I
.~
ON THE: DISTRIBUTION OF AVERAGES OVER THE VARIOUS
LAGS OF CERTAIN STATISTICS RELATED TO THE
SERIAL CORRELATION COEFFICIENTS*
by
V. K. Murthy
University of North Carolina
This research was supported by the Office of Naval
Research under Contract No. Nonr-855(09) for research in probability and statistics at Chapel Hill.
Reproduction in whole or in part is permitted for
any purpose of the United States Government.
Institute of Statistics
Mimeograph Series No. 262
August 1960
ON THE DISTRIBUTION OF AVERAGES OVER THE VARIOUS
LAGS OF CERTAIN STATISTICS RELATED TO THE
SERIAL CORRELATION COEFFICIENTS*
By
V. K. Murthy
University of North Carolina, Chapel Hill
SUMMARY:
The sum of all the different circular serial corre-
lation coefficients defined in the usua,l manner, with lags of 1, 2,
3, ••• 3-1 time units and a sample of N successive observations,
turns out to be identically equal to
-1 while the corresponding
sum of the non-circular serial correlation coefficients, defined with
the sum of squares of deviations from the mean as common denominator,
is identically equal to
-1/2
The customary definitions of the
circular and non-circular serial correlation coefficients are slightly modified hereby by dropping the correction term due to the
sample mean.
It is shown in this note that a certain function of
the average of these modified circular serial correlation coefficients and another function of the average of modified non-circular
serial correlation coefficients based on a random sample of size
N
from a normal distribution with zero mean and a fixed variance have
F-distributions with N-l
and 1
degrees of freedom.
~ be a random sample of size
Let Xv x2 '
normal distribution with zero mean and unit variance.
0
•
N from a
*This research was supported by the office of Naval Research
und.er Contract No. Nonr-855 (09) for Researeh in probability and statistics
Reproduction i.n whole or in part for any purpose of the
United States Government is permitted.
&
Then one can see that
x
( OK + ON-K + 0'.K + 0' N-K)
(4)
where
,1.
v
X
=
1\
~r
"'"2
l'
"
I
,
and
,Xul
I
C
denotes the transpose of the matricJ( C.
Now
It
(5)
N-1
Z
K=l
EK
""
X
i
N-1
Z
K=l
. d\. c·1\
(-'
")
+
X
-3
e
-
But
N-l
{
z
(6)
K=l
(011 •
I 101 •
llK"1
K
C + C .....
_ _i
Therefore
N-l
l
EK =
1:
K=l
X
.•
• 1
•
•
-1 1
°
)
=
D(say)
D X.
The characteristic roots of the matrix
and -1
• 1
D are easily seen to be
N-l
the later repeated N-l times.
But
N-l
=
E
K=l K
(8 )
1:
f
X D X
•
N---;K~ ~
are related to Xl' X2 ".
transformation we find from (8) that
N-l
r
1:
K=l
(N-l)
K
is distributed like
yi - Y~
-
0
•
•
o
• ...
+
y2
N
(I
N by
X
a v'-0r°i;ain orthogonal
- 4Now definLlf;;
r
'= 1, we get
O
111...1
L
r
K"'O
K
is distributed like
where Yl ' Y2, • • IN
are normally and independently distributed variates
witlJ; zero mean and a common variance.
Hence
(10)
=
r
N-l
2:
•
KeO
is distributed like
(11)
•
e
From (le) and (11)
(12)
F
::1
,,~t~
1 -
__
•
2
YN
one can easily see that
r
..-....-~_._
(N-l)
+
.. _
I
r
has an F distribution with N-l
and 1 degrees of freedom.
Now, consider the non-circular serial correlation coefficient of lag K
defined by
(13)
N-K
r~ = Z x. 'i-"'Jo+k
'"
j=l
-..
v
~--""f'_~~H
j=l
J
--
.. 5 ..
It is easily seen that
~l-
N-K
E =Z
K
j=l
Therefore,'
.
(14)
N-l
Z
K=l
r
x x
j
*
K
2
N-l
=t
~;./ ~. x~
Z
K=l
x
)
j=l J
1
I
N- l
-- 1\=1
~r I
A
(Kc- T)
+ C 11.
= Z X Z
1
= 2
C~·K }
1
=.~
j+k
X
)
D X
..... t •
X X
Hence
is distri"tm'c,ed like
(17)
-l.-.
2N
+
where Y1 '
1
,
2"
Y2~
• • Y
N
are normally independently distributed ran... '
dom variables Hi+h 7.,ero mean and a C01TlJ.llon variance.
~~om
(18)
(16) and (17) one can easily see that
•
'
)
.l,--
- 6has an F - distribution with
1
F- '2'...
N-l
and
1
degrees of freedom.
The function
of either (18) or (12) has the student distribution with
'll
•
-.--
N-l
degrees of freedom.
ACKNO(JL~;DGN:::I:NT:
I thank Prof. Harold Hotelling for his comments and criticisnls.