Effective Number of Observations and Unbiased Estimators of Variance for Autocorrelated Data - an Overview

Abstract

When observations are autocorrelated, standard formulae for the estimators of variance, s 2 , and variance of the mean, ) ( 2 x s , are no longer adequate.They should be replaced by suitably defined estimators, 2 a s and ) ( 2 x s a , which are unbiased given that the autocorrelation function is known.The formula for 2 a s was given by Bayley and Hammersley in 1946, this work provides its simple derivation.The quantity named effective number of observations n eff is thoroughly discussed.It replaces the real number of observations n when describing the relationship between the variance and variance of the mean, and can be used to express 2 a s and ) ( 2 x s a in a simple manner.The dispersion of both estimators depends on another effective number called the effective degrees of freedom ν eff .Most of the formulae discussed in this paper are scattered throughout the literature and not very well known, this work aims to promote their more widespread use.The presented algorithms represent a natural extension of the GUM formulation of type-A uncertainty for the case of autocorrelated observations.

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