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7/21/2019 Brillinger 1975 b
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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
measure. 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
IOPab(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
distributed
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 of 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
OO
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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 would 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
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