ON ES'l'llVlATION Olt' 'I'HE SPECTRUM
.';.
by
V. K. Murthy
VnivE:rsity of' North Carolina and Andhru University
Sponsored by thE: Office of Naval Research under
thE: contract 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. 185
N~)vember, 1957
Ii."i...
ON ESTIMATION OF THE SPECTRUM*
by
V. K. Murthy
University of North Carolina and Andhra University
The object of this note is to extend some results of Grenander
!-11
L..,.
rela-ting to discrete real stationary normal process with abso-
lutely continuous spectrum to the case in which the spectrum also
contains a step function with a finite number of saltuses.
It is shown by Grenander
to. "
L~J
that the perlodogram is an as-
ymptotically unbiessed estimate of the spectral density
f(~)
that its variance is [f(~)12 or 2 Lf(~il2 according as ~
:f 0
~
= O.
and
or
In the present note the same results are established at a
point of continuity.
The consistency of a sUitably weighted periodogram for estimating f(~) is established by Grenander
I..lJ.
In this note a
weighted periodogram estimate similar to that of Grenander [1]
(except that the weight function in this case is more restricted)
is constructed which consistently estimates the spectral density
at a point of continuity.
It appears that this extended result leads to a direct approach
to the location of a single periodicity irrespective of the presence
of others in the time series.
*
This work was assisted in its final stages by the Office of Naval
Research through the contract for research in probability and
statistics at Chapel Hill, North Carolina, Project NR 042031.
Reproduction in whole or in part is permitted for any purpose of
the United states Governm~nt.
- 2 ..
We shall now proceed to establish our results.
Let x(n) be a discrete, real, stationary, normal process.
is known (Karhunen
[1])
It
that the process can be decomposed into
two mutually orthogonal stationary processes as x(n)
= x1 (n)+ x2 (n)
where xl(n) 1s a purely periodic process and x2 (n) is a purely
non-periodic process.
Let
CX( . N),
x(-N + 1), ••••• x(-l), x(O), xCl), •••• x(N - 1),
X(Ni] be a realization of size
2N + 1 from the process x(n).
us consider the statistic proposed by Grenander
INP.. )
=
N
1
21t(2N+l)
[1]
I
I _I
2
x( v)e -1 \I'" I
=
(1)
j\l--N
This is the usual periodosram based on observations.
IN(h)
We have
. N
1 ",' 2
l' N
i
i r. x (\I)e- v I' +
I' =E_ x (v)e1
2
N
21t(2N+l) ! v=-N
21t(2N+l) v
1
+
1
21t(2N+ 1)
+
1
2rr(2N+ 1)
N
t," V=~N
Xl (v)e
-1vh\
N
(V=~N
Let
x 1 (v)e-
.J
iV
v"'I' 2
t' ~
() -i v"')
\ v=-N x 2 v e
(~
() -iV"')
"') , v=-N x 2 v e
(2)
The two stationary parts x (n) and x2 (n) have the spectral repre1
sentations
where zl("') and Z2(A) are orthogonal processes.
We shall have to use the following two lemmas in our further
work.
- 3 •
Lemma 1:
If z(s) is an orthogonal process with the associated measure
o(s) on the subsets s of the elements (A) of W, and if Sl(A) and
g2(~)
are complex valued functions of the real variable A such.
that each of them 1s quadratically integrable on Wwith respect
to the a-measure, then we have (Karhunen [\])
-
1
Elf 81 (~)dz(h) J g2(~)dZ(~) I = J gl (~)82(~)do(~)
/-"W
J
W
where
(3)
W
dcr(~) = E {dZ(~) dZ(~)}
,
Lemma 2:
For any discrete, real, stationary, normal process with absolutely continuous spectrum, Grenander
(iJ
has shown that
1
a)
r·~
2
D ! I~(~)l =
b)
L
_J
+f""
I
1f
_.
