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This article discusses point estimation of the parameters in a linear measurement error (errors in variables) model when the variances in the measurement errors on both axes vary between observations. A compendium of existing and new regression methods is presented. Application of these methods to real data cases shows that the coefficients of the regression lines depend on the method selected. Guidelines for choosing a suitable regression method are provided.

The coefficient of determination (R 2 ) is used for judging the goodness of fit in a linear regression model. It gives valid results only when the observations are correctly observed without any measurement error. The R 2 provides invalid results in the presence of measurement errors in the data in the sense that sample R 2 becomes an inconsistent estimator of population multiple correlation coefficient between the study variable and explanatory variables. The corresponding variants of R 2 which can be used to judge the goodness of fit in multivariate measurement error model have been proposed in this paper. These variants are based on the utilization of information on known covariance matrix of measurement errors and known reliability matrix associated with explanatory variables. The asymptotic properties of the traditional R 2 and proposed R 2 have been studied.

Abstract. An adjusted least squares estimator, introduced by Cheng and Schneeweiss (1998) for consistently estimating a polynomial regression of any degree with errors in the variables, is modi ed such t h a t i t s h o ws good results in small samples without losing its asymptotic properties for large samples. Simulation studies corroborate the theoretical ndings. The new method is applied to analyse a geophysical law relating the depth of earthquakes to their distance from a trench where one of the earth's plates is submerged beneath another one.

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.