The Empirical FT
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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 .

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The empirical ft

The Empirical FT.

What is the large sample distribution of the EFT?




The empirical ft

Proof. Write

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments



The empirical ft

Comments.

Already used to study rate estimate

Tapering makes

Get asymp independence for different frequencies

The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0

Also get asymp independence if consider separate stretches

p-vector version involves p by p spectral density matrix fXX( )


The empirical ft

Estimation of the (power) spectrum.

An estimate whose limit is a random variable


The empirical ft

Some moments.

The estimate is asymptotically unbiased

Final term drops out if  = 2r/T  0

Best to correct for mean, work with


The empirical ft

Periodogram values are asymptotically independent since dT values are -

independent exponentials

Use to form estimates






The empirical ft

Estimation of finite dimensional .

approximate likelihood (assuming IT values independent exponentials)



The empirical ft

Crossperiodogram.

Smoothed periodogram.





The empirical ft

Large sample distributions.

var log|AT|  [|R|-2 -1]/L

var argAT [|R|-2 -1]/L


The empirical ft

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

...


The empirical ft

Networks.

partial spectra - trivariate (M,N,O)

Remove linear time invariant effect of M from N and O and examine coherency of residuals,


The empirical ft

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


The empirical ft

Realizable case.

a(u) = 0 u<0

A() is of special form

e.g. N(t) = M(t-)

arg A( ) = -