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s ta t i s t i ca l Inference
fo r
Stationary Point.
Processes
by David R.
B r i l l i n g e r
The
University
of C a lifo rn ia , Berkeley
I n t
rOd ct
ion
This
work i s
divided into
t h r e e
p r i n c i p a l
sections
which a l s o
correspond to th e
t h r e e
l e c t u r e s
given a t Bloomington.
The t o p i c s cO ver, some u s e f u l
point process
parameters and the i r p r o p e r t i e s ,
estimation
o f time
domain parameters
and th e
estimation
o f
f r e ~ ~ e n e
domain
parameters.
The
work
may
be
viewed
as
an
extension o f
some
o f
th e
r e s u l t s in
Cox and Lewis
19.66, 1972) to apply ~
vector-vall1ed
processes and
to higher order
parameters. t w i l l proceed
a t a
h eu ris tic le v el
r a t h e r than formal.
A
fo rm al ap pro ach
may
be
f o ~ n
in
Da.ley
and ·Vere-Jones 1972) fo r example.
The
notation J f w i l l be used
fo r J
f x ) d ~ x , U being
Lebesgue
meas·ure. A
general
lemma
concernin g
th e
e ~ -
istence of
c o n s i s t e n t
estimates
is
given in Section
IV.
Point Process
Parameters
Consider i so l a t e d points of
r
d i f f e r e n t types
randomly
d i st r i b u t e d along th e
r e a l l ine
R.
Prepared
while
th e
alxthor was a M i l l e r
Research
Professor
and
with
th e
support
o f
N.S.F.
Grant
GP-
31411.
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D VID R. RILLINGER
x ~ p l e s that w
have
in mind include,
the
times
of
heart
beats or
earthqua.kes
in
the
case
r
=
1,
the
times
of nerve
pulses
released by a network of r
nerve
eel1s
the
case of general r . Let
Na A
denote
the
number of points
of
type
a fa l l ing
in
the
inter val A R and
le t
Na(t)
=
Na(O,tJ for
a
=
l , ••• , r .
1 .
Suppose
Pr ob
[point
of type a in t , t +h]} lp a t h
as
h 0 • Pa(t)
provides
a
measure of
the intensity
with which
points
of type a occur near t .
can
often conclude that
2.
Suppose,
for t
1
f
t
2
.
Prob
[point of
type
a
in
t
1
,
t
l
+ h
l
J and point
of
type
b
in
t
2
,
t
2
+
h
2
J}
as h
l
, h
2
IO·Pab(t
1
, t
2
)
provides ameas ure
of the
intensi ty with w aich points of
type
a occur near ·
t
l
and
simultaneously points of type b occur
near
t
2
•
A related useful measure is provided by
Prob[point
of type a
in t1 , t l+h
J I point
of
type
b
a t
t
2
}
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fo r
a
b
STATISTICAL INTERFERENCE
as h 0
0
The r a t i o
P ab t
1
, t
2
) / P b t
2
)
i s seen to
provide a measure of
the
i n t e n s i t y with which type
a
point,s
occ ur:
near
t
1
, given t h a t
t h e r e
i s
a type
b
point a t
t
2
•
In th e
case t h a t type a
points are
distribu·ted
independently of
type
b
point
s ,
Pa b t
1
, t
2
) =
P a t
1
)P b t
2
) ,
and
th e r a t i o e o ~ e s
Pa t
1
) , th e f i r s t order i n t e n s i t y .
The
function
Pa b t
1
, t
2
) i s
l i k e the
second
order
moment runction
of
ordinary time
s e r i e s ; however in
p r a c t i se i t
seelns
t o be
ml ch more u s e f u l as
i t
has a
f u r t h e r
i n t e r p r e t a t i o n as
a
p r o b a b i l i t y .
Often
i t
i s
true
t h a t
t t
= J J
P a b t l , t2 )d tld t2
o 0
t t
= fa f o Pa b t l , t 2 ) d t l d t 2
t
J
Pa t)
d t
fo r a
=
b
o
3. Suppose next t h a t ,
f o r
t1,
•• •
, t
k
d i s t i n c t an d
v 1 , . o . , v
r
non-negative i n t e g e r s wit.h
S urn
k
Prob
[type
a
p o i n t
in
each
of
t . ,
t
h . ] ,
J J J
j = L v 1 , ••• , ~ v and a = l , ••• ,r }
b
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D VID RILLINGER
Prob{type a . point in t . , t .+h . l j = l , ••• ,k}
J
J J
- PaI ak(tI , · · · , tk) hI · · ·h
k
as
h1,
•••
,h
k
0;
k
=
1,2 .0
• •
The function
P (\)1)· · · (\)r)
is
ca.lled pr.o.d-uct d E l ~ 1 - - i · : t - y o·f or4er k.
Such
a function was
introduced by
S.
O Rice in
a
pa rt icu la r s it ua ti on and by A. ~ k r i s h n n in a
general s i t u a t i o ~ ,
see
Srinivasan
1974 . No claim
i s
made that
the
probabili ty in
(1) always .depends
on
hI
••. ,h
k
in such
a direct
manner. Rather i t is
the
claim tha t
this
happens
for
an
interest ing
class
of e x a ~ p l e s . B r ~ l l i n g e r 1972 gives an expression
for
4.
The Erobabili ty g e ~ ~ r a t i n g f u n c t i o ~ a l of the
process
~ t = [Nl(t) , ••• ,Nr(t)} i s defined
by
E[exp[J log
Sl(t) dNl(t)
J log Sr(t) dNr(t)}]
for
suitable
functions
Sl
••
o ~ r .
Writing
i t
as
r
er
{I
I; ( f)-I)}]
a:=l type a
point
a
and
expanding,
we can see
that i t
is given by
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STATISTICAL INTERFERENCE
where we define
0
v
~
t • • •
t
v -
This
fUnctional
is of use in computing probabi l i t ies
of
int r st for
the
process. For example
sett ing
~ a t
=
z
for
t
E A
a
=
for
t
and
deterlnining
the coefficient
of
j l
j r
z l
we
see
that
•
v1-j l
•••
v jo
-1 r r
2
e m y l ikewise determine condit ional product
densit ies such as
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E
\ ~ j
r r
D VID
R
RILLINGER
Yl) • • • Yr)
p t
l
···
, t
k
N
l
A = j l
•••
,Nr A) =
jr )
1
••
0)/ 2)
These
c onditiona l
product densities
are
u s e f u l
in
s ta t i s t i ca l inference. They provide likelihood
functions and also
o ~ the
i n v e s t i g a t i o n
of the
,distribution
of
s ta t is t ics
c onditiona lly the
observed
number
of
p o i n t s .
Were
N A
=
0 ,
one
wouldn t want
to
claim much.)
