Robust Estimation under Heavy Contamination using Enlarged Models

In data analysis, contamination caused by outliers is inevitable, and robust statistical methods are strongly demanded. In this paper, our concern is to develop a new approach for robust data analysis based on scoring rules. The scoring rule is a discrepancy measure to assess the quality of probabilistic forecasts. We propose a simple way of es-timating not only the parameter in the statistical model but also the contamination ratio of outliers. Estimating the contamination ratio is important, since one can detect outliers out of the training samples based on the estimated contamination ratio. For this purpose, we use scoring rules with an extended statistical models, that is called the enlarged models. Also, the regression problems are considered. We study a complex heterogeneous contamination, in which the contami-nation ratio of outliers in the dependent variable may depend on the independent variable. We propose a simple method to obtain a robust regression estimator under heterogeneous contamination. In addition, we show that our method provides also an estimator of the expected contamination ratio that is available to detect the outliers out of train-ing samples. Numerical experiments demonstrate the effectiveness of our methods compared to the conventional estimators.