Modelling Count Data with Heteroscedastic Measurement Error in the Covariates

This paper is concerned with the estimation of the regression coefficients for a count data model when one of the explanatory variables is subject to heteroscedastic measurement error. The observed values W are related to the true regressor X by the additive error model W=X+U. The errors U are assumed to be normally distributed with zero mean but heteroscedastic variances, which are known or can be estimated from repeated measurements. Inference is done by using quasi likelihood methods, where a model of the observed data is specified only through a mean and a variance function for the response Y given W and other correctly observed covariates. Although this approach weakens the assumption of a parametric regression model, there is still the need to determine the marginal distribution of the unobserved variable X, which is treated as a random variable. Provided appropriate functions for the mean and variance are stated, the regression parameters can be estimated consistently. We illustrate our methods through an analysis of lung cancer rates in Switzerland. One of the covariates, the regional radon averages, cannot be measured exactly due to the strong dependency of radon on geological conditions and various other environmental sources of influence. The distribution of the unobserved true radon measure is modelled as a finite mixture of normals

Different Nonlinear Regression Models with Incorrectly Observed Covariates

We present quasi-likelihood models for different regression problems when one of the explanatory variables is measured with heteroscedastic error. In order to derive models for the observed data the conditional mean and variance functions of the regression models are only expressed through functions of the observable covariates. The latent covariable is treated as a random variable that follows a normal distribution. Furthermore it is assumed that enough additional information is provided to estimate the individual measurement error variances, e.g. through replicated measurements of the fallible predictor variable. The discussion includes the polynomial regression model as well as the probit and logit model for binary data, the Poisson model for count data and ordinal regression models

Fitting a Finite Mixture Distribution to a Variable Subject to Heteroscedastic Measurement Error

We consider the case where a latent variable X cannot be observed directly and instead a variable W=X+U with an heteroscedastic measurement error U is observed. It is assumed that the distribution of the true variable X is a mixture of normals and a type of the EM algorithm is applied to find approximate ML estimates of the distribution parameters of X

Extreme value analysis of Munich airpollution data

We present three different approaches to model extreme values of daily air pollution data. We fitted a generalized extreme value distribution to the monthly maxima of daily concentration measures. For the exceedances of a high threshold depending on the data the parameters of the generalized Pareto distribution were estimated. Accounting for autocorrelation clusters of exceedances were used. To get information about the relationship of the exceedance of the air quality standard and possible predictors we applied logistic regression. Results and their interpretation are given for daily average concentrations of ozone and of nitrogendioxid at two monitoring sites within the city of Munich

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

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 modified such that it shows good results in small samples without losing its asymptotic properties for large samples. Simulation studies corroborate the theoretical findings. 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