1
1
2
IN(~)
2
Sin (2N+l)(¥)
)
L~1f (2N+ 1 )
. 2 i.A
T
::1f
Sln
sin I. ;~
denotes the variance of IN(A); also that
IN(~~l =
c)
RN(A,
~)
sin(2N+l) (!;f)
£-~
sin '2
-
+
[
1
,,~1t (2N+ 1)
1f",.
f
,.,
-:It
ain(2,Nfrl) (¥)
.
Sln
-£-'A.
2
.e.~
ain(2N+l)T
-
i-~
sin ""'2
f(.e)~~l
~.).
(Sin(2N+l) (~)
L~1f (2N+ 1)!.1f
where D
f(£)d.e,
2
- 4 where f(X) is the spectral density.
Using lemmas (1) and (2),it is easily seen that for our proceases, i.e. for discrete, real, stationary, normal processes
whose spectrum includes besides the absolutely continuous part, a
step part with a finite number of saltuses,
1
=
23f(2N+l)
+ -1- -
(4)
21C(2N+l)
3f
sln2(2N+l)~
-rc
2 i-X
sin ""2
sin(2N+l)~
sin i.x
2
it
+[2n(~1) 1.
,...
i1 =
cov LIN(X), IN(IJ~J
RNP.·, IJ)
==
(1)
RN
(2)
(X, IJ) + RN (X, IJ),
where
R(l)(X, IJ)
N
R(2)(X, IJ)
N
= 1=__
· __1____
~rc(2N+l)
= /-
..
3f
r
Sin(2N+l)1.;",
rc
sin(2N+l)~
-1f
1. ."
sin .....!::.
2
~
-1£
1
-2rc(2N+l)
i. ... X
sin -;=;-.
II;
(6)
- 5 From the nature of the two parts xl(n) and x (n) of the process
2
x(n), their spectra
al(~)
and
a2(~)
are respectively a pure step
function and an absolutely continuous bounded measure function.
Also it is evident that the spectrum
the sum of
01(~)
cr(~)
of the process X(N) is
and 02(h), the spectra of the two parts.
We sball now prove
Theorem 1:
For any real, discrete, stationary, normal process whose
spectrum consists of the absolutely continuous part and a step
function with a finite number of saltuses, IN(h) is an asymptotically unbiassed estimate of
f(~)
at a point of continuity of O(h).
.
Proof: '
Let Sl' 8 2 , ••• , Sp be the steps of
al(~)
tbe values Xl" h2 ' "., hp of h in (-1t, 1f).
r
-,
E~N(h~ =
r;.. Sin2(2N+l)l~h
61n 2 ~
1
J
21f(2N+l)
-n:
+ __1_,_ _
2n:(2N+l)
where d02 (i)
corresponding to
'"I: ~ h:;,;;::
from (4)
dOl (£)
2
J
:TC
-n:
Sin2(2N+l)!.?
2 i -h
f ( £)d£,
( 7)
sin 2
= f(.e)d£.
The first term on the right-hand side (R.H.S.) of (7) can be written as
If
~
is a point of continuity of the spectrum a(h) it does not
coincide with anyone of
~
k=l,2, ••• ,p , and hence all the p
- 6 As N -->
terms in the above expression are finite.
~re8sion
tends to zero.
00
the above ex-
By Fejerts theorem the second term on the
R.H.S. of (7) tends to f(A) as N -->
00.
We have thus established
that
N .:..~
E
00
[rN(A~
= f(A) at a point of continuity.
Theorem 2:
For any discrete, real, stationary, normal
pro~ess
whose
spectrum consists of the absolutely continuous part and a step
2-
-,
function with a finite number of saltuses, the variance D lIN(A~
of Grenander's statistic is equal to
-
12
Lf(~)J
[...
~2
or 2 f(A)J
accord-
ing as A I 0 or A = 0 at a point of continuity of the spectrum.
-Proof:We have from (5)
2
d(Ol(1) <.
+
r
02(llD
1
_21t(2N+l)
By an
argu~~nt
similar to the one used in the proof of the
previous theorem, the first term on the R.H.S. of (8) tends to
[f(A~ 2 at
a point of continuity of 0'( A).
