The Robustness of Estimator Composition
We formalize notions of robustness for composite estimators via the notion of a breakdown point. A composite estimator successively applies two (or more) estimators: on data decomposed into disjoint parts, it applies the first estimator on each part, then the second estimator on the outputs of the first estimator. And so on, if the composition is of more than two estimators. Informally, the breakdown point is the minimum fraction of data points which if significantly modified will also significantly modify the output of the estimator, so it is typically desirable to have a large breakdown point. Our main result shows that, under mild conditions on the individual estimators, the breakdown point of the composite estimator is the product of the breakdown points of the individual estimators. We also demonstrate several scenarios, ranging from regression to statistical testing, where this analysis is easy to apply, useful in understanding worst case robustness, and sheds powerful insights onto the associated data analysis.
The paper considers the robustness of composite estimators, estimators that apply one estimator to sections of the data set and another estimator to the output of the first estimator. The problem is important due to the fact that big data sets are hard to curate and composite estimators are often applied to them to simplify data analysis. The notion of robustness is quantified by the breakdown point, which is the minimum fraction of data points that need to be significantly modified in order to significantly modify the output. The breakdown point of a composite estimator is usually the product of the breakdown points of its component estimators. The authors apply this result to analyze the robustness of existing composite estimators (such as median of medians) and propose ways to increase their robustness. 
Technical Quality: The technical quality is good. The proofs appear to be sound and the remarks touch on important issues. The main text does not give experimental results. My main concern is how to couple this work with qualities of estimators that are traditionally desirable in statistics. For example, people often want unbiasedness or statistical efficiency. The applications section talks about how to make estimators more robust, but can we get estimators that are robust and have a low MSE?  Novelty: The paper is fairly novel. It builds upon previous basic work on the robustness of estimators via the breakdown point by defining a new version of the breakdown point and extending to composite estimators.   Potential impact: I think that this work is important for real-world data analysis. In some fields existing algorithms have many steps giving complicated composite estimators, but the problem of error propagation and robustness may not be given due attention. This work may also be useful for companies that don't spend the time to curate their business data. My only concern is the difficulty of computing the breakdown point for more complicated estimators. The paper only considers percentiles and other median-based estimators.   Clarity and presentation: The paper is well-organized and well-written, and the authors communicate the ideas clearly. The proofs were easy to follow. However, there are a few typos and grammar errors. Also, to better show the applicability of the work, I think that the simulations ought to be added to the paper and the complete proofs pushed to the supplement if needed.