inference. estimates stationary p.p. {n(t)}, rate p n, observed for 0
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Inference. Estimates
stationary p.p. {N(t)}, rate pN , observed for 0<t<T
First-order.
Tf
dfT
p
dZiiT
TTNp
NN
NNN
N
N
/)0(2 ~
)()2/
2/sin(ˆvar
)(/]1}[exp{N(T)tion representa spectral via
unbiased /)(ˆ
2
Asymptotically normal.
1)1/|)(|(
/)1(1/|)(|...
11/|)
11(|
...1
)...1
(|)(|...|)11
(T|... .4
)...1
(1)2(
...1
)...1
()()...11
(T... .3
)(2)()( .2
22|)(| .1
)2//()2/}(sin2/exp{/]1}[exp{)(
)()()(
kdkkT
kkk
kdkk
kkTdkkT
kdd
kkkT
kTk
kdd
kkkT
TdTT
TdT
TTTiiTTiTdZTTN
cumulantshigher Next,
/)0(2 ~
)(2)2/
2/sin( (T)} var{N
NTp E{N(T)} :haveAlready
1)-1/(kp 1),-k/(kp
)1(|)2//()2/(sin||)(| .5
TNN
f
dNN
fT
pTOdpTdpT
Theorem. Suppose cumulant spectra bounded, then N(T) is asymptotically N(TpN , 2Tf2 (0)).
Proof.
)(
|(||.|
...)...,()...()()...(...
{N(T)}k
cum
1)1/(
111111
T
TO
dM
ddf
kkkT
k
kkkkkk
T
The normal is determined by its moments
Nonstationary case. pN(t)
Second-order.
)(/)(ˆ)(
|)|/()(ˆ)(
)(|)|(
)()(
)()()(
histogram a
},2/2/|,{#)(Let
heConsider t
}1)(|1)(Pr{)(
,2/||
,2/||
TNuIuh
uTuIup
upuT
dsdttspuEI
tdNsdNuI
kjuukjuI
tdNutdNuh
T
NN
T
NN
NN
tsutsNN
T
tsuts
T
kj
T
kj
NN
Bivariate p.p.
)(/)(ˆ)(
|)|/()(ˆ)(
)(|)|(
)()(
)()()(
histogram a
},2/2/|,{#)(Let
heConsider t
}1)(|1)(Pr{)(
2/||
2/||
TMuIuh
uTuIup
upuT
dsdttspuEI
tdNsdMuI
kjuukjuI
tdMutdNuh
T
NMNM
T
NMNM
NM
ustNM
T
NM
ust
T
NM
kj
T
NM
kj
NM
Volkonski and Rozanov (1959); If NT(I), T=1,2,… sequence of point processes with pN
T 0 as T then, under further regularity conditions, sequence with rescaled time, NT(I/pN
T ), T=1,2,…tends to a Poisson process.
Perhaps INMT(u) approximately Poisson, rate TpNM
T(u)
Take: = L/T, L fixed
NT(t) spike if M spike in (t,t+dt] and N spike in (t+u,t+u+L/T]
rate ~ pNM(u) /T 0 as T
NT(IT) approx Poisson
INMT(u) ~ N T(IT) approx Poisson, mean TpNM(u)
Variance stabilizing transfor for Poisson: square root
For large mean the Poisson is approx normal
by separated lags ift independen
normalally asymptotic is )(ˆ then 2
upTIf