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A polynomial functional relationship with errors in both variables can be consistently estimated by constructing an ordinary least squares estimator for the regression coef®cients, assuming hypothetically the latent true regressor variable to be known, and then adjusting for the errors. If normality of the error variables can be assumed, the estimator can be simpli®ed considerably. Only the variance of the errors in the regressor variable and its covariance with the errors of the response variable need to be known. If the variance of the errors in the dependent variable is also known, another estimator can be constructed.

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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.

Using rounded data to estimate moments and regression coefficients typically biases the estimates. We explore the bias-inducing effects of rounding, thereby reviewing widely dispersed and often half forgotten results in the literature. Under appropriate conditions, these effects can be approximately rectified by versions of Sheppard's correction formula. We discuss the conditions under which these approximations are valid and also investigate the efficiency loss caused by rounding. The rounding error, which corresponds to the measurement error of a measurement error model, has a marginal distribution, which can be approximated by the uniform distribution, but is not independent of the true value. In order to take account of rounding preferences (heaping), we generalize the concept of simple rounding to that of asymmetric rounding and consider its effect on the mean and variance of a distribution.