The integrated product
densities
give the
factoria l moments o f
the
process. For e.xaJ.ilple,
if N v) =
N N-l)
N - v + l ) , then
\)1) • • • \ r)
E Nl A)
)
•• oN
A) ) =
\ 1 r \)r k
A
Also
of
use are
the c umulant d e n s i t i e s ,
\)l) · · · \)r)
q t1 ,
••• , t
k
)
given by
3
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ST TISTIC L INTERFERENCE
They measure
the degree of
dependence
of
increments
of the process a t d if fe r e n t t
j
•
Certain o th er c on di ti on al
product
d e n s i t i e s
are
of us e . We mention
Prob{type a point in each of
( t j , t j+h jJ ,
j
=
vb
vb and a = 1,.g. ,rtNl{O} l} /
b a
bsa
and f o r rrl, ••• ,rr
k
Prob{type
1 point in
t , t + h ] \
po in ts of
type
1 , v
2
p oints of type 2 , • • • a t
1
,
2
,
• • •
,
k
respectively}/h
l
+l v
2
· · · v
- p r ( t ,
1 1
'1 2 ' ' ' . ' r
k
vI)g·· v
r
P rr1, ••• ,rr
k
I f a l l points up to t a re in clud ed, this becomes the
complete i n t e n s i t y
lim Prob{type 1 point in t , t + h ] , (u) , u
s
~
5.
Certain
p r o b a b i l i t i e s and mOlnents
a re of
s p e c i a l
in teres t . We
l i s t
some o f
t h e s e .
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DAVID R BRILLINGER
i)
th e renewal fUnctions
Uab. t
=
E{Na t)
Nb
{
=
I}
fo r
t
>
0
t
J Pab u,O) du / Pb O a b=l • •• r
o
The renewal
density is Pab t,O)/pb O)
•
i i th e forward
recurrence
time d is trib u tio n
is
given by
Prob[event
before or
a t
t}
= Prob[time of
next event
from
0 is
t}
1 - Prob[N t) O}
1 _ ~ l lV r
p v)
\ J ~
\
6 t]\J
i i i
th e survivor function or d is trib u tio n of
l ifetime)
Prob[time of
next event
from 0 is >
t
N[O}
l}
Prob{N t) 0 \ N{O} I}
= p O -l r ~ ~ V S p V+l O,o.o
~ • O,t]\J
1 - F t) say.
iv) the hazard function
or force
of mortali ty
~ t
=
f t / l
- F t)).
Prob {point in t , t+h ) t N
{O}
N t)
=
O}/h
where
F t)
is
given
in
i i i
and
f t is
i t s
derivative.
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STATISTICAL
INTERFERENCE
(v) the variance time curve
var
N(t)
= E/N(t)(N(t) - 1) + E N(t) - E
N(t))2
t t
(2)
t
=
J J
p t
l
, t
2
)dt
l
dt
2
+ J p(t)
dt
a a t a
p
t
dt)2
a
(vi) the Palm functions
Ql(j1, •••
, j r
; t
=
prob{N
1
( t)
=
j1, · ·o,N
r
(t)
=
j r
I
NI{a}
=
I}
vl-j l+· ·o+v
- j
= 1 (-1) r r
j1
J
••• j r P1(6)v (v
l
-3
1
••• (vr-J
r
J
1 1 r r
1
+1 v
2
· · · v
r
p r
J
v
1
+••• +v
r
(0 •• 0
(0,
6.
We
next
indicate the
values of
a few
of
these
parameters for some examples
of in teres t .
Example
1 .
The Poisson process with mean intensity
p(t)oThe
numbers
of
points in dis joint intervals
I
1
,o
•• , I
k
are independent Poisson variates with
means P I l , .o . ,P I
k
respectively where
P(I) =
J
p(t) dt.
Here
I
and so
G[E]
= exp[ (s(t)
- 1
p(t)
dt}
Prob
{N A
j} =
P(A)j
exp{-P(A)}
J •
k
I
_
jJ
t l , . oo , t
k
I
N A
=
J
= Z j k l ~ o
••
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DAVID R BRILLINGER
I f
pet)
=
stp( t)
dt and N (s),
s E R is
a
o
Poisson
process
with mean
intensity
1, then the
general
process
may be represented
as
N t = N (P(t))
Example
2.
The
doubly
stochastic
Poisson process
o
Suppose
[ x l t ) , ~
•• , x r t ) ~ t E R+, is a process with
non-negative sample
paths,
moments
m v1 · · · vr t1 , · ·· , t
k
=
E{Xl(tl)· · ·Xl(tvl)
X2(tvl+l)· · ·Xr(tk)}
and moment generating functional
M[Sl,o
•• ,9
r
J = E[exp[J8
l
( t )x
1
( t )dt
jer(t)Xr(t)d t})
Suppose af ter a real izat ion
of
th is
process is
obtained, independent Poissons with mean intensi t ies
x1(t ) , ••• ,xr t a re genera ted . Then
v1 o.o v
r
v1 ·o. v
r
p t
1
,
•••
, t
k
m (t1,o •• , t
k
G [ ~ l , · . o ~ r J
=
M[Sl-1 ••• S r-
1
]
= E[exp{ (Sl(t)-1)x1(t)dt+•••
}]
I f Xa(t) = xa(t)dt,
and
Nl s , •••
, N ~ s
are
independent Poissons with mean intensi t ies
1,
then
th is
process may
be
represented as
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ST TISTI L
INTERFEREN E
N:L .Xl t))
••
8 , N ~ X r t ) )
This
process
seems
to
be
u s e fu l fo r
checking out
general
formulas
t h a t
have b ee n d ev el op ed , such
as
2)
an d (3),
among
other things.
EXaJ;llple 3: The c l u s t e r
process.
Suppose N
t) , ••• ,
N ~ t )
i s
a primary process of c llls te r
centers
with
p r o b a b i l i t y
generating
functional
G [ ~ l , •• o ~ r J .
Suppose t h a t
secondary points are gener.ated
in
independent
c l u st e r s
centered
a t
th e
points
of
•
Suppose t h a t th e p g f
o
fo r c l u s t e r points o f
type
a
centered a t t
is
G a [ s l t J .
Then
th e p g f
o
of the o v e r a l l process is
G [ ~ l
, l ; r J
= E r n l ; l [ c r ~
J ~ k J n E ; r [ c r ~ + J ~ k J }
j k
j k
=
E { ~ G l [ g l l c r ~ J
•••
~
G r [ g r l c r ~ J }
J J
= [ l [ S l \ · J · · · r [ ~ r \ · J J
I f r =
2 ,
and
th e f i r s t
component
is th e primary
process and
th e
second component
corresponds
to
c lu s te rs of one member, then we have a process of
the
character
of the
G G oo
queue.
Example ·4. The renewal process.
Here
th e
points
correspond to th e
par t ia l
sums of a random walk
with
p os itiv e s te ps .
Suppose r = 1 , t
l
< t
2
t
k
, then
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DAVID R. BRILLINGER
As
the
process
has stat ionary
increments,
i t has a
spectral representat ion
and i f SU
denotes
the shi f t transformation,
S U ~ t ~ t + u ,
then
Pa
Pab t
l
- t
2
\}l · · · \}r
p .
t l - t
k
···
t k_ l - t
k
Pa t
Pab t
l
t
2
\}l · · · \}r
p
tl ··· t
k
p t
... t
1 .
which
the process
is
stationary,
tha t i s probabil-
i ty dis tr ibut ions
are
invariant under t rans la t ions
of t . This means for example,
p 1 t
1
p 2 t
2
t
1
p 2 t
3
t
2
p 1 t
1
p 1 t
2
P 2 t
k
, t
k
_
1
p l t
k - l
where p l
and p 2
sat is fy renewal
eq :uations, see
p. 5 in Srinivasan
197
4
.