In the second term the
contribution of the term containing 0'1(£) tends to zpro as N -->
00
so that we have only to investigate the nature of
1
2:rr( 2N+~.·,
1t.
J
-:rr
ein(2N+1)£;A sin(2N+1)£~A
,
p. -'A.
• £+ 'A.
f (£ j'1..e
sin ~
Sln ~
'. (9)
- 7 •
Case I:
"A:;; O.
Then in view of FeJer's theorem it is easily seen that (9)
tends to
[f(XB ~o
Case II:
"A:I o.
as N ->
00 •
We divide the ranse of integration (-n, n) into six parts as
Let "A> 0 and (-n, -A-e), (-"A-£1 -A+€)1 (-A+e, 0),
follows.
(0, A-e l
),
("A-e', A+£'), ("A+e', n) be the six parts into which the
range is divided where
e' are small, arbitrary, positive con-
€,
stante and denote the corresponding integrals by 1 , 1 , 1 , 1 41
2
1
3
1 and 16. Applying the first mean value theorem, it is easily seen
5
that 1 , 1 31 14 and 1 tend to zero as N --> 00. Consider
6
1
A+£'
1->".
.e+,,sin(2N+l )
sJ.n(2N+l)T
I
1
---:6,:-"
..':""X.........·
•
£+ A.
f ( £) d t •
5 - 2n(2N+l)
sin
T
s~n ~
A.-E'
-r-
J
Putting i-A. = t, we have
iE'
1
5
=
1
2rc(2N+l)
J
e'
=
1
2rc(2N+l)
.J'0
Sin(2N+l)~
'2t
sin
Sin(2N+l)~2"t+2"-
6in(2N+l)~ sine 2N+ l)( 2~-t)
.
6J.U
t
'2
Sin(2N+l)~
t
sin -2
.
2A.-t
sin ~
k
2rc (2N+-1)
f(,,--t)dt
81n(2N+1) t+i'"
t+2"A
sin
~
• •
<
t(t+X)dt
sin T""
dt ,
f(A+t )dt
(10)
- 8 which can be written (Zygmund [lJ) as
5
Hence N
• '.
1:!~ 00 1
5 =
O.
k
<
1
0 (log N) •
21t (2N+1)
Similarly N
.!!~ 00 1 2 = O•
the expression (9) in the case A
r 0 tends
to zero as N
->
co •
~
lave
thus e3taibHshe~ that" at a Wint 0:1 continuiV.AtotAo IONe~D2[IN(Aij
= [f{AO~2;
while at A = 0,
Nl-~ D2[r-N(~~
except in the trivial case f(A)
~
2~(A.U~0'
=
Thus
0, IN(A) is not a consistent
estimate of the spectral density at a point of continuity of
cr(~).
We will now try to construct a weighted estimator which estimates consistently the spectral density at a point of continuity.
Consider
1
~ x (v)e. i VA ~ N x (v) e -1 v~
\1=- N
.
v=.
xCv) being real it is easy to verify that IN{A)
= IN(-~)'
i.e. IN(A)
is an even function of A in (-1t, 1t).
Let W(A) be an even function of A such that in (0, 1t),
vanishes c1utside
w(~)
(~
+ h) and .1:1 is so chosen that the h--neighborohood of Ao does not contain any saltus of cr(A) •
(12)
Consider
f;(~o)
1t
=
~,tIN{£)W{i)d£
(13)
Taking expectations on both sides of (13) we have
- 9 -
Taking limits as N -> co we have at a point of' continuity
;>. +h
o
=
J"
2
f(i)w(i)d£
(14)
;>. -h
o
Putting the condition for f * (;>") to estimate asymptotically unN
biassedly f(;>..) at a point of continuity;>.. of cr(A), we have
2 )~h
f(l)w(t)dl
=
f("o) •
A -h
o
If f(;>") does not vary too much in the neighborhood of A the apo
proximate condition for asymptotic unbiassedness, is
(16)
Theorem 3:
Let w(;>..) be a continuous weight function satisfying (12) and
(16).
