the empirical ft
DESCRIPTION
The Empirical FT. What is the large sample distribution of the EFT?. The complex normal. Theorem. Suppose X is stationary mixing, then. Proof. Write. Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments. Consider. Comments . - PowerPoint PPT PresentationTRANSCRIPT
The Empirical FT.
)()0( Note
real }exp{)()(:
1)}-TX(1),...X( {X(0),
1 0
tXd
titXdEFT
Data
T
N
N TT
X
What is the large sample distribution of the EFT?
The complex normal.
2var
)1,0( ...
onentialexp2/2/)(|)1,0(|
2/)()2/1,0()1,0(
||var
:Notes
)2/,(Im ),2/,(Ret independen are V and Uwhere
V i U Y
form theof variatea is ),,( normal,complex The
22
22
1
2
2
2
2
2
2
1
2
21
22
22
2
E
INZZZ
ZZN
iZZINN
YEY
EY
NN
N
j
C
C
C
Theorem. Suppose X is stationary mixing, then
))(2,0( asymp are
~/2 with 0 integersdistinct ,..., ),2
(),...,2
( ).
)(2,0(
asymp are 0 anddistinct ,..., ),(),...,( ).
0 )),(2,0(
0 )),0(2,(
allyasymptotic is )( ).
1
1
L11
XX
C
lL
LTT
lXX
C
L
T
X
T
X
XX
C
XXX
T
X
TfIN
TrrrT
rd
Tr
diii
TfIN
ddii
TfN
TfTpN
di
Evaluate first and second-order cumulants
Bound higher cumulants
Normal is determined by its moments
Proof. Write
)()()( X
TT
XdZd
Consider
)(2)()(
have We
)()(2~
)()()(
)()()()()}(),(cov{
TTT
XX
T
XX
TT
NN
TTT
X
T
X
d
f
df
ddfdd
Comments.
Already used to study rate estimate
Tapering makes
dfHdXX
TT
X)(|)(|~)(var
Get asymp independence for different frequencies
The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0
Also get asymp independence if consider separate stretches
p-vector version involves p by p spectral density matrix fXX( )
Estimation of the (power) spectrum.
22
2
2
2
2
2
2
2
)(}2/)(var{ and
)(}2/)({ notebut
0)( unlessnt inconsiste appears Estimate
lexponentia ,2/)(~
|))(2,0(|2
1~
|)(|2
1)(
m,periodogra heconsider t ,0For
XXXX
XXXX
XX
XX
XX
C
T
X
T
XX
ff
ffE
f
f
TfNT
dT
I
An estimate whose limit is a random variable
Some moments.
2222
0
22
|)(|)(|)(||)(|
)(}exp{)()(
|| var||
T
NXX
TT
X
T T
X
T
X
pdfdE
so
ctitXEd
EE
The estimate is asymptotically unbiased
Final term drops out if = 2r/T 0
Best to correct for mean, work with
)()( TT
X
T
Xcd
Periodogram values are asymptotically independent since dT values are -
independent exponentials
Use to form estimates
sL' severalTry variance.controlCan
/)(}2/)(var{ )(}2/)({
Now
ondistributiin 2/)()(
gives CLT
/)/2()(
Estimate
near /2
0 integersdistinct ,...,Consider .
22
2
2
2
2
2
1
LfLffLfE
Lff
LTrIf
Tr
rrmperiodograSmoothed
XXLNNXXLXX
LXX
T
XX
l l
TT
XX
l
L
Approximate marginal confidence intervals
LVT
L
ff
fT
T
data,split might
FT or taperedmean, weightedmight take
estimate consistentfor might take
ondistributi valueextreme viaband ussimultaneo
levelmean about CIset
logby stabilized variance
Notes.
)}2/(/log)(log)(log
)2/1(/log)(Pr{log2
2
More on choice of L
biased constant,not is If
radians /2
(.) of width affects L of Choice
)()(W
)2/)/(()2/)(sin2
11
/2/2 },/)({)}({
Consider
T
2
T
T
T
lll
lll
l
T
ff
TL
W
df
dfTTL
TrTrLIEfE
Approximation to bias
0)( symmetricFor W
...2/)()(")()(')()(
...]2/)(")(')()[(
)()(
)()(
)()( Suppose
22
22
11
dW
dWfBdWfBdWf
dfBfBfW
BfW
dfW
BWBW
TTT
TT
T
T
TT
T
Indirect estimate
duucuwui T
XX
T )()(}exp{21
Estimation of finite dimensional .
approximate likelihood (assuming IT values independent exponentials)
)}/2(/)/2(exp{)/2()(
in ),;( spectrum
1 TrfTrITrfL
f
r
T
Bivariate case.
)( )(
)( )(
matrixdensity spectral
)()()}(),(cov{
)(
)(}exp{
)(
)(
YYYX
XYXX
XYYX
Y
X
ff
ff
dfdZdZ
dZ
dZit
tY
tX
Crossperiodogram.
T
YY
T
YX
T
XY
T
XXT
T
Y
T
X
T
XY
II
II
ddT
I
)( formmatrix
)()(2
1)(
I
Smoothed periodogram.
LlTrLTr ll
l
TT
NN ,...,1 ,/2 ,/)/2()( If
Complex Wishart
ΣW
XXWΣ
0XX
nE
nW
IN
n T
jj
C
r
rn
squared-chi diagonals
~),(
),(~,...,
1
1
Predicting Y via X
)()(}exp{)(
}exp{)()2()(
)()(/)()( :coherency
)(]|)(|1[ :MSE
phase :)(arggain |:)(|
functionfer trans,)()()(
|)(-)(|min
)(by )( predicting
1
2
1
2
X
YYXXYX
YY
XXYX
XYA
XY
dZAtitX
diuAua
fffR
fR
AA
ffA
AdZdZE
dZdZ
Plug in estimates.
LRE
R
LL
RRLLFR
R
fffR
fR
AA
ffA
T
TL
T
T
YY
T
XX
T
YX
T
T
YY
T
TT
T
XX
T
YX
T
/1||
)-(1-1
point %100approx 0|| If
)1()1()(
)||||;1;,()||1(
|| ofDensity
)()(/)()(
)(]|)(|1[ :MSE
)(arg |)(|
)()()(
2
1)-1/(L
2
22
12
2
2
2
1
Large sample distributions.
var log|AT| [|R|-2 -1]/L
var argAT [|R|-2 -1]/L
Advantages of frequency domain approach.
techniques formany stationary processes look the same
approximate i.i.d sample values
assessing models (character of departure)
time varying variant
...
Networks.
partial spectra - trivariate (M,N,O)
Remove linear time invariant effect of M from N and O and examine coherency of residuals,
)(
)()()()()(
M of effects
invariant lineartime removed having O and N of spectrum-cross partial
),()()()(
),()()()(
|
1
|
1
|
1
|
T
MNO
MNMMNMNOMNO
MMOMMOMO
MMNMMNMN
R
fffff
ffBdZBdZdZ
ffAdZAdZdZ
Point process system.
An operation carrying a point process, M, into another, N
N = S[M]
S = {input,mapping, output}
Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt
System identification: determining the charcteristics of a system from inputs and corresponding outputs
{M(t),N(t);0 t<T}
Like regression vs. bivariate
Realizable case.
a(u) = 0 u<0
A() is of special form
e.g. N(t) = M(t-)
arg A( ) = -