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0.04

|Ï„ |

0.0

1âˆ’ |Ï„ | < M

0.02

M

-0.2

0 otherwise.

0.0

0 10 20 30 40 50 60 0.0 0.01 0.02 0.03 0.04 0.05 0.06

Lag Frequency

â€¢ Parzen

ï£±

M ï£´1 âˆ’ 6

|Ï„ | 2

+ 6 |Ï„ |

3

|Ï„ | < ï£´

Figure 12.18: Lag windows (left) and spectral ï£´

ï£²

M M 2

windows (right) for four periodogram derived M

|Ï„ | 3

ï£´2 1 âˆ’ M â‰¤ |Ï„ | â‰¤ M

spectral estimators. ï£´

ï£´ 2

ï£³

Solid: Chunk estimator with m = 11.

0 otherwise.

Short-dashed curve: Daniell estimator with n =

11. The chunk and Daniell spectral windows

Examples are shown in the left hand panel of

coincide.

Figure 12.18.

Medium-dashed curve: Bartlett estimator with

Spectral Windows W (Ï‰)

M = 32.

Long-dashed curve: Parzen estimator with M =

â€¢ Chunk

40.

ï£±

All estimators have approximately 22 equivalent

ï£²1 Ï‰=0

degrees of freedom and an approximate bandwidth

m

of 0.047 when the times series is of length T = ï£³

0 otherwise.

240.

â€¢ Daniell (n odd)

ï£±

Periodogram derived spectral estimators also

ï£´1 n

ï£²

have an equivalent bandwidth (see [12.3.17]) that

|Ï‰| â‰¤

indicates, roughly, the width of the frequency n 2T

ï£´

ï£³0

band of which an estimate (Ï‰ j ) is representative. otherwise.

Estimates at frequencies separated by more

than an equivalent bandwidth are asymptotically â€¢ Bartlett

independent.

M sin(Ï€ Ï‰M)

The spectral estimators we have discussed are 2

.

summarized below. In the following, m is the Ï€ Ï‰M

T

number of chunks used by the chunk estimator,

â€¢ Parzen

M is either the length of a chunk or the cutoff

point of the Bartlett or Parzen lag windows, T is

3M sin(Ï€Ï‰M/2) 4

the length of the time series and n is the number .

Ï€Ï‰M/2

4T

of periodogram ordinates that are averaged to

produce the Daniell estimator.

Examples are shown in the right hand panel of

Lag Windows w(Ï„ ) Figure 12.18.

Equivalent Degrees of Freedom (EDF) and

â€¢ Chunk

Equivalent Bandwidth (EBW)

|Ï„ | â‰¤ M âˆ’ 1

1 â€¢ The following table lists the EDF and EBW

for the various spectral estimators.

0 otherwise

12: Estimating Covariance Functions and Spectra

278

Estimator EDF EBW Chunk Estimate

20

Chunk 2m 1/M

Daniell 2n n/ T

10

3 T /M

Bartlett 1.5/M

3.71 T /M

0

Parzen 1.86/M

-10

The chunk estimator is generally suitable for

problems in which there is a natural chunk length.

-20

However, if a contiguous time series is available, 0.0 0.1 0.2 0.3 0.4 0.5

Frequency

the use of a smoothed periodogram estimator is

Bartlett Estimate

preferred because it better uses the information

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