Autocorrelation - Estimation

Estimation

For a discrete process of length defined as with known mean and variance, an estimate of the autocorrelation may be obtained as

for any positive integer . When the true mean and variance are known, this estimate is unbiased. If the true mean and variance of the process are not known there are a several possibilities:

• If and are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate.
• A periodogram-based estimate replaces in the above formula with . This estimate is always biased; however, it usually has a smaller mean square error.
• Other possibilities derive from treating the two portions of data and separately and calculating separate sample means and/or sample variances for use in defining the estimate.

The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of, then form a function which is a valid autocorrelation in the sense that it is possible to define a theoretical process having exactly that autocorrelation. Other estimates can suffer from the problem that, if they are used to calculate the variance of a linear combination of the 's, the variance calculated may turn out to be negative.