стр. 151 |

1

4 about the true spectrum. Both are equally rough

2

E |ZT j |2 в‰€ 2 (П‰ j ) HT (П‰ в€’ П‰ j )2 dП‰

and have peaks scattered randomly amongst

T в€’1

2

the Fourier frequencies. Doubling the sample

4

= (П‰ j ). length has not produced a smoother estimate

T

of the spectrum; rather, it has produced almost

independent spectral estimates at twice as many

Consequently frequencies. It is this property of the periodogram,

its ability to extract increasing amounts of roughly

E IT j в‰€ (П‰ j ). independent information about the spectrum with

increasing sample length, that is exploited by

That is, the jth periodogram ordinate IT j is an the spectral estimators described in the following

asymptotically unbiased estimator of the spectral subsections.

density at frequency П‰ j .13 The third panel in Figure 12.9 shows the

periodogram, computed from our now familiar

AR(2) time series, on the decibel scale.14 The

12.3.7 The Distribution of the Periodogram.

amplitude of the variations in the periodogram

We need to know the distribution of an estimator

reп¬‚ects the magnitude of the underlying spectrum,

to understand its properties and to use it for

but the periodogram itself is at best a poor

making inferences about the true spectrum by

estimator of the spectrum.

constructing conп¬Ѓdence intervals and developing

One other comment about equation (12.31) is

testing procedures.

in order. The statement for j = 0 applies in the

Initially, the periodogram would appear to be

present circumstances because we assumed that

a reasonable estimator of the spectrum, since

the process has mean zero and therefore did not

it is nearly unbiased and estimates at adjacent

bother to remove the sample mean from the data.

frequencies are nearly uncorrelated. However, as

In fact, the j = 0 statement means that, with the

we have seen before, unbiasedness is only one

assumptions we have made,

attribute of a good estimator. Efп¬Ѓciency and

consistency (i.e., low variance that decreases with

X в€ј N (0, (0)).

1

T

increasing sample size) are also very desirable

attributes. Unfortunately, the periodogram lacks In general, the mean of a time series taken

both of these properties. from an ergodic weakly stationary process will

When {Xt : t в€€ Z} is an ergodic, weakly asymptotically be a normal random variable with

mean Вµ X and variance T (0). However, the

1

stationary process, the periodogram ordinates are

periodogram can not be used to estimate (0).

asymptotically proportional to independent П‡ 2 (2)

Ordinarily IT,0 = 0, since the sample mean

13 When X is a white noise process, it is easily shown that the

t

periodogram ordinate I T j is an unbiased estimator of (П‰ j ) = 14 That is, we plot П‰ versus 10 log (I ). See [11.2.13] for

j 10 T j

Var(Xt ), regardless of sample size. a discussion of plotting formats.

12: Estimating Covariance Functions and Spectra

268

10

4

8

3

6

2

4

1

2

0

0

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

Frequency

4

Figure 12.10: The periodogram of a time series of

length 240 generated from process Yt = Xt +

3

10 sin(2ПЂ0.3162t + 0.5763) where Xt is an AR(2)

2

process with (О±1 , О±2 ) = (0.3, 0.3). The continuous

part of the true spectrum of Yt is depicted by the

1

dashed curve. The true spectrum of Yt also has a

0

spectral line at П‰ = 0.3162, which is not shown.

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

[423]. The problem is discussed in some detail in

20

Section 6.6.

10

0

стр. 151 |