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Unbiased estimates for linear regression with roundoff error

Withers C. S., Nadarajah S.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

177--182

We consider the linear regression model, where the residuals have zero mean and an otherwise unspecified distribution F. Suppose that least squares estimates are formed by using rounded values of the dependent variables.We show that these are still unbiased, and that unbiased estimates for the moments and cumulants of F are given by applying Sheppard’s corrections to their estimates.

Effective number of observations and unbiased estimators of variance for autocorrelated data - an overview

Zięba A.

Metrology and Measurement Systems

2010

Vol. 17, nr 1

3-16

When observations are autocorrelated, standard formulae for the estimators of variance, s², and variance of the mean, s²(x), are no longer adequate. They should be replaced by suitably defined estimators, sa² and sa²(x), which are unbiased given that the autocorrelation function is known. The formula for sa² was given by Bayley and Hammersley in 1946, this work provides its simple derivation. The quantity named effective number of observations neff 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 sa² and sa²(x) in a simple manner. The dispersion of both estimators depends on another effective number called the effective degrees of freedom veff. 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.