Estimating the endpoint of a uniform distribution under normal measurement errors with known error variance

The paper studies the problem of estimating the upper end point of a finite interval when the data come from a uniform distribution on this interval and are disturbed by normally distributed measurement errors with known variance. Maximum likelihood and method of moments estimators are introduced and compared to each other

Abraham Wald

This paper grew out of a lecture presented at the 54th Session of the International Statistical Institute in Berlin, August 13 - 20, 2003, Schneeweiss (2003). It intends not only to outline the eventful life of Abraham Wald (1902 - 1950) in Austria and in the United States but also to present his extensive scientific work. In particular, the two main subjects, where he earned most of his fame, are outline: Statistical Decision Theory and Sequential Analysis. In addition, emphasis is laid on his contributions to Econometrics and related fields

The polynomial and the Poisson measurement error models: some further results on quasi score and corrected score estimation

The asymptotic covariance matrices of the corrected score, the quasi score, and the simple score estimators of a polynomial measurement error model have been derived in the literature. Here some alternative formulas are presented, which might lead to an easier computation of these matrices. In particular, new properties of the variables $t_r$ and $\mu_r$ that constitute the estimators are derived. In addition, the term in the formula for the covariance matrix of the quasi score estimator stemming from the estimation of nuisance parameters is evaluated. The same is done for the log-linear Poisson measurement error model. In the polynomial case, it is shown that the simple score and the quasi score estimators are not always more efficient than the corrected score estimator if the nuisance parameters have to be estimated

On the Estimation of the Linear Relation when the Error Variances are known

The present article considers the problem of consistent estimation in measurement error models. A linear relation with not necessarily normally distributed measurement errors is considered. Three possible estimators which are constructed as different combinations of the estimators arising from direct and inverse regression are considered. The efficiency properties of these three estimators are derived and analyzed. The effect of non-normally distributed measurement errors is analyzed. A Monte-Carlo experiment is conducted to study the performance of these estimators in finite samples and the effect of a non-normal distribution of the measurement errors

Relative Efficiency of Maximum Likelihood and Other Estimators in a Nonlinear Regression Model with Small Measurement Errors

We compare the asymptotic covariance matrix of the ML estimator in a nonlinear measurement error model to the asymptotic covariance matrices of the CS and SQS estimators studied in Kukush et al (2002). For small measurement error variances they are equal up to the order of the measurement error variance and thus nearly equally efficient

Comparing the efficiency of structural and functional methods in measurement error models

The paper is a survey of recent investigations by the authors and others into the relative efficiencies of structural and functional estimators of the regression parameters in a measurement error model. While structural methods, in particular the quasi-score (QS) method, take advantage of the knowledge of the regressor distribution (if available), functional methods, in particular the corrected score (CS) method, discards such knowledge and works even if such knowledge is not available. Among other results, it has been shown that QS is more efficient than CS as long as the regressor distribution is completely known. However, if nuisance parameters in the regressor distribution have to be estimated, this is no more true in general. But by modifying the QS method, the adverse effect of the nuisance parameters can be overcome. For small measurement errors, the efficiencies of QS and CS become almost indistinguishable, whether nuisance parameters are present or not. QS is (asymptotically) biased if the regressor distribution has been misspecified, while CS is always consistent and thus more robust than QS

A comparison of asymptotic covariance matrices of three consistent estimators in the Poisson regression model with measurement errors

We consider a Poisson model, where the mean depends on certain covariates in a log-linear way with unknown regression parameters. Some or all of the covariates are measured with errors. The covariates as well as the measurement errors are both jointly normally distributed, and the error covariance matrix is supposed to be known. Three consistent estimators of the parameters - the corrected score, a structural, and the quasi-score estimators - are compared to each other with regard to their relative (asymptotic) efficiencies. The paper extends an earlier result for a scalar covariate

The Effect of Microaggregation Procedures on the Estimation of Linear Models: A Simulation Study

Microaggregation is a set of procedures that distort empirical data in order to guarantee the factual anonymity of the data. At the same time the information content of data sets should not be reduced too much and should still be useful for scientific research. This paper investigates the effect of microaggregation on the estimation of a linear regression by ordinary least squares. It studies, by way of an extensive simulation experiment, the bias of the slope parameter estimator induced by various microaggregation techniques. Some microaggregation procedures lead to consistent estimates while others imply an asymptotic bias for the estimator

Some Recent Advances in Measurement Error Models and Methods

A measurement error model is a regression model with (substantial) measurement errors in the variables. Disregarding these measurement errors in estimating the regression parameters results in asymptotically biased estimators. Several methods have been proposed to eliminate, or at least to reduce, this bias, and the relative efficiency and robustness of these methods have been compared. The paper gives an account of these endeavors. In another context, when data are of a categorical nature, classification errors play a similar role as measurement errors in continuous data. The paper also reviews some recent advances in this field

Estimation of a Linear Regression under Microaggregation with the Response Variable as a Sorting Variable

Microaggregation is one of the most frequently applied statistical disclosure control techniques for continuous data. The basic principle of microaggregation is to group the observations in a data set and to replace them by their corresponding group means. However, while reducing the disclosure risk of data files, the technique also affects the results of statistical analyses. The paper deals with the impact of microaggregation on a linear model in continuous variables. We show that parameter estimates are biased if the dependent variable is used to form the groups. Using this result, we develop a consistent estimator that removes the aggregation bias. Moreover, we derive the asymptotic covariance matrix of the corrected least squares estimator