Chi-Lun Cheng

Editorial: Total Least Squares and Errors-in-variables Modeling

The total least squares method is a numerical linear algebra tool for finding approximate solutions to overdetermined systems of equation s Ax = b, where both the vectorb as well as the matrixA are assumed to be perturbed. Since its definition by Golub and Van Loan in 1980, the classical total lea st squares method has been extended to solve weighted, structured, and regula ized total least squares problems and was applied in signal processing, system ident ification, computer vision, document retrieval, computer algebra, and other field s.

On Estimating Linear Relationships When Both Variables Are Subject to Heteroscedastic Measurement Errors

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.

A Small Sample Estimator for a Polynomial Regression with Errors in the Variables

An adjusted least squares estimator, introduced by Cheng and Schneeweiss for consistently estimating a polynomial regression of any degree with errors in the variables, is modified such that it shows good results in small samples without losing its asymptotic properties for large samples. Simulation studies corroborate the theoretical findings.

Robust calibration

This article presents robust methods for the random calibration problem. Many calibration techniques are based on regression models or measurement-error models. Prediction from such models is known to be highly nonrobust, and robust techniques should prove quite valuable. Robustcalibration procedures are procedures that work well even if there is some contamination in the data or if the model assumptions used in deriving the procedure are not quite true for the given data. Several approaches to robustifying calibration are compared theoretically, by Monte Carlo simulation, and on real data.

Coefficient of determination for multiple measurement error models

The coefficient of determination (R^2) is used for judging the goodness of fit in a linear regression model. It is the square of the multiple correlation coefficient between the study and explanatory variables based on the sample values. It gives valid results only when the observations are correctly observed without any measurement error. The conventional R^2 provides invalid results in the presence of measurement errors in the data because the sample R^2 becomes an inconsistent estimator of its population counterpart which is the square of the population multiple correlation coefficient between the study and explanatory variables. The goodness of fit statistics based on the variants of R^2 for multiple measurement error models have been proposed in this paper. These variants are based on the utilization of the two forms of additional information from outside the sample. The two forms are the known covariance matrix of measurement errors associated with the explanatory variables and the known reliability matrix associated with the explanatory variables. The asymptotic properties of the conventional R^2 and the proposed variants of R^2 like goodness of fit statistics have been studied analytically and numerically.

ON THE POLYNOMIAL MEASUREMENT ERROR MODEL

This paper discusses point estimation of the coefficients of polynomial measurement error (errors-in-variables) models. This includes functional and structural models. The connection between these models and total least squares (TLS) is also examined. A compendium of existing as well as new results is presented.

The Invariance of Some Score Tests in the Linear Model With Classical Measurement Error

The linear model with classical measurement error is an alternative to the standard regression model, in which it is assumed that the independent variables are subject to error. This assumption can cause statistical inferences and parameter estimators to differ dramatically from those obtained by the standard regression model. However, in some cases, inferences remain unchanged even though the independent variables are assumed to be subject to error. This article investigates the invariance property of score tests for assessing heteroscedasticity, first-order autoregressive disturbance, and the need for a Box–Cox power transformation. Under specific constraints, we show that the score tests for measurement error models are identical to the corresponding well-established tests derived from standard regression models. Hence practitioners can assess assumptions of constant variance and independent errors, as well as the need for a Box–Cox transformation, irrespective of whether or not the variables are measured with error. We also discuss some possible generalizations.

Goodness of fit in restricted measurement error models

The restricted measurement error model is employed when certain study variables are not observable by direct measurement and if some information about the unknown regression coefficients is available a priori. In this study, we present a method for checking the goodness of fit in the restricted measurement error model. We obtain the goodness-of-fit statistics based on the concept of coefficient of determination and their asymptotic distributions are derived. The results of simulations are also presented to demonstrate the finite sample behaviour of the estimators.

Errors in Variables in Econometrics

This article discusses the use of instrumental variables and grouping methods in the linear errors-in-variables or measurement error model. Comparisons are made between these methods, standard measurement error model methods with side conditions, least squares methods, and replicated models. It is demonstrated that there are close relationships between these apparently diverse estimation techniques.