Robust estimation in very small samples

In experimental science measurements are typically repeated only a few times, yielding a sample size n of the order of 3 to 8. One then wants to summarize the measurements by a central value and measure their variability, i.e. estimate location and scale. These estimates should preferably be robust against outliers, as reflected by their small-sample breakdown value. The estimator's stylized empirical influence function should be smooth, monotone increasing for location, and decreasing–increasing for scale. It turns out that location can be estimated robustly for n3, whereas for scale n4 is needed. Several well-known robust estimators are studied for small n, yielding some surprising results. For instance, the Hodges–Lehmann estimator equals the average when n=4. Also location M-estimators with auxiliary scale are studied, addressing issues like the difference between one-step and fully iterated M-estimators. Simultaneous M-estimators of location and scale (‘Huber's Proposal 2’) are considered as well, and it turns out that their lack of robustness is already noticeable for such small samples. Recommendations are given as to which estimators to use.     

RectEuler: Visualizing Intersecting Sets using Rectangles

Euler diagrams are a popular technique to visualize set-typed data. However, creating diagrams using simple shapes remains a challenging problem for many complex, real-life datasets. To solve this, we propose RectEuler: a flexible, fully-automatic method using rectangles to create Euler-like diagrams. We use an efficient mixed-integer optimization scheme to place set labels and element representatives (e.g., text or images) in conjunction with rectangles describing the sets. By defining appropriate constraints, we adhere to well-formedness properties and aesthetic considerations. If a dataset cannot be created within a reasonable time or at all, we iteratively split the diagram into multiple components until a drawable solution is found. Redundant encoding of the set membership using dots and set lines improves the readability of the diagram. Our web tool lets users see how the layout changes throughout the optimization process and provides interactive explanations. For evaluation, we perform quantitative and qualitative analysis across different datasets and compare our method to state-of-the-art Euler diagram generation methods.