Accounting for outliers in optimal subsampling methods

Nowadays, in many different fields, massive data are available and for several reasons, it might be convenient to analyze just a subset of the data. The application of the D-optimality criterion can be helpful to optimally select a subsample of observations. However, it is well known that D-optimal support points lie on the boundary of the design space and if they go hand in hand with extreme response values, they can have a severe influence on the estimated linear model (leverage points with high influence). To overcome this problem, firstly, we propose a non-informative "exchange" procedure that enables us to select a "nearly" D-optimal subset of observations without high leverage values. Then, we provide an informative version of this exchange procedure, where besides high leverage points also the outliers in the responses (that are not necessarily associated to high leverage points) are avoided. This is possible because, unlike other design situations, in subsampling from big datasets the response values may be available. Finally, both the non-informative and informative selection procedures are adapted to I-optimality, with the goal of getting accurate predictions.