.'iJ.) 2 -I .
1.
likelihood cstL~ato (2.7.10) for
If we usc the maximum
g0t, by sGttinc
J..+
A~l(a._l) "" m - 1
J
J
1
1.
and
2
we
65
we get
(for prO)
m-l 1
Z A.: (a.-I)
j=l J
J
(2.7.13)
Thcl'cforo,
m
since
A.
-1
~ A.. (a.-I)
j=l J
J
Now for largo
m
=1
=p-
1
-1 7
= p -1 L-m-l- m-l
~ A.. _
j=1 J
-
p .
and
/-m-l-
m-l
a ~ 0
m
1
~ A.:
j=l J -
,
7 _ (l_p)-l
•
m we have given the approximate value of
m-l -1
~
A..
j=l J
in
(1.8.6) by an integral approximation.
Hcnce
or for largo
(?7.16)
m
-1 m-l -1
m
~ A..
j=l J
1
p-l
1
~1-(1-p)-2(1+3p)2_7
Usine this approximation in (2.7.11) wo get
By alGebraic simplification we got
66
1:
1:
(l- p ) 2 (1+3p) 2
(2.7.18)
Squarin~
= 1 + p ( 1)0-1)
both sides and simplify.ing
vTO
get
2 2 2
(2.7.19)
+ 2(1)0-1) P - 2p+ 3p
P (TlQ-l)
= 0
Hence
This moans for large
Now for
ro
0
~
'lo
<
4,
m the maximum likelihood ostimate of
"'-
P is a monotonic decroasing function of
and si1co this is the intorval for
justified in using
1)0
p
~rhich
1)0
is dofined we arc
for tests concerning p
8. Remarks.
T.
il,
Anderson ["2_7 showed that if the model matrix was
2
1+ p - p
-p
-p
2
l+p
o
0
-p
o
(2.8.1)
/\ ,.
1
o
1+p2
o
_p
2
-p l+p -p
,
67
tho Von Neumann ratio led to a libost tl test.
It can bo shown that if the model i,latrix (2.8.1) is used, tho
asymptotic ,-;1aximum likelihood estimato
function of the Von Neumann ratio.
Part I, section 9, hold
als~for
p
is a monotonic decreasing
All the proporties of P ik
of
this model.
Further
where
1
-1
-1
2
A=
Hcmco
A1
2
-1
-1
1
2
has roots
A.
=(1-p)
j
2.
+
4p cos
¥m
and
A.
1
commutes
wi th tho nULlcrator and denominator of the Von NGllffiann ratio,
There-
fore all the power functions of the statistics given for tho model
of Part I, scction 9 can be given for
chango of the latent roots
is the model matrix.
A.., when
J
T. 11. Anderson f s model with the
68
This seems to further advance the usc of the Von Neumann ratio
to test for dependence between succossj.ve observations, since for two
different models, both hav ing the same Scneral properties in common
with the stationary r1arkov process, the Von Neumann ratio provides a
'1best lf taste
T. F. Anderson
£2_7
showed that tho Von Neumann ratio
led to a unifonnly most powerful test for one-sided alternatives if
the model (2.8.1) is used.
This property also holds if the model in
tho previous section is used, since it satisfies Anderson's
concerning uniformly most powerful tests given in
~2_7.
theorer~
69
BIBLIOGRAH-IY
"Distribution of the serial correlation coefficient," Annals of Mathematical Statis~, XIII (1942), pp. 1-13.
liOn the theory of testing serial correlation,"
Skandinavisk Aktuariettdskrift, XY-XI (1948),
pp. B8-1lr---'-
L- 3_7
T.
1fT.
Anderson and R. L. Anderson, rrDistribution of the circular serial correlation coefficient for residuals from a fitted Fourier series," Annals
of l1athematical 3tatisti.cs, XXI (1950 ), pp.
59-81.
-----
D. Dierens DeHaan, Nouvelles Tables Dllntegrales Definies,
G. E. Stecnert and Company, New York (1939).
L- 5_7
J. Durbin and G. S. Watson, "Exact tests of serial correlation using non-circular statistics," Annals
of lJiathematical Statistics, XXII (1951), pp.
446-451.
"Distribution of rCltios of quadratic forms, It em-des
Commission Discussion Paper, Statistics, No.
354 (1951).
rllnversion formulae for the distribution of ratios,1I
Annals of Mathematical Statistics, XVIX (1948),
PP. 228-237.
r 8 7 u.
-
-
Hoeffding and H. Robbins, rIThe central limit theorem for dependent random variables,n Duke Mathematical
Journal, XV (1948), pp. 773-780.
liOn the asymptotic distribution of certain statistics
used in test~n~ the independence between successive observations from a normal population,"
Annals of Mathematical Statistics, XVII (1946),
pp. 350-354.
"on the mean successive difference and its ratio
to the root mean square," Biometrika, XL
-(1953), pp. 116-127.
"Serial correlation and quadratic forms in
normal variables," Annals of Hathematical
Statistics, XIII (194?), pp. 14-23.
70
on the distribution of the serial correlation coefficient ~ II Annals of l'1athematical
Statistics, XVI ~1945), pp. j08-310•
r~ote
.1:13_7
B. Hatern,
fllndependence of non-negative quadratic forms in
normally correlated variables, II Annals of
Mathematical Statistics, XX (1949), pp. 119-120.
11. D. IIcCarthy,
liOn the application of the z-test to randomized blocks," X (1939), pp. 337-359.
liOn the distribution of quadratic forms ll (Unpublished thesis), Library of the University of
North Carolina, (1953).
I. J. Schwatt,
L-17_7
J. Von Neumann,
An Introduction to the operations with Series,
The Press of the University of Pennsylvania,
Philadelphia (1924).
'~istribution
of the ratio of the mean square
successive difference to the variance, II Annals
of Mathematical Statistics, XII (1941), pp. 367-
395.
J. Von Neumann,
"A further remar]( concerning the distribution
of the ratio of the mean square successive
difference to the variance, II Annals of IvIathematical Statistics, XIII (1942), pp. 86-88.
Neumann, R. H. Kent, H. R. Bellinson and B. I. Hart,
TIThe mean square successive difference,1t Annals
of Mathematical Statistics, XII (1941), pp.
153-162.
-;-20-7
G. S. Watson,
IISerial correlation in regression analysis II
(unnublished Thesis)~ Library of North Carolina
State College, (1951).
The Theory of Groups and Quantum Mechanics, MGthuen
and Co., Ltd., London (1931).
L-22_7
E. T. T1hittaker and G. N. \o,!atson, Modern Analysis, Cambridge
University Press, (1952).
L-23_7
J. D. 1!illiams,
"Moments of tho ratio of the mean square
successive difference to the variance in samples
from a normal universe, II Annals of l1athematical
Statistics, XII (1941), pp. 239-241.