### 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:

#### 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

*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