### 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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Product of Two Gaussian PDFs

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Independent Implies Uncorrelated