Example 5. Zero cross ing processes . Expressions
may
be
se t down
for
the
product
densit ies
of point
processes corresponding to the zeros of random
d b _ e t t e : : r 1 ~ 7 2 .
7. We now turn
to
a
consideration
of
the
case in
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ST TISTI L INTERFEREN E
Na t = JC exp{itA}-l / iA ] dZa A
_co
~ o r a
l r
We
may
define
cumulant spectra of
order k by
V1 ··· V
r
o Al+.o.+Ak
f Al,
••• ,Ak_l dAl
••• dAk
=
cum{dZl Al ,··G,dZl AVl ,···,dZr Ak }
with o e
the
Dirac delt.a. ~ u n t i o n Alternately,
making use
of
product densi t ies , we might d e ~ i n e the
power
spectra by
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DAVID BRILLtNGER
mixing condition,
.Assumption
~ t ,
t E R,
i s
an
r
vector-valued
stat ionary
point
process sat isfying
1), whose
c tunulant
densi t ies
of 3 sat isfy
The second-order
spectra
of the
process,
fab A , possess
many of the same propert ies as the
spectra
of
ordinary
time
ser ies .
There
are
however
some
differences,
we mention
that
for mixing point
processes instead of
the
l imi t
for ffilxlng ordinary time seriea.
The spectral
representation ~
be used to
relate the point process
to the
associated ordinary
time series
h
f. t) =
h
t - ~ , t + ~ = exp[iAt}[ sin h
A
2
hA/2 J d ~ A
t This shows, for example,
tha t the
cross-
spectrum
of
the a-th
and b-th cOlnponents of
. ~ t
i s
8. A
key indicator of
the appearance of the process
of
points
of
type
s a y ~
is
provided
by
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ST TISTI L INTERFEREN E
h small
the
empirical
intensi ty with
which
points of type
1
are
seen
to occur
near
to Models
for
the
process
may
usefully involve models for th is ·variate.
A
simple
statement says
Prob[point of type 1
in (t , t+h]}
~ Plh
for
h
small.
A
more
complicated statement i s
Prob[point
of
type 1
in
( t , t+h]
\
point
of
type
a at
1 }
..
PI
(t-1 )h/p
a a
In the
case
that the process 1,
near
t is
independent of the
process a,
near 1 th is las t
is
~ l h the marginal intensi ty . This
happens
often
as It-1 l
00 .
An even more
complicated
statement
involves
Prob[point of
type
1 in
( t , t+h]
vI
points of type
v
2
points of
type 2
• • • a t
1 1
-2 • • • ,
1 k
respectively}
(v
+l)(V
2
)···(v
r
)
p t - 1 k
1-
1 k ··· , 1 k_l- 1 k)h/
· · · v
r
p (rr
l
-1 k,···,1 k_l-1 k)
Suppose r ;
2.
A useful simple model here is ;
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DAVID R BRILLINGER
Prob{point
of
type
1 in ( t , t+h]
N
2
(U),
C X < U ~ J
{fJ.
a(t-u)
d
2
(U)}h
4
l:
a(t- T .)}h
j
J
where
the
T .
are
the times
of the
events of
the
J
second process.
This
model
allows
the intensi ty ,
near t of
points
of
type
1
to
be affected in a
direct
manner
by
points
of
type
2.
f
the
system
is c ~ l s l then a(u) = 0, u < O The second
process
may excite or
inhibi t
the f i r s t process depending
on the sign of a(u) .
The model
implies,
for
example,
5
showing that ~
may be
i n t e r p r ~ t e d as
the intensi ty
with
which type
1 points woul·d occ ur
where
P2
=
o.
Also
f
A A) = J a(u) exp{-iAu}du
then 5
and
6 lead to
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STATISTICAL INTERFERENCE
suggesting
how the
para.meters ~ A A might be
ident if ied
o
I f P22 u
is constant,
as in the
Poisson
case,
then 6
leads
to
and
a t may
be
measured direct ly .
As an example of
the model
4 we mention
the
G/G/oo
queue
with
N
l
referrin g to the
process
of
exi t
t imes,
N
2
to the
process
of entry times, a -u
referring to the density of service times and
~ =
O.
Clearly, here
r o ~ { c u s t o m e r leaves in
the interval
t, t h] t
N
U ,
_ X · < t }
,.. [ t a t-rr . }h
j
An
interest ing
problem
is
that of measuring the
degree
of
association of two point
processes.
A
measure
suggested
by
the
preceding
model
is
the
o ~ r n
see Bri l l inger 1974a . This
parameter
also
appears
as a measure
of the
degree of l inear
predictabi l i ty
of
the
proeess
N
l
by
the
process
N
2
• I t
sa t i s f ies
a R
I2
A) 1
2
s 1.
Other
measures
of
association
could
be
based on the
nearness of
the
function
P12 u PIP2 to O.
We
mention next the self-exci t ing processes
introduced by
Hawkes see Hawkes
1972 .
For
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DAVID R BRILLINGER
r
= 1, these sat isfy
Prob{point
in
t , t+h] ,
N u ,
u
~
t}
t
+ a t-u dN u }h
_ 0 0
~ + l a t-rr. }h
.
~ t
J
J
I f we have more
than
one
p rocess, then w
could also
set
up multivariate l inear models and
define
par t ia l
parameters.
As
another
extension,
we
could consider non-linear models such
as
Prob{point
of type
I
in t , t+h]
I N
2
U , _ o o ~ u o o }
-faa
+
J a1 t-u dN
2
U +uU
a
2
t-u,t-v dN
2
U
dN
2
V }h
More
detai ls concerning such extensions may be
found
in B rillin ge r
197
4b
9
We
end by mentioning
that
some, possibly
unexpected, relationships exist between certain of
the
parameters
that
have been defined. These are
the
Palm-Khinchin
relat ions,
00
Prob{N t
S
j}
= p Prob{N u = j I
N{ol
=
l ldu
t t
= l p
o
Prob N u = j t N{O} =l}du
Prob{N t >
j
N{O}
= I} =
1 + D+{p-l
j+l-k •
j=O
Prob [N t .k:}}
EtN t N t -l •••
N t - k } =
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k+l P J t E{N u) N u) - 1
•••
N u) - k +
1
I
o
{ }
= I}
du
Such relat ionships are discussed in Cramer,
Leadbetter and
Serfl ing
1971).
In
th is
f i r s t
section
of
the paper we
have
sought to
provide
a framework
within
which
stat ionary
point processes may be handled when the
only
element of s ta t i s t ica l independence is
asymptotic.
I I . Estimation of
Time Domain
Parameters
for
Stationary
Processes
We consider
th e estimation of certain time
domain parameters g iven a realizat ion of a
process
t ) over the interval O,T], i . e . given
the
observed
times of events
in
O,TJ. We
begin
with the
f i r s t
order mean in tens i t ies p ,
·a
a
z l , ••• , r .
1.
Obvious estimates of the P
a
, a = 1 , • •• , r , are
the
a l , ••• r In connection with these we have,
Theorem 1.
