ANALYSIS OF THE SCALE-CONTAMINATED NORMAL MODEL: DIAGNOSTICS AND ROBUSTNES

I study the linear regression model with additive errors when these errors are drawn independently from the same mixture of two normal distributions. These normal distributions have mean zero but differ in scale. The mixture is a simple instance of a scale-contaminated normal distribution and two parameters characterize the contamination. I refer to the model as the scale-contaminated normal (SCN) model.;I discuss two ways to analyze the SCN model. The first uses maximum likelihood; parameter estimates are calculated via the EM algorithm. Despite recent controversy about convergence of this algorithm, I show it to be a trustworthy tool for analyzing the SCN model and use EM to study the sensitivity of the likelihood to different values of the contamination parameters. I also justify and state the joint asymptotic distribution of the maximum likelihood estimators.;The second way to analyze the SCN model uses the entire likelihood rather than maxima of the likelihood. Using a suitable prior density, the posterior density of the regression parameters may be derived conditional on fixed values for the contamination parameters. (While it is difficult to approximate the marginal posterior density for the regression parameters, in many cases it suffices to calculate a series of posterior densities conditional on the contamination parameters.) Under the assumptions I use it has been shown that this conditional posterior density is a weighted sum of t densities. I show that the weights resemble certain regression diagnostics justified on other grounds. I then show how to approximate the posterior density of the regression parameters by adapting algorithms used to calculate regression diagnostics and I give several examples.;The form of the weights indicates that the SCN model appears to combine information from certain regression diagnostics to yield an estimate of the regression parameters that is not sensitive to outliers. Thus, the SCN model links certain regression diagnostics with robust regression. The interplay of the two ways to analyze the SCN model shows this model to be a sensible and simple alternative to popular robust regression approaches.