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.
The simplest case to study first is the sample mean:
Then the variance of our sample-mean estimator can be calculated as follows:
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 .
Consider now the sample variance estimator
When is assumed to be Gaussian white noise, we have
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.
Product of Two Gaussian PDFs
Independent Implies Uncorrelated