### Estimator Variance

As mentioned in §5.12, 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*:

*i.e.*,

**Var**

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

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
,

*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

**Var**

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 [191,115,91].

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Product of Two Gaussian PDFs

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Independent Implies Uncorrelated