#### 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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Uniform Distribution

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Sample-Mean Variance