### Estimator Variance

As mentioned in §6.12, the pwelch function in Matlab and Octave offer confidence intervals'' for an estimated power spectral density (PSD). A confidence interval encloses the true value with probability (the confidence level). For example, if , then the confidence level is .

This section gives a first discussion of estimator variance,'' particularly the variance of sample means and sample variances for stationary stochastic processes.

#### Sample-Mean Variance

The simplest case to study first is the sample mean:

 (C.29)

Here we have defined the sample mean at time as the average of the successive samples up to time --a running average''. The true mean is assumed to be the average over any infinite number of samples such as

 (C.30)

or

 (C.31)

Now assume , and let denote the variance of the process , i.e.,

 Var (C.32)

Then the variance of our sample-mean estimator can be calculated as follows:

where we used the fact that the time-averaging operator is linear, and denotes the unbiased autocorrelation of . If is white noise, then , and we obtain

We have derived that the variance of the -sample running average of a white-noise sequence is given by , where denotes the variance of . We found that the variance is inversely proportional to the number of samples used to form the estimate. This is how averaging reduces variance in general: When averaging independent (or merely uncorrelated) random variables, the variance of the average is proportional to the variance of each individual random variable divided by .

#### Sample-Variance Variance

Consider now the sample variance estimator

 (C.33)

where the mean is assumed to be , and denotes the unbiased sample autocorrelation of based on the samples leading up to and including time . Since is unbiased, . The variance of this estimator is then given by

where

The autocorrelation of need not be simply related to that of . However, when is assumed to be Gaussian white noise, simple relations do exist. For example, when ,

 (C.34)

by the independence of and , and when , the fourth moment is given by . More generally, we can simply label the th moment of as , where corresponds to the mean, corresponds to the variance (when the mean is zero), etc.

When is assumed to be Gaussian white noise, we have

 (C.35)

so that the variance of our estimator for the variance of Gaussian white noise is

 Var (C.36)

Again we see that the variance of the estimator declines as .

The same basic analysis as above can be used to estimate the variance of the sample autocorrelation estimates for each lag, and/or the variance of the power spectral density estimate at each frequency.

As mentioned above, to obtain a grounding in statistical signal processing, see references such as [201,121,95].

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