Adaptive robust regression with continuous Gaussian scale mixture errors

Model based regression analysis always requires a certain choice of models which typically specifies the behavior of regression errors. The normal distribution is the most common choice for this purpose, but the estimator under normality is known to be too sensitive to outliers. As an alternative, heavy tailed distributions such as t distributions have been suggested. Though this choice can reduce the sensitivity to outliers, it also requires the choice of distributions and tuning parameters for practical use. In this paper, we propose a class of continuous Gaussian scale mixtures for the error distribution that contains most symmetric unimodal probability distributions including normal, t, Laplace, and stable distributions. With this quite flexible class of error distributions, we provide the asymptotic property and robust property of the proposed method, and show its successes along with numerical examples.