On Estimating Linear Relationships When Both Variables are Subject to Errors

Abstract

This paper discusses the problem of estimating a linear relationship between two variables observed with errors. It focuses on confidence intervals for the slope parameter and confidence regions for the intercept and slope parameters in Gaussian models. A compendium of existing results as well as new results are presented.is a generalization of the structural and functional models; if fJ.1 = ... = fJ.n, the ultrastructural model reduces to the structural model, whereas, if a 2 = 0, the ultrastructural model becomes the functional model. In this paper, the term tAddress for correspondence: Programs in Mathematical Sciences, University of Texas at Dallas, Box 830688, Richardson, TX 75083-0688, USA.

Polynomial Regression With Errors in the Variables

“…Whereas the linear functional relationship has found extensive treatment in the literature Ð for some reviews see, for example, Madansky (1959), Kendall and Stuart (1979), chapter 29, Schneeweiss andMittag (1986), Fuller (1987), chapter 2, Cheng and Van Ness (1994) Ð the polynomial functional relationship has received only a little attention despite the fact that it is perhaps the most natural extension of the linear model towards a non-linear model.…”

Bayesian Volumetric Calibration: An Application of Reversible Jump Markov Chain Monte Carlo Methods

“…We also chose to parameterize the true volumes v ij rather than the aliquots a ij . Finally, because of the innate unidenti®ability of the errors-in-variables problem with no information on the parameters (see, for example, Moran (1971), Fuller (1987 and Cheng and Van Ness (1994)), we assume that the ratio of the aliquot measurement error variance 2 a to the level measurement error variance 2 l is known and is ®xed throughout. Working without this assumption resulted in non-convergence problems in a simpler Gibbs sampling approach (McKnespiey, 1996) presumably due to the non-identi®ability of the model.…”

Robust estimation in the simple errors-in-variables model

“…Let us consider the empirical meanZ n ¼ ðX n ;Ȳ n Þ 0 , and the empirical covariance matrix V n ¼ V 11;n V 12;n V 12;n V 22;n of the Z i s'. The usual orthogonal regression estimators a n and b n of a and b are functions of the components of Z n and V n (Cheng and Van Ness, 1994). More precisely, we have, according to the identification hypothesis,…”