Suppose
the
process sa t i s f ies
Assumption I . Then [Pl,
•••
,PrJ is asymptotically
as T .. CX .
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DAVID BRILLINGER
T his theorem , as
are
those given la te r , is
proved in th e f ina l sect i on
of
th e
paper.
The
e stim ate s a re a sy mp to tic ally
normal.
The
s ~ n p t o t i
variance o f
p i s 2n
T - l f
0). Were increments of
a aa
- 1
th e
process uncorrel at ed, th is wO uld be T Pa. We
w i l l
see
how to estimate f A)
next sect i on.
Were
aa
T
l a r g e , we might
s e t T = J U
an d
take
The r a t i o
2nf
O)/p is
u s e f u l
in describing
aa a
c e r t a i n asp ec ts o f th e process N
a
• I f i t is
g r e a t e r than 1 , th e process is
said
to be cl ust ered
o r u n d e r d i s p e r s e d .
I f i t is l e s s than 1 , th e
process is c a lle d
overdispersed.
2 .
In th e
second
order
case
we are
in te re ste d in
estimating
Pab u) ~ r o b {type a in t+u,t+u+h
l
J and type b i n
t , t + h
2
]}/ h
l
h
2
fo r u fO and
Pab U)/Pb
Prob[type
a in t+u,t+u+h] 1 type b
a t
t }/h
fo r
u f
I t seems n a t u r a l
to
base
e stim ates o f the s e
on
J;b U)
=
[ j ,k) such tha t u -
S <
-
<
U
and
7)
fo r
some small
b in width 26
> o. On th e
6400,
th is
s ta t i s t ic
can
be
computed abo-u.t twice
as f a s t
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STATISTICAL INTERFERENCE
as
a
direct convolution
based
on N T
values.
In
connection with
th is
variate
we have,
Theorem
2.
Suppose the
process
~ sa t isf ies
AssuJnption
I
and
that p b
. is
a
continuous
a
function
for
a,b
= l , ••• , r . Suppose Jab(u) is
given
by 7 with
a = depending on
T. Suppose
u ~ uk
with
l u ~ -
u ~ / I T ~ 2S
T
for
1
S
k <
k ~
K.
Then
as
T
~ 00 (i)
i f ST
= LIT,
fixed,
the
variates J ; l b 1 u i , · · . , J ~ b K U ~ are a s ~ n p t o t i c a l 1 y
independent
Poissons
with
means
2S
T
T
Pakbk(u
k
) ,
k
= 1 ,
••• ,K and
i i)
i f
aT ~ 0,
but
STT
~
00 the
variates are
s ~ n p t o t i l l y
independent normals with
variances
2 ~ T T P b
(uk)
k =
1 ,
••
o,K.
a
k
k
The
two
asymptotic
dis tr ibut ions
are
consistent
for
large ~ T T ,
becrolse
a Poisson
variate with
large
mean
is approximately normal.
The
resul t
in
( i)
is
not unexpected beCa1 1Se we a re counting
rare
events
o
I t is surprising that such
a
general
resul t
is so simple however.
The
theorem leads us to estimate
Pab(u)
by
and to approximate the
distr ibution of
th is variate
by
2S
T
T -1
P 2S
T
TP
ab
(u))
or N P
ab
(u), (2s
T
r)-1
Pab (u) ) ,
where P ~ here
denotes a Poisson distr ibution with
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DAVID
R
BRILLINGER
mean
This estimate
should
prove
reasonable so
long
as 1u In th e case tha t u
has
not i ceabl e
magnitude
compared
to T
i t
might
be
bet te r
to
replace
J;b u) by
T J ~ b u
T -
lui) or by
J ~ b u
P a P b l u l ~ S T 8
The use
of th e v a r i a t e
o f B
i s
suggested by
th e
u su al estim ate o f the autocovariance function o f an
ordinary time s e r i e s . I ts
const ruct i on is based
on
th e
observation
t h a t
qab u)
0
as lu
00
fo r
many
processes.
I t should have
bet te r
o v e r a l l mean
squared
e r r o r p ro pe rtie s fo r such
processes.
e remark t h a t
we
a re here e s s e n t i a l l y
c a r r ying
o ut histo gra m const ruct i on.
Considerations
o f t h a t
t opi c
are
rel evant
he r e .
For e x ~ p l e we
j i g h t
choose to
construct a
rootogram based
on
J ~ b u
to
get
stable
variance.
I f
there m y
be
some cel l s with
low
c o u n t s, we
might
follow Tukey
and
use
}2
4
JT
b
u)
. The variate P b u) w i l l
a a
1
have approximately s t a b l e vari,ance of BaTT - •
The theorem
lik ew ise lead s
us to estimate Pab u)/Pb
by
J;b u)/ 2S
T
N
b
T)) and to approximate the
d i s t r i b u t i o n
of th is
estimate by
2S
T
T P
b
-1 P 2S
T
T Pab u)) or N Pab u)/P
b
,
2S
T
T p ~ - l p a b u .
The variance
of
J J ~ b u / 2 S T N b T
w i l l
be
approximately s t a b l e and
may be estimated by
8S
T
N
b
T -1.
The
above
resul ts
may
be
used
to
s e t
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ST TISTI L INTERFEREN E
approximate confidence in tervals and multiple
confidence
in tervals
for
the
estimates. In the
case
tha t
the
increment
of the
process
N
i s
a
independent of
the
increment
of
the
process N
b
, U
time
units
away, Pab(U)/P
b
= Pa.
We may
examine
th is
hypothesis
by plot t ing
on the
same
graph
for
x ~ ~ p l
This
sor t
of graph
i
is useful in
checking for
some degree of
associat ion
between
the
process N
a
and the process Nbo
What
we have been doing
may
be viewed as
es timating the probabil i ty
density
function
of the
times between
a events
and
b events from
the
observed
differences
a <
a
f
J
Cox (1965) suggested th t one could also
consider
window estimates. If Let W(u) be bounded
and
absolutely integrable.
Let
WT u) = W(u/S
T
)
for
the
sequence
of
scale
factors
ST
T
=
1,2,
• • • •
t i s now natural
to
base
estimates
on
T b
Jab(u)
= at:. b W
(u
- f
j
+ f
k
)
o < fj;a fkST
II
WT u-{Y+ f) dNa
a
dN
b
(
f)
O < T ~ O ~
(The previous J ~ b ( u ) corresponds
to
W(u)
1 for
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DAVID
R
BRILLINGER
lut
< 1 .) The
variances
of
the
asymptotic
dist r ibut ions of
i i
of
Theorem 2 are now replaced
by STT JW U)2
du
Pab (Uk) k
=
1 ,
0
,K. By di rec t
computation
we see
tha t
T
E J ~ b U = J (T_lp )wT(u_p) Pab(p)dP
-T
J ST T - lul) [Pab(u) J w P)dp - e T P ~ b u )
SPW
p dp
S i p ~ b
(u)
J
p
2
W
I
dp/2
}
suggesting tha t
bias
may become a problem when
Pab(p)
varies
substantial ly
i ~
the neighborhood
of
u
or
when u is
of appreciable
magnitude
compared
to
To We have
already discussed
one modification
to handle th is las t case.
The
asymptotic
dis tr ibut ion
determined
in
Theorem 2
is
an
unconditional one.