Let the spectral density f(A} be continuous. Then at a
point of continuity ;>"0 of cr(A) the variance of the weighted estima-
*
tor fN(;>"o)
goes to zero as N -:;>
00
•
Proof:
We have from Grenander
[iJ
N
=
n,m;lt, i r(n+m)r(k+.e)W (n+k)w{m+ £)
-N
N
+ n,m;k,.e r(n+m)r(k+£)W(ml-i)W(n+k) ,
-N
(17)
- 10 -
where
r(n)
(18)
J
n
wen)
==
n
=
eini\W(i\)di\
-n
j
cos ni\ w(i\)di\
-n
Since w(i\) is an even function we have
.
N
41t2(2N+l)~~;(i\0») = 2 n,m;k,£ r(n+m)r(k+t)W(ntk)W(mH)
(19)
-N
Again following Grenander
2
2
21t (2N+ 1)D
[1]
F;(i\o~] < aJ
I
we have
r r(a)r(p )W( r )w(a+t3-r)
2N
2N
l~
=
t
r(n)W(n+v)
n=-2N
v=-2N
(20)
Case 1:
dO'(i\)
= t(i\)di\
where f(i\) is an even function being the spectral density of a real
process. We have
r(i\)
=
w(i\)
= N->00
lim
N-:>co
1im
f(i\)W(A.)
=
N
1:
-N
co
r ( n )e -ini\ = Z r(n) cos ni\ ,
-00
N
1:
ex>
w(n)e-ini\ = r. Wen) cos ni\
-CD
-N
N
lim
N-;>oo
_~
00
where
d( 11 )
==
.E
-00
d(n)e -ini\ ,
00
r (n )W ( n+ 11 )
= -00
1:
W( n )r (n+ 11 )
•
I
of
- 11 -
=
Let us write
2N
-~N
r(n)W(n+v) •
We have from (20)
2.2(2NtllD21;;U\ol] <
Taking the limit as N --->
since
00
E
-co
00
2
.t fd2N (Ylj2
we have,
d (v)
< 00
,
that
Case 2:
where OleA) is a step function with a finite number of saltuses
8 1 , 82 , ••. , 8p at A , A , ••• , Ap respectively.
2
l
We have from (18)
p
r(n) = r 1 (n) + i~l 8 i cos nAi
We have from (20) in another form
(21)
- 12 p
+
I
E
1,J=1
S S
2N
i j n=~2N
t
W(n) eos n Ai cos v Ai
+ sin n Ai s in v
y~n=-~N2N w(n)f~os
L
/l.J
n /I.. cos v A
J
• Bin n
j
~j
Bin v
r
~DJ'
(22)
But we have, in view of the conditions imposed on the weight function, that
00
E Wen) cos n /I..J. = W(/I..1 ) = 0
-00
eo
1: Wen) sin n /1.
-co
00
and
1
}
)
=0
(23)
1
2
E d (v)
-00
I
< co
>-..J
Taking limits on both sides of (22) as N -->
00
and taking into ac-
count (23) we have, at a point of continuity of o(A), that
lim
N-;>oo
D2
*
fN(/I.)
0
=
0,
which proves the theorem.
Acknowledgement
I wish to express my indebtedness to Professor K. Nagabhushanam
for suggesting this problem to me and for his guidance during the
preparation of this note and I thank Professor Harold Hotelling for
going through the manuscript and for his comments and criticism.
References
1. Ulf Grenander:
I10n Empirical Spectral Analysis of Stochastic
Processes" Arkiv for Mathematik, Vol. 1
(1951) PP. 503-531.
"Dber Lineare Methoden In DeI' Wahrscheinlichkeitsrechung" Ann. Acad. Bc. Fenn. Soc. AI, MathsPhys No. 37, 1947.
3. Antoni Zygmund: I1Trigonometric Series II page 172.
2.
Kart Karhunen: