point processes brillinger

8/20/2019 point processes brillinger

1/45

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.

55


2/45

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

}

56


3/45

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


4/45

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

58


5/45


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

59


6/45

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

60


7/45

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 .

61


8/45

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.

62


9/45

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

••

63


10/45

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

64


11/45

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

65


12/45

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


13/45

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

OO


14/45

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


15/45


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 ;

69


16/45

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

70


17/45


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

71


18/45

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 } =

72


19/45


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 .

73


20/45

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

74


21/45


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

75


22/45

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

76


23/45


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


24/45

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

78


25/45

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 )

79


26/45

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


27/45

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}

81


28/45

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

82


29/45


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

83


30/45

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

84


31/45

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,

•••


32/45

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


33/45

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


34/45

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


35/45

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

89


36/45

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


37/45

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


38/45

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


39/45


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

93


40/45

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

94


41/45

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


42/45

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


43/45

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)

97


44/45

DAVID R. BRILLINGER

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

a

method

of calculat ion of

semi- invar iants .

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.

98


45/45

ST TISTI L

INTERFEREN E

WHITTLE P

1953 .

Estimation and

information

in

stat ionary

time s r i s

rk

Math

stron

y ~

2. 423-434.

point processes brillinger

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