In pract ise the
worker may feel
tha t the
conditional d i s t r i u t i o ~
condit ional on the
o s e ~ v e
Na(T), Nb T) is
the
appropriate one.
In
I .4 we set d o w ~ the form of
product densities in the conditiona l case. I t
should be
possible
to make use of these to determine
the form
of
the large sample conditional dist r ibu
t ion .
Cox
and
Lewis 1972)
discuss
some aspects of
the problem of estimating second-order product
densit ies
for
a vector-valued process.
30 In the k-th order
case
we
might consider
the
s ta t i s t ic
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STATISTICAL
INTERFERENCE
T T ~
J a l o o O a k U l ~ o . O ~ U k - l =
J
o; . J W u l - a l + a k ~ · · · ~
uk_l-ak_l+ak) dNal(al) •••
dNak(ak)
9)
where
W T U l ~ . o . ~ U k _ l
= W U l / ~ T ~ •• o ~ u k _ l / a T
and
the ; l
in
9)
indicates
that
the
ra.nge of
in tegrat ion
i s over
dis t inct cr_.
J
Theorem 3. Suppose the process sa t i s f ies
.Ass·ump
..
t ion
I and that P a (e)
is
continuous a t
a
l
•••
k
)
Th T
1
-) l-f STk-1T =
L,
1 ••• ,u
k
_
1
• en as
~
00
L f i x e d ~ i f W u l ~ ••• ~ u k _ l = 1 for
lUjl
< l ~ the
variate of 9) is asymptotically Poisson with mean
2 ~ T k - I T
P (u1,
••• ,u
k
_
l
) and
i i )
i f
a l · · · a
k
ST O ~ but S ~ - l T the
variate
i s asymptotically
normal
with mean
T T T
TIT· · ·
ITw u l - P l ~ ·
••
~ u k - l - P k - l P a l ••• a
k
(Pl , •••
,Pk_l)dPl
•••
dPk_l
10
k-l w )
nd
variance
T P
a
a u l ~ · · u k _ l ·
•
•• k
The in tegra l
of
10
may
be
expected
to
be
near
k-l k-l
)
T W
Pa a
u l , · · · , u
k
_
l
•• • k
suggesting the
consideration
of the estimate
A T
P
a
a (u
l
Uk_I)
= J
a
a (u
l
,u
k
_
l
) /
•
••
k
•
••
k
S ~ - l
T
k - l
W )
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DAVID R. BRILLINGER
4.
Let
A denote the
in terval
O,TJ and
suppose tha t
the
points observed in A are: Y
I
of type I a t
t
l
,
•••
, t
y
Y
r
of
type
r
a t
t
k
•
Then,
using the
~ x p r s s o n s
of Section
I
the l ikelihood
function
here
is B/C where
and
Let us consider the approximate
value
of
the
l ikelihood function, B/C, for large T. In the
case
of
large T
J
vI ··· v
P
r
t
t
J
1 - y1
+. • • V r
Y
r
1 •
••
Y
l
• • •
YI · · · Y
r
p tl,···,t
+ +
YI •••
Y
r
vI-YI+··.+v
Y vI-Y
I
v -Y
T
r r
p p
r r
r
suggesting tha t for
large T,
the
l ikelihood function
i s
approximately
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STATISTICAL
INTERFERENCE
4. In th is sect ion we wil l
propose
estimates of
the
parameters described in
Section
I .5
in
the
case
tha t a
real izat ion
of a stat ionary process
i s
available for the time
in te rva l (O,TJ.
i We begin
with
the
renewal
function,
t
Uab t = E{Na t
Nb O}
= = JPab U dU/P
b
A
natural estimate to consider is
Uab t = t ~ j T ~
O}/Nb T
T-t
t
=
J
J
dNa U+W) dNb(U)/Nb(T)
o
0
To
determine
the asymptotic dist r ibut ion of
U
b t
a
we wil l need tffi
jo int
asymptotic dist r ibut ion of
{.} and Nb(T). I t i s fa i r ly c lear tha t
under
Assumption
I ,
the
var ia te i s s y m p ~ o t i l l y norma.l
with
asymptotic
variance
tha t is
O(T-
l
.
However
the
form of
the asymptotic variance
seems very
messy.
In pract ise one
would
probably have
to
estimate
by segmenting
the data.
i i
Let
us next estimate the survivor function
1
F(t)
Prob{N(t)
= 0 t
N{O}
=
I}
Prob
{ r
i
l
r
i
>
t}
1
Prob{ r
i
l
r
i
s t}
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D VID
RILLINGER
This
l as t
suggests the estim ate
;'It,
F(t)
=
{
1 i+l -
1
l
~
t
i
N(T) -l}/N(T)
This
estimate is
based
on the in te rar r iva l
times
x
= 1 .+1- 1 .. The
process X.,
i =
O,± 1 , . . . is
l l l l
sta t ionary.
I f
i t
is
mixing in some
sense then
;'It,
1 -
F(t)
wil l be s y ~ p t o t i l l y normal, see Deo
(1973), for example. This l as t suggests the
in teres t ing
problem of
re la t ing
a mixing
condit ion
for
a sta t ionary point
process
to some mixing
condition
for the corresponding process
of
in te r
a r r iva l times.
i i i
The
following i s
a
plausible
estimate for the
hazard
function,
with
ST a
small posi t ive number,
~ ( t ) = { t - a T < f i + l - f
i
< t+6
T
; i l •••
,N(T)
- I}
2
S
T {
i +I -
i
>
t ; i
=
1,
• • • , N(T) - 1 }
(iv)
Next
consid er th e estimation
of
the forward
recurrence
time dis t r ibu t ion
G(t)
1 - Prob{N(t) = O}
P (1 - F (
u))
du t
P[
(1
- F (
t ) ) t
J
P
J
U
dF
(
U )
where we use
a
Palm-Khinchin relat ion from
Section
1.9
and in tegra te
by
par ts . The l as t re la t ion
suggests the
estimate
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ST TISTI L INTERFEREN E
G t) = ~ [ l - F t t J + p }: ( 1 i+1- 1 i) / N T )- l )
-i+l- -i ~ t
,..
t
[
i +1 - i > t }IT + L i +1- rr i IT
i+l- -i ~ t
j = 0 , • . •
,
J - l
xp[-iAt} dN t
a
I I I .
Estimation
of Frequency
Domain Parameters
1. We begin with a discussion of
f i r s t
order
s ta t i s t i c s . Suppose T = JU, J an
in teger .
Set
j+l)U
d ~ O j jJ
~
jU<
a
~ j + l U
Sexp
[ - i
(A-O:)
j
+
~
} s in
(A-Cl
u 2
A
1 ) )
dZ
a )
a
using
the
spect ra l
representat ion
a t
the l a s t
s tep.
In
the
case tha t J = 1, U = T, we sha l l
write
d ~ A .
We
ha.ve,
Theorem
4.
Let the
process
~ t
sa t is fy Assumption
I . Suppose A
~
o.
Then
~ U A , j ,
j
=
O, •••
, J - l
are asymptotically
independent
r var iate complex
normal with mean 0 and covariance matrix
2TTU[f
ab
A)] as T
...
co. Also
var ia tes a t frequencies
of
the form
2TTu/u, are
asymptotically
independent
for u
dis t inct
posit ive in tegers .
