Efficient and Robust Estimation of Linear Regression with Normal Errors

Abstract: Linear regression with normally distributed errors - including particular cases such as ANOVA, Student's t-test or location-scale inference - is a widely used statistical procedure. In this case the ordinary least squares estimator possesses remarkable properties but is very sensitive to outliers. Several robust alternatives have been proposed, but there is still significant room for improvement. This paper thus proposes an original method of estimation that offers the best efficiency simultaneously in the absence and the presence of outliers, both for the estimation of the regression coefficients and the scale parameter. The approach first consists in broadening the normal assumption of the errors to a mixture of the normal and the filtered-log-Pareto (FLP), an original distribution designed to represent the outliers. The expectation-maximization (EM) algorithm is then adapted and we obtain the N-FLP estimators of the regression coefficients, the scale parameter and the proportion of outliers, along with probabilities of each observation being an outlier. The performance of the N-FLP estimators is compared with the best alternatives in an extensive Monte Carlo simulation. The paper demonstrates that this method of estimation can also be used for a complete robust inference, including confidence intervals, hypothesis testing and model selection.