2. Suppose
we are
in teres ted in estimating the
second order spectrum f b A).
Various
procedures
a
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DAVID R. BRILLINGER
suggest themse lves , based ei ther on the expression
or
the expression
co
P
a
J
Paa U)
-
p;} exp -iAu} dU}/ 2TT
-co
co
{Pab u) - PaPb} exp{-iAu} dU} 1 2rr
=
fab A)
~
-co
Procedure I . Set
IU A,j)
=
2 n U - l d U A , j d U A , j ~
-
for A ~ 0 and
consider
the
estimate
J l
U A)
= J
l
IU A,j)
j=O
From Theorem 4,
as
T ~ co , but remains fixed
£U A tends to J - I W ; J ~ f A)) where W; denotes the
complex
Wishart. ~
Procedure
I I . Set rT A) = 2 ~ T - 1 £ T A £ T A •
For
2TTS./T
dis t inc t ~
0, non-negative and a l l
~
A
set
J
T -1 J l T
f A =
~ I
2ffS
j
/T).
j=O
T
From Theorem 4, as T ~ co , A
tends
to
J - l w ; J ~ f A
Both of the above e stimates are a symp to tic ally
normal i f the l imit ing conditions are
as
T
co ,
~ co, but
J/T
~
O.
In
the
above procedures we sometimes choose
to
weight the periodogram ordinates unequally.
For example in Procedure
I I
we might
take
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ST TISTI L
INTERFEREN E
fT
A)
=
TT ~
W
T
A
_
TT
T
S
IT TT
T
S
- T s ~ 0
with WT a =
B ~ l w a / B T
where
=
1.
I f
B
T
~
0,
BTT ..
00
~ s T ..
0 0 ,
th is estimate is
asy-mptotically
normal, s ~ e Bri l l inger 197
2
.
Procedure I I I .
Let
P
a
, Pab u be
given
by the
expressions
of
I I . l ,
I I .2 respect ively. Let
wT u
=W BTU
be
a
convergence fac tor . Set
f ~ b A .
=
f 2 ~ T
~ f P a b 2 ~ T j - P a ~ b } e x p f - i A . 2 S T j }
J
W
T
2S
T
j }/ 2ff a b
fp
a
+
2S
T
~ f P a a 2 S T j
-
p; , lexpf-H2STjl
J
W
T
2S
T
j }/ 2ff
a
=
b
Because
of the
per iodic i t ies
involved,
i t
only
makes
sense to compute
th i s estimate
for
A1 s
T T / ~ T .
The
choice of bin width
.2S
T
i s
seen
to show i t s e l f
in
the
Nyquist
frequency
~ 8 T
This estimate
i s
asymptotically normal
under
conditions including
B
T
,
~ T
..
0,
BTT .. 00 as T ..
0 0 .
This estimate
is
the
one computed
most rapidly.
I t has the disadvantage
of
possibly
leading
to
negative
power
spectrum
estimates and coherences b ig ger than 1, even i f
W a ~ o.
Procedure IV. Compute the spectrum of the ordinary
process
-1
h
h
X t = h N t-
2
,t
but
remember
tha t
85
t=0,±h,±2h,
•••
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DAVID BRILLINGER
Problems
of
al iasing clear ly
ar ise
here.
Tapering
and
pref i l te r ing play
essen t ia l roles
in the
estimation
of the spectra
of ordinary
time
ser ies .
I t is
not
ent i re ly obvious
how to apply
these
techniques in the poin t process
case
with
the
excep tio n of tapering fo r Procedures I
and
I I .
I f
the
complete intensi ty
A t h
Prob[point in t , t+h l 1
N u ,
u t}
ex is ts
and
can
be
evaluated, then with
t
A t A t
dt
the
t ransformation N t N A t carr ies
N
over
in to a Poisson process with unit in tens i ty ,
and
constant power
spectrum.
This t ransformation is
analagous to
the
condi t ional probabil i ty in tegra l
t ransformation to
uniform varia tes
in the case of
ordinary
time
ser ies .
For the
doubly stochast ic
Poisson process A t =
x t .
Pref i l t e r ing
procedures
carr ied
out
ent i re ly
in
the
frequency
domain,
for
ordinary
time ser ies ,
clear ly have point process
analogs.
For example,
i f
we can think of a g A near f A ,
then
we
might
form
the
estimate
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ST TISTI L
INTERFEREN E
Detrending
can
be very
important.
Lewis
1972)
contains
important adv ice
on
these matters.
3.
~
next
turn
to
a
br ie f
discussion
~
the estim a
t ion of the
parameters
of the model
Prob[type
1
event
in t , t+h ] N
2
u),
u
~
t}
~ ~ + J
a t-u) dN
2
U))h
as
h O
I f
P22 u)
i s
not constant , then
we
estimate
a u),
a
time
domain
parameter
by going
through
the frequency
domain. We have the relat ions
A ~
J a u) e x p [ - i ~ u } d u
PI ~ + A O P2
f 1 2 ~ A ~ f 2 2 ~
a u)
2 )-1
A a)
exp{iua}da
suggesting
the estimates
A T T
A ~ f 1 2 ~ / f 2 2 ~
o P
- A O
P
a u)
=
2 )-1 B
T
A kB
T
exp{iukB
T
} vT kB
T
k
where vT a)
v CTa
is convergence
factor. More
detai ls on th is procedure may
be found in
Bri l l inger
1974
a .
4. On
occasion
we may be
led
to model the
process in
some manner involving a f in i te
dimensional parameter
e We would then
li.ke
to be a.ble to estimate e
Sometimes
such
a
model
wil l
lead
to
a
t rac t ib le form
87
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DAVID BRILLINGER
fo r
the
second-order spectra . For example, suppose
we have a clus ter process with p r ~ ~ y process
Poisson
and
the
secondary
process
independent
exponentials from the clus ter centers , then the
power spectrum of the process has
the
form
involving the
three dimensional parameter
e
a ,b ,c .
We
now describe
one method
o f estimating
Related methods
are
given in Whittle (1953),
Walker
(1964),
Hawkes
and
Adamopoulos
(1973).
Let the t rue value of e be St. Suppose
lim f(A;S) = ~ S
co
and ~ e t =
p/(2n)
where p
i s
the
mean intensi ty of
the process. ~ e t
may
be
estimated
by 0 = p / 2 ~ .
The periodograms I T 2 ~ S / T , s
1 2 • • are
asymptotically
independent
exponentials
with means
f(2TTS/T; a t , s
1 2 The
scaled
varia tes
T
I (2TTS/T)/ ~
s 1 2
•••
88
This
resul t
suggests our
set t ing down the following
approximate log l ikel ihood function
are
therefore
asymptotically
independent exponentiaJs
with mea.ns
s
1 2 • ••
(2TTS/T; 8 )
IJ
(et
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STATISTICAL
INTERFERENCE
A
and taking
as
an
estimate of 8,
the value
8 tha t
maximizes 11).
In
the
theorem below we set g A; 8) =
f A; 8 / -l e)
an
d
S T A
I\T e)
= _ TT r:
T
{log
g 2TTS/T;
e) +
I
b
TTs
/
T
/ J
T
s=l
g TTs/T;
12)
The
e maximizing 11) also
maximizes 12).
T·heorem
5.
I f
a) the process N t) , co <
t
< co has
mean
intensi ty ~ 8 t
and power spectrum f A; e t
b) f A;8),
8 E @
C
RL,is
non-negative
and
l
8) =
lim
f
A;
A
I
A
1
....
exis t s , c ) with g A;
e) =
f X;
A /lJ e) ,
A e
= -
J{ log g A; e) + ~ t ~ ~ - 1 } d A
exis ts as a Lebesgue in tegra l has a unique maximum
a t at and
i s
such tha t
max B
) ... e
ft EU
as
the
neighborhood U
of
a shrinks
to
[a}, d)
AT e) A 8)
a t
e and uniformly
near
other 8 e)
8E@
maximizing
11) is bounded in probabi l i ty , then
A
e
e
t in probabil i ty
as
T
...
co
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D VID
BRILLINGER
Condition
(d) i s
s a t i s f i e d
for
processes
satisfying, Assumption
I , provided
g(A;8)
i s
a
sUfficiently
regular
function of A.
We next turn
to the
large
sample distr ibution
of 8.
To
t h i s
end
s e t ,
13
Because
of
(c) above,
generally A
j
(6 )
=
oA(6 )/o6
j
=
o.
90
The above
procedure provides us
with a
fur ther
estimate, ~ A = f(A;8)
of the
power spectrum.
Under the
conditions
of
the
theorem, t h i s estimate
w i l l
be asymptotically normal
with
mean f(A;S ) and
variance
d log g(a;S ) a log
g(a;sf)
da
oS} o e ~
a log g(a;Sf) a
log g(S;Sl)
oSj
o e ~
f
4
) C -a. - · 6 )
f a ; G f ~ f{S;8f)
da
dS
00 00
TTJ J
Theorem 6.
Suppose
the conditions of Theorem 5 are
s a t i s f i e d . Suppose also
(f) the
derivat ives
of
13
e x i s t ,
(g) ~ k
C
T
) A
jk
for any
sequence ,T of
varia tes tending to
Sf
in probabil i ty, (h) with
~
=
[A
jk
J,JT{
ft
i 6 ) , . •• ,Ai:(6 )1-4 NL Q, ~ + ~ , then
S i s a,s ymptotically normal
with
mean S and co-
variance ma tr ix T-
l
- 1 ~ + ~ A - l •
For
processes sat isfying Assumption I
and
g(A;8)
a
s u f f i c i e n t l y regular
function of A
we
have
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STATISTICAL
INTERFERENCE
L
j ,k
of A;
8
of A;
8 )
08 .
08
J
k
In the
case
of
a
vector-valued process, instead of
maximizing
11
we
would maximize
S T
-
L [log Det ~ 2 T I S / T ; 8 t r ~ 2TIS/T) ~ 2 T I S / T ; 8 }
s=l
where
g
A;
8) j k = f
A;
8) j k / ~ j ll k 8)
T T IA A
H A) jk
=
I A) jk / \
U j ~ k
5.
We mention br ie f ly tha t the p r ~ m t r s of
a
se l f -
excit ing process may be estimated via a frequency
domain analysis . Such
a
process i s defined by
a
re la t ionship
t
E[dN t
\
N u ,
u
~ t} =
tJ
Sa t-u dN u
dt
-co
where ~ a u
~
0;
a u
du <
1 ; a u = 0 for
u
~
O. Let
ex
A ~
=
J a u e x p [ - i ~ u } ~
For th is
process
~
p
[1
-
A
0
J
and
2
f
A
=
p/ 2TT \ 1 - A A \ )
Because A A)
is the Fourier
transform
of
a one
sided
function,
the problem of
est imating
A A)
from
fT A ,
is
seen to
involve
the
fac tor iza t ion
of
f T ~ .
Rice
1973)
carr ied
out
th is
empir ical ly
and
91
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DAVID R BRILLINGER
found the asymptotic dis t r ibut ion of the
est imate.
This
procedure also provides a fur ther
spect ra l
A A A
2
est imate, namely f(A) = p/ 2TI11 -A A)
I ).
6.
We next turn
to
the
problem
of e stima ting th e
variance time curve given by
var
N(t) as a function
of t . Using the spect ra l representat ion, we see
tha t
V(t)
va.r N(t)
J O O S i ~ 7 2 t / 2 2 f a
da
co
t
+ Sco(sin
crt/2)2(f(
)_
p
)da
p o 2
2Tr
co
The fo llowing type o f estim ate is considered by
Torres-Melo
(1974),
V(t)
=
tp
+ Bi
l
t
2
f
T
O _-i 2 (Sin Bst/2)2
2TI
+ Bs 2
s=l
T
P
)'
f (Bs) - 2TI
J
He f inds the asymptotic dis t r ibut ion of th is
est imate.
7.
Product dens i t ies may be estimated
in
s imi la r
manner
to th e v ariance
time
curve. We
have
p(u)
=:
JOO f a -
- )exp[iua}
da
+
p2
co
suggesting the estimate
A S
p(u) =: B [ f T O - 2 ~
+
2
(fT(BS)
-tTT COS Bs]
s=l
This estimate
would undoubtedly
be improved by the
inser t ion of
convergence
factors .
92
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STATISTICAL INTERFERENCE
Fina.lly
we remark tha t we may
sometimes wish
to estimate the spectral measure
F A = J A f ~
o
The
obvious
estimate
is
F( A
IV. Proofs
B
1. Proof
of
Theorem
1.
The jo in t fac tor ia l cumulant
of N
a
T , , N
a
T) i s
1
k
T T
q t l , · · · , t k d t l · · · d t
k
=
O T)
a a a 1 • • • a
k
.
in
view
of
Assumption
I .
The
ordinary
joint
cumulant of these same var ia tes
i s
a sum
of
multiples
of lower order
fac tor ia l cumulants.
It.
follows
tha t i t too
i s
O T)
as
T 00 . This means
that the standardized jo in t cumulants of order k
of these variates are
O T
1
-
k
/
2
) 0
as
T 00
for
k
>
2,
and
so the
variates are
asymptotically
jo in t ly
normal.
2.
Proof of
Theorem
2. The variate J ~ b U may be
represented as
J dNa
J dN
b
rr
G
where
G
i s
the
se t
[u - ~ t
<
(J
- < u + T J ~ rr}.
I t
follows
from
th i s
representat ion,
Assumption
I
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S
k.
J
c j )
D VID
RILLINGER
and the
rules of
Leonov and
Shiryaev
1959
t h a t
the j o i n t f a c t o r i a l
moment
of order
k
of
JTb u
i s
k
a
of order
O STT .
An
ordinary
cumulant
of order
k,
c
k
i s
connected
to
corresponding
f a c t o r i a l
cumulants, c k )
through
k
c
k
= L
j==l
where
s ~ i s
a S t i r l i n g
number.
I f
L/T, then
T T T
E Jab u
-
2L
Pab u as
T
-
00 when u
-
u. I t
follows,
t h a t
in t h i s case
the
cumulant
of order
k
T T
of
Jab u
-
2L
Pab u and
so the var ia te i s
asymptotically Poisson.
In the case STT
-
00 the
standardized j o i n t cumulant of
order
k i s
O STT 1-k/2
0 fo r k
>
2. I t o ~ s t h a t the
var ia te i s
asymptotically
normal. The indicated
asymptotic
independence
follows
on
evaluating
j o i n t
second-order cumulants.
3.
Theorem 3 i s
proved in
the
same
manner t h a t
Theorem
2 i s
proved.
4. Theorem 4 i s proved by
evaluating
the . joint
cumulants of
the
dUe A related
r e s u l t ,
Theorem
4.2,
a
i s proved in B r i l l i n g e r
1972 .
5.
Before proving
Theorem
5,
we prove
a lemma
of
some
independent i n t e r e s t .
Lemma
1. I f i )
B
i s local ly compact, complete,
s eparable , me tr ic , i i ) O , G ~ p i s a probabil i ty
space
with [ complete,
separable,
metric , i i i )
QTCS,w
i s
real-valued, Borel
measurable
for
S,w
E B x
2 and a l l T, iv
Q S i s real-valued, lower
semi-continuous,
Q S
Q St
fo r S Sf,
v
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ST TISTIC L INTERFERENCE
QT 6 ,w) = Q 6 )
0 p l ) ,
QT 6,w) ;:: Q 6)
0 p l ) ,
S S as T
0 0
Vi)
given
e ,h
> 0 ,
8
1
Sf, t h e r e
exis ts
U
l
a
neighborhood
o f
Sl and
t h e r e
exis ts
TO
such tha t
A
Vii) f o r
each w a n d T t h e r e exis ts S such
tha t
Q
S,w)
=
in f
Q
S,w)
e E ®
V i i i ) g iv en h >
0 , t h e r e exis ts
a compact se t C 8
and T such tha t r o { ~ ~
}
< h, fo r T > T ,
the n
A
0 0
S
= Sf
0 1 ).
P
A
Proof. The
m e a s u r a b i l i t y
of A
resu l t s from Theorem
2 o f Brown and Purves 1973). Let U C be an open
neighborhood
o f
Sf.
From
iv )
t h e r e
exis t s
y
>
0
such tha t Q 8
1
) - Q 8
f
) ~
y
fo r e E C\U. Suppose
Sl E C\U. Then from
v)
T T
lim Prob{Q
S l w ) - Q 8 , w ) S
2y} =
0
T 00
From th i s
and
v) t h e r e exis ts a neighborhood U
l
o f
Sl
such
tha t
lim Prob{ in f
T ro
8
E U
l
QT e,w) _ QT
S
,w ) y}
=
0
14)
Using th e f a c t
tha t
C
is
compact, se lec t a f in i t e
number o f p o i n t s S , s
=
1 , •• ,N w i t h
neighborhoods
s
U s s =
1,
,N c o v e r i n g
C\U.
From
14)
9
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DAVID
R.
BRILLINGER •
li m
Prob{ in f QT S,w) - QT B ,W) Sy}=O 15
T-+oo 8EC \U
Now
from
v i i )
prob{6 U or B\C} Prob{ in f Q T S , w ) _
8 E C\ U
QT 8
t
,W
y}
From
15
th is
l as t
tends t o O. From v i i i ,
prob[8
E
8\C}
tends to O. This gives the resu l t .
Theorem
5
now follows
from
th is Lemma.
6.
Theorem 6
follows
from th e
r e l a t i o n
with a between
n t •
96
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ST TISTI L
INTERFEREN E
REFERENCES
BRILLINGER, D R. 1972 . The spectra l analysis of
s ta t ionary
in terval
functions. pp. 483-513
in
Proc.
Sixth
Berkeley
Syrup. Math. Stat Prob.,
Vol. eds. L. M LeCam, J . Neyman, E. L.
Scott .
Berkeley University of California
Press.
BRILLINGER, D R. 1974a .
Cross-spectral
analysis
of processes with
stationary
increments
including G/G/oo
queue.
Ann. Prob. ,
2,
815-827.
BRILLINGER, D R.
1974b).
The ident i f icat ion of
point process systems.
Special
Invited Lecture
presented to the ns t i tu te of Mathematical
Sta t i s t ics
at
Edmonton.
COX
D R.
1965 . On
the
estimation
of the
in tensi ty
function of
a
stationary
point
process.
J R. Sta t i s t
Soc. , B, 27.
332-337.
COX D R. and LEWIS, P. A.
W
1966 . The
Sta t i s t i ca l
Analysis of Series of
Events.
London,
Methuen.
COX
D R. and
LEWIS
P. A. W
1972 .
:Multivariate
point
processes. pp.
401-448 in Proc. Sixth
Berkeley Symp. Math.
Stat Prob. , Vol.
eds. L. M LeCam, J. Neyman, E. L. Scott .
Berkeley, University
of
California Press.
CRAMER H.,
LEADBETTER
M R.
and
SERFLING, R. J .
1971 . On distr ibution
function
- moment
relat ionships
in
stat ionary poin t p roce sse s.
Zeit .
Wahrschein.,
18. 1-8.
DALEY
D J .
and
VERE-JONES, D
1972 .
A summary of
the theory of poin t p roce sse s. pp.
299-383 in
STOCHASTIC
POINT
PROCESSES ed.
P.
A.
W
Lewis)
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New Ycrrk, Wiley.
DEO C.
M.
1973 . A
note
on
empir ical processes of
strong-mixing sequences.
Ann.
Prob., 1 .
870
875.
HAWKES
A.
G.
1972 .
Spectra of
some
mutually
excit ing point
p roce sses with
associated
variables . pp.
261-271 in t o ~ h s t i c
Point
Processes ed. P.
A. W. Lewis .
New
York, i l ~
HAWKES
A. G. and
ADAMOPOULOS
L. 1973 . Cluster
models
for
earthquakes
-
regional
comparisons.
Bul. In te r
Sta t i s t Ins t 39.
LEONOV
V.
P.
and SHIRYAEV A.
N.
1959 .
On
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method
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Theory
Probe
Appl. ,
4. 319-329.
LEWIS,
P.
A.
W. 1972 . Recent resul ts
in
the
s t a t i s t i c a l
analysis
of
univariate poin·t
processes. pp. 1-54
in
Stochastic Point
Processes ed.
P. A. W. Lewis .
New
York,
Wiley.
RICE, J A. 1973 .
Sta t i s t i ca l analysis of
se l f -
excit ing point processes and
related
l inear
models. Ph.D. Thesis, U niversity of Cali fornia ,
Berkeley.
SRINIVASAN, S. K. 1974 . Stochastic
Point
Processes.
Ne w
York,
Hafner.
TORRES-MELO L. 1974 . Stat ionary poin t p roce sse s.
Ph.D.
Thesis, University
of
Cali fornia ,
Berkeley.
WALKER
A. M. 1964 .
Asymptotic
propert ies of leas t
squares
estim ates of parameters of
the
spectrum
of
a
s ta t ionary nondeterministic time ser ies
J
Austral .
Math.
Soc. ,
4.
363-384.
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ST TISTI L
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WHITTLE P
1953 .
Estimation and
information
in
stat ionary
time s r i s
rk
Math
stron
y ~
2